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Page 1: Formal Ontology and Conceptual Realism
Page 2: Formal Ontology and Conceptual Realism

FORMAL ONTOLOGY AND CONCEPTUAL REALISM

Page 3: Formal Ontology and Conceptual Realism

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, DenmarkJOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor:

JAAKKO HINTIKKA, Boston University, U.S.A.

Editors:

DIRK VAN DALEN, University of Utrecht, The NetherlandsTHEO A.F. KUIPERS, University of Groningen, The Netherlands

TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A.PATRICK SUPPES, Stanford University, California, U.S.A.JAN WOLENSKI, Jagiellonian University, Kraków, Poland

VOLUME 339

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FORMAL ONTOLOGYAND CONCEPTUAL

REALISM

by

Nino B. CocchiarellaIndiana University, Bloomington,

IN, U.S.A.

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A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-6203-2 (HB)ISBN 978-1-4020-6204-9 (e-book)

Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

Printed on acid-free paper

All Rights Reserved© 2007 Springer

No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recording

or otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being entered

and executed on a computer system, for exclusive use by the purchaser of the work.

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Dedicated tothe village of Fragneto L’Abate,

Provincia di Benevento,and

to my family and friends in Italy

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Contents

Preface xi

Introduction xiii

I Formal Ontology 1

1 Formal Ontology and Conceptual Realism 31.1 Formal Ontology as a Characteristica Universalis . . . . . . . . . 41.2 Radical Empiricism and the Logical Construction of the World . 61.3 Commonsense Versus Scientific Understanding . . . . . . . . . . 81.4 The Nexus of Predication . . . . . . . . . . . . . . . . . . . . . . 101.5 Univocal Versus Multiple Senses of Being . . . . . . . . . . . . . 121.6 Predication and Preeminent Being . . . . . . . . . . . . . . . . . 141.7 Categorial Analysis and Transcendental

Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 The Completeness Problem . . . . . . . . . . . . . . . . . . . . . 171.9 Set-Theoretic Semantics . . . . . . . . . . . . . . . . . . . . . . . 191.10 Conceptual Realism . . . . . . . . . . . . . . . . . . . . . . . . . 211.11 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 23

2 Time, Being, and Existence 252.1 Possibilism versus Actualism . . . . . . . . . . . . . . . . . . . . 262.2 Logics of Actual and Possible Objects . . . . . . . . . . . . . . . 282.3 Set-theoretic Semantics . . . . . . . . . . . . . . . . . . . . . . . 302.4 Axioms in Possibilist Logic . . . . . . . . . . . . . . . . . . . . . 312.5 A First-order Actualist Logic . . . . . . . . . . . . . . . . . . . . 342.6 Tense Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7 Temporal Modes of Being . . . . . . . . . . . . . . . . . . . . . . 402.8 Past and Future Objects . . . . . . . . . . . . . . . . . . . . . . . 422.9 Modality Within Tense Logic . . . . . . . . . . . . . . . . . . . . 442.10 Causal Tenses in Relativity Theory . . . . . . . . . . . . . . . . . 482.11 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 522.12 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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viii CONTENTS

3 Logical Necessity and Logical Atomism 593.1 The Ontology of Logical Atomism . . . . . . . . . . . . . . . . . 613.2 The Primary Semantics of Logical Necessity . . . . . . . . . . . . 643.3 The Modal Thesis of Anti-Essentialism . . . . . . . . . . . . . . . 663.4 An Incompleteness Theorem . . . . . . . . . . . . . . . . . . . . . 693.5 The Semantics of Metaphysical Necessity . . . . . . . . . . . . . 703.6 Metaphysical Versus Natural Necessity . . . . . . . . . . . . . . . 74

3.6.1 The Concordance Model of a Multiverse . . . . . . . . . . 763.6.2 The Multiverse of the Many-Worlds Model . . . . . . . . 76

3.7 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 78

4 Formal Theories of Predication 814.1 Logical Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Nominalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Constructive Conceptualism . . . . . . . . . . . . . . . . . . . . . 874.4 Ramification and Holistic Conceptualism . . . . . . . . . . . . . . 914.5 The Logic of Nominalized Predicates . . . . . . . . . . . . . . . . 944.6 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 99

5 Formal Theories of Predication Part II 1015.1 Homogeneous Stratification . . . . . . . . . . . . . . . . . . . . . 1015.2 Frege’s Logic Reconstructed . . . . . . . . . . . . . . . . . . . . . 1045.3 Conceptual Intensional Realism . . . . . . . . . . . . . . . . . . . 1085.4 Hyperintensionality . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 118

6 Intensional Possible Worlds 1216.1 Actualism Versus Possibilism Redux . . . . . . . . . . . . . . . . 1236.2 Intensional Possible Worlds . . . . . . . . . . . . . . . . . . . . . 1276.3 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 1336.4 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

II Conceptual Realism 137

7 The Nexus of Predication 1397.1 Predication in Natural Realism . . . . . . . . . . . . . . . . . . . 1417.2 Conceptualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.3 Referential Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 1447.4 Singular Reference and Proper Names . . . . . . . . . . . . . . . 1477.5 Definite Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 1487.6 Nominalization as Deactivation . . . . . . . . . . . . . . . . . . . 1517.7 The Content of Referential Concepts . . . . . . . . . . . . . . . . 1537.8 The Two Levels of Analysis . . . . . . . . . . . . . . . . . . . . . 1567.9 Ontology of the Natural Numbers . . . . . . . . . . . . . . . . . . 1607.10 Ontology of Fictional Objects . . . . . . . . . . . . . . . . . . . . 1637.11 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 166

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CONTENTS ix

8 Medieval Logic and Conceptual Realism 1698.1 Terminist Logic and Mental Language . . . . . . . . . . . . . . . 1698.2 Ockham’s Early Theory of Ficta . . . . . . . . . . . . . . . . . . 1748.3 Ockham’s Later Theory of Concepts . . . . . . . . . . . . . . . . 1768.4 Personal Supposition and Reference . . . . . . . . . . . . . . . . 1788.5 The Identity Theory of the Copula . . . . . . . . . . . . . . . . . 1808.6 Ascending and Descending . . . . . . . . . . . . . . . . . . . . . . 1838.7 How Confused is Merely Confused . . . . . . . . . . . . . . . . . 1918.8 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 193

9 On Geach Against General Reference 1959.1 Geach’s Negation Argument . . . . . . . . . . . . . . . . . . . . . 1969.2 Disjunction and Conjunction Arguments . . . . . . . . . . . . . . 1999.3 Active Versus Deactivated Concepts . . . . . . . . . . . . . . . . 2019.4 Deactivation and Geach’s Arguments . . . . . . . . . . . . . . . . 2049.5 Geach’s Arguments Against Complex Names . . . . . . . . . . . 2079.6 Relative Pronouns as Referential Expressions . . . . . . . . . . . 2099.7 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 213

10 Lesniewski’s Ontology 21510.1 Lesniewski’s Logic of Names . . . . . . . . . . . . . . . . . . . . . 21610.2 The Simple Logic of Names . . . . . . . . . . . . . . . . . . . . . 21910.3 Consistency and Decidability . . . . . . . . . . . . . . . . . . . . 22210.4 A Reduction of Lesniewski’s System . . . . . . . . . . . . . . . . 22410.5 Pragmatic Uses of Proper and Common Names . . . . . . . . . . 22810.6 Classes as Many as the Extensions of Names . . . . . . . . . . . 23010.7 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 233

11 Plurals and the Logic of Classes as Many 23511.1 The Logic of Classes as Many . . . . . . . . . . . . . . . . . . . . 23611.2 Extensional Identity . . . . . . . . . . . . . . . . . . . . . . . . . 24111.3 The Universal Class . . . . . . . . . . . . . . . . . . . . . . . . . 24411.4 Intersection, Union, and Complementation . . . . . . . . . . . . . 24611.5 Lesniewskian Theses Revisited . . . . . . . . . . . . . . . . . . . 24811.6 Groups and the Semantics of Plurals . . . . . . . . . . . . . . . . 25111.7 Plural Reference and Predication . . . . . . . . . . . . . . . . . . 25511.8 Cardinal Numbers and Plural Quantifiers . . . . . . . . . . . . . 25911.9 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 26411.10Appendix 1: A Set-Theoretic Semantics . . . . . . . . . . . . . . 26611.11Appendix 2: Bell’s System M . . . . . . . . . . . . . . . . . . . . 270

12 The Logic of Natural Kinds 27312.1 Conceptual Natural Realism . . . . . . . . . . . . . . . . . . . . . 27312.2 The Problem with Moderate Realism . . . . . . . . . . . . . . . . 27512.3 Modal Moderate Realism . . . . . . . . . . . . . . . . . . . . . . 27912.4 Aristotelian Essentialism . . . . . . . . . . . . . . . . . . . . . . . 281

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12.5 General Versus Individual Essences . . . . . . . . . . . . . . . . . 28712.6 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . 28812.7 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29012.8 Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Afterword on Truth-Makers 295

Bibliography 297

Index 307

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Preface

This book is based on a series of lectures given in Rome in 2004 at the LateranPontifical University under the auspices of the Science, Theology, and theOntological Quest (STOQ) project, which was supported by the PontificalCouncil for Culture with the support of the John Templeton Foundation. Theproject was aimed at developing a dialogue between science, philosophy andtheology at a time when many theoretical, ethical and cultural challenges arebeing raised by new developments in science. The director of the project isProfessor Gianfranco Basti of the Pontifical Lateran University, whom I wantto thank for inviting me to give the lectures and for his many helpful comments.I also want to thank Professor Michele Malatesta and Professor Philip Larreyand Mr. Ciro De Florio for their comments and participation in the lectures.I also want to thank Professor Greg Landini for comments and corrections ofmy discussion of hyperintensionality in chapter five.

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Introduction

The history of philosophy is replete with different metaphysical schemes of theontological structure of the world. These schemes have generally been describedin informal, intuitive terms, and the arguments for and against them, includingtheir consistency and adequacy as explanatory frameworks, have generally beengiven in even more informal terms. The goal of formal ontology is to correct forthese deficiencies. By formally reconstructing an intuitive, informal ontologicalscheme as a formal ontology we can better determine the consistency and ade-quacy of that scheme; and then by comparing different reconstructed schemeswith one another as formal ontologies we can better evaluate the arguments forand against them, and come to a decision as to which system it is best to adopt.

This book is divided into two parts. The first part is on formal ontologyand how different informal ontological systems can be formally developed andcompared with one another. An abstract set-theoretic framework, which we callcomparative formal ontology, can be used for this purpose without assuming thatset theory is itself a superseding ontological system. The second part of thisbook is on the formal construction and defense of a particular ontological schemecalled conceptual realism. Conceptual realism is to be preferred to alternativeformal ontologies for the reasons briefly described below, and for others as wellthat are given in more detail in various parts of the book. Conceptual realism,in other words, is put forward here as the best ontological system to adopt.

1. Formal Ontology

Formal ontology, as we explain in chapter one, is a discipline in which the for-mal methods of mathematical logic are combined with the intuitive, philosoph-ical analyses and principles of ontology, where by ontology we mean the studyand analysis of being qua being, including in particular the different categoriesof being and how those categories are connected with the nexus of predicationin language, thought and reality. The purpose of formal ontology is to bringtogether the clarity, precision and methodology of logical analyses on the onehand with the philosophical significance of ontological analyses on the other.

The phrase ‘formal ontology’ was first used by Edmund Husserl in his Formaland Transcendental Logic (1929). For Husserl a formal ontology is supposed tobe a system of logic taken as a universal theory of science, and in particular as

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the justifying discipline for science. Instead of actually constructing a formalontology, however, Husserl turned to the study of the categorial structures ofwhat he took to be a transcendental subjectivity for the justification of such alogic. That is not the kind of study that is presented here.

The phrase ‘formal ontology’ was again used in 1969 at a memorial confer-ence for Bertrand Russell at Indiana University.1 Later, in 1972, a conference onformal ontology was held at the University of Victoria in Victoria, BC, Canada.2

Today the subject of formal ontology has become more widely discussed, espe-cially in Europe, by many philosophers, logicians, and cognitive and computerscientists working on programs for knowledge representation. In March of 1993,for example, an international conference was held in Padova, Italy, on the roleof formal ontology in information technology.3 Since then there have been atleast three other international conferences on formal ontology in informationsystems.4

The idea of a formal ontology, even if not the phrase itself, goes back toGottfried Leibniz’s notion of a characteristica universalis , the general featuresof which are explained in chapter one. Leibniz was a mathematician and he didmake preliminary attempts at constructing a system of logic that would functionas a characteristica universalis. Unfortunately, logic had still not progressedbeyond Aristotle’s theory of the syllogism, and we have only fragments and thegeneral idea of what such a system should achieve. In addition, because his logicwas strictly algebraic, Leibniz did not deal with the central feature of a formalontology, namely how the different ontological categories are connected with thenexus of predication. This is also true of many of the systems constructed inanalytic metaphysics, where a logical analysis of different aspects of language,thought and reality are represented. These systems are really fragments orcomponent parts of implicit, or as yet unspecified, formal ontologies with whichthey are compatible.

The first full system of logic that was designed to function as a character-istica universalis and that also provided a formal theory of predication wasGottlob Frege’s system in his 1893, Der Grundgesetze der Arithmetik.5 Thissystem was subject to Russell’s paradox, but, as we explain in Part I, Frege’slogic can be reconstructed and shown to be equiconsistent with the theory ofsimple types. Russell’s own attempt in his 1903 Principles of Mathematics wasalso subject to his paradox, but, Russell’s implicit system at that time can alsobe consistently reconstructed. Both Frege’s and Russell’s early ontology were

1See Cocchiarella 1974. The papers for that conference were published in Nakhnikian 1974.2The Victoria Conference on Formal Ontology was held in October of 1972. The papers

for the conference were later published in the Israeli journal, Philosophia, vol, 4, no.1, 1974.3The papers from that conference, including my paper, “Knowledge Representation in

Conceptual Realism,” were published later in a special issue of the International Journal ofHuman-Computer Studies, vol. 43 (5/6), 1995.

For more general information on formal ontology see the web sitehttp://www.formalontology.it.

4See, e.g., Guarino 1998, Welty and Smith 2001, and Varzi and Vieu 2004.5Frege was adamant that his logic was “not a mere calculus ratiocinator, but a lingua

characteristica in the Leibnizian sense” (Frege 1972, p. 90).

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based on a modern form of Platonism, which in this book we call logical realism.Frege’s ontology was extensionalist, however, whereas Russell’s was intension-alist, and as a result their respective accounts of the nexus of predication werequite different. Nevertheless, as reconstructed here, both Frege’s and Russell’searly systems of logic can be taken as variants of a logical realist type of formalontology. Russell’s later theory of ramified types was also originally describedby Russell as a version of Platonism, but in fact it is more suitable to a construc-tive form of conceptual realism.6 In part 1 of this book we describe a system oframified second-order logic with nominalized predicates as a formal ontology forconstructive conceptual realism, which is a subsystem of the larger frameworkof conceptual realism.

2. Time, Being and Existence

A criterion of adequacy for any formal ontology is that it should provide alogically perspicuous representation of our commonsense understanding of theworld, and not just of our scientific understanding. A central feature of ourcommonsense understanding is how we are conceptually oriented in time withrespect to the past, the present and the future, and the question arises as tohow we can best represent this orientation. It is inappropriate to represent itin terms of a tenseless idiom of moments (or intervals of time) of a coordinatesystem, as is commonly done in scientific theories; for that amounts to replacingour commonsense understanding with a scientific view. A more appropriaterepresentation is one that respects the form and content of our commonsensespeech and mental acts about the past, the present and the future. Formally,this can best be done in terms of a logic of tense operators, or in what is nowcalled tense logic, which we discuss in chapter two.

Now the most natural formal ontology for tense logic is conceptualism, ora formal ontology such as conceptual realism that contains conceptualism as acomponent. This is because what tense operators represent in conceptualism arecertain cognitive schemata regarding our orientation in time, cognitive schematathat are in fact fundamental to both the form and content of our conceptualactivity. Thought and communication are inextricably temporal phenomena,and it is the cognitive schemata underlying our use of tense that structures thatphenomena temporally in terms of the past, the present and the future.

A second criterion of adequacy for a formal ontology is that it must explainand provide an ontological ground for the distinction between being and exis-tence, or, if it rejects that distinction why it does so. Put simply, the problemis: Can there be things that do not exist? Or is being the same as existence?Different formal ontologies will answer these questions in different ways.

The logic of time in conceptual realism provides the clearest ontologicalground for such a distinction in terms of the tense-logical distinction betweenpast, present and future objects, i.e., the distinction between things that didexist, do exist, or will exist, or what in the proposed book we call realia, as

6See Cocchiarella 1991 for a defense and explanation of this claim.

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opposed to existentia, which is restricted to the things that exist at the timewe speak or think, i.e., the time we take to be the present of our commonsenseframework. The present, in other words, unlike the tenseless medium of ourscientific theories, is indexical, and refers at any moment of time to that mo-ment itself. We explain in chapter two how the categories of time, being andexistence can be represented in tense logic.

3. Ontology and Modality

Another criterion of adequacy for a formal ontology is that it must explainthe ontological grounds, or nature, of modality, i.e., of such modal notions asnecessity and possibility. A set-theoretic semantics for modal logic may beuseful for showing consistency or completeness, but it does not of itself providean ontological ground for modality. Chapter two of the proposed book dealsnot just with the fundamental categories of time, being and existence, but withmodality as well. Some of the earliest views of necessity and possibility aregrounded in such a framework. As we explain in chapter two, even the temporaland modal distinctions of the special theory of relativity theory can be bestunderstood within the framework of conceptual realism.

There is more to the distinction between being and existence than thatbetween past, present and future objects, of course, and there is also more tomodality than what can be grounded in time and the special theory of relativity.The abstract intensional objects of conceptual realism, for example, do notexist as concrete objects, but, as we explain later, they also are not Platonicforms. They do not exist in an independent Platonic realm, in other words,but rather have a mode of being dependant upon the evolution of culture andconsciousness. Similarly, the natural or causal modalities of the logic of naturalkinds and natural properties and relations that we describe below for conceptualrealism go beyond the modalities that can be grounded on the logic of tenses,including the causal tenses of the special theory of relativity. But both ofthese developments, i.e., the abstract being of intensional objects and the causalmodalities of the logic of natural kinds, are extensions of the distinctions madein chapter two and not rejections of those distinctions. Indeed, the logic ofactual and possible objects described in chapter two is retained, or only slightlymodified, throughout the remainder of the book.

Finally, it should be noted that all too often it has been assumed that theonly modalities appropriate for a formal ontology are the logical modalities,e.g., logical necessity and possibility. In chapter three we explain how logicalnecessity and possibility can be understood as modalities, but only within theontological framework of logical atomism. It is only in an ontology of simpleobjects and simple properties and relations as the bases of logically independentatomic states of affairs that logical necessity and possibility can be made sense ofas modalities, as opposed to semantic properties of sentences. Logical atomism,of course, is not an adequate formal ontology for either our commonsense or ourscientific framework, and we are not putting it forward as such here. Rather,

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the point is that it serves to explain why logical necessity and possibility cannotbe ontologically grounded on a formal ontology of complex objects and complexstates of affairs that are not logically reducible to simple objects and logicallyindependent atomic states.

4. Formal Theories of Predication

Part I of this book, as already noted, is on formal ontology, and especially com-parative formal ontology. Comparative formal ontology is the preferable, if notalso the proper, domain of many issues and disputes in metaphysics, epistemol-ogy, and the methodology of the deductive sciences. For just as the constructionof a particular formal ontology lends clarity and precision to our informal cate-gorial analyses and serves as a guide to our intuitions, so too comparative formalontology can be developed so as to provide clear and precise criteria for con-structing and comparing different formal ontologies so that ultimately we canmake a rational decision about which such system we should ourselves adopt.

Different formal ontologies are primarily based on different formal theories ofpredication, which in turn are based on different theories of universals, the threemost important being nominalism, conceptualism, and realism. A basic featureof a formal ontology, in other words, is a formal theory of predication based ona theory of universals. A key aspect of such a theory is how the categories ofbeing, especially the category of objects and the category of universals, are re-lated to one another, and how the unity of the nexus of predication is explainedin terms of those categories. Such a categorial analysis indicates another basicfeature of a formal ontology, namely, how it represents the categorial structureof the world, and in particular whether it can represent the categorial structureof our commonsense understanding of the world as well that of our scientific the-ories, without the two being in conflict. A formal ontology is not just a formalaxiomatic development, in other words, but rather it is a system in which onto-logical categories are represented by logical categories, and ontological analysesby logical analyses.

We have already constructed and compared a number of such systems inCocchiarella 1986 and 1987, and in fact a detailed consistent reconstructionand analysis of nominalism and Gottlob Frege’s and Bertrand Russell’s formsof logical realism have already been given in those previous books. For thatreason we do not go into great detail of their views in the present book, butsome coverage of that reconstruction is necessary in order explain the core part ofconceptual realism, which has a number features in common with those systems,and which is the framework we will develop and defend in part II of this book.That reconstruction is given in chapters four and five.

Nominalism, which denies that there are any universals, whether real orconceptual, is logically the weakest formal ontology. From a logical point ofview what is interesting about nominalism is the kind of constraint it imposeson a theory of predication. That constraint, however, has a more interestingcounterpart in constructive conceptualism and represents an important stage of

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cognitive development, a stage that is an essential part of conceptualism andthe more general framework of conceptual realism. The more general frameworkallows the constraint to be acknowledged and yet also transcended, whereas thesimilar constraint in nominalism leaves no room for such transcendence. Wediscuss and explain the nature of this constraint and why it can be transcendedin constructive conceptualism but not in nominalism in chapter four. Thisdifference between nominalism and conceptualism is one of the reasons whyconceptualism, as a formal ontology, is to be preferred over nominalism.

There is another reason as well, namely, that predication in language, whichis the only form of predication acknowledged in nominalism, depends uponour cognitive capacity for language, including in particular our rule-followingcognitive capacities underlying the use of referential and predicable expressions.A cognitive theory of predication is needed to explain predication in language,in other words, and that is precisely what the form of conceptualism we defendhere is designed to do. In fact, the referential and predicable concepts of ourform of conceptualism are none other than the rule-following cognitive capacitiesunderlying the use of referential and predicable expressions, and the unity ofthe nexus of predication in conceptualism is what underlies and accounts forthe unity of predication in language. Conceptualism is to be preferred overnominalism because, unlike the latter, which is based on an unexplained accountof predication in language, it is framed in terms of a theory of predication, i.e.,a theory of predication about the cognitive structure of our speech and mentalacts, and therefore a theory of thought that underlies and explains predicationin language.

5. Conceptual Realism

The purpose of comparative formal ontology, we have noted, is to provide clearand precise criteria by which to judge the adequacy of a particular proposedsystem of formal ontology. Conceptual realism, the system we think is best andhave adopted, contains, in addition to a conceptualist theory of predication, anintensional realism that is based on a logic of nominalized predicates, and anatural realism that is based on a logic of natural kinds.

Unlike the a priori approach of the transcendental method, which claims tobe independent of the laws of nature and our evolutionary history, i.e., of ourstatus as biological beings with a culture and history that shapes our languageand much of our thought, conceptual realism is framed within the context ofa naturalistic epistemology and a naturalistic approach to the relation betweenlanguage and thought, thought and reality, and our scientific knowledge of theworld. This naturalistic approach is one of the reasons why conceptual realismis to be preferred to a logical realism as a modern form of Platonism as wellas to a transcendental idealism such a Husserl’s. The following are some of thefeatures of conceptual realism that we will cover in this book.

The realism part of conceptual realism, we have said, contains both a naturalrealism and an intensional realism, each of which can be developed as separate

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subsystems, and both of which are not only consistent in themselves but com-patible with each other within the larger framework. We call these two subsys-tems conceptual natural realism and conceptual intensional realism. Conceptualnatural realism represents a modal form of moderate realism, which, by beingextended to include a logic of natural kinds, can be developed into a modernform of Aristotelian essentialism. Conceptual intensional realism, on the otherhand, represents a modern counterpart of Platonism based on the intensionalcontents of our referential and predicable concepts. These two subsystems cap-ture the more important ontological features of both logical realism and naturalrealism as theories of universals while also explaining our epistemic access to theabstract intensional objects of logical realism on the one hand and the naturalkinds and natural properties and relations of natural realism on the other. Thetwo subsystems are compatible in the larger framework of conceptual realism,as we have said, because each is constructed on the basis of a different logicalaspect of that framework.

Conceptual intensional realism, for example, is based on a logical analysisof nominalized predicates and propositional forms as abstract singular terms,i.e., a logical analysis of the abstract nouns and nominal phrases that we usein describing the intensional contents of our speech and mental acts. The in-tensional objects that are denoted by these abstract singular terms serve thesame purposes in conceptual intensional realism that abstract objects serve inlogical realism as a modern form of Platonism. The difference is that, unlikePlatonic Forms, the intensional objects of conceptual realism do not exist in-dependently of mind and the natural world, the way they do in logical realism,but are products of the evolution of culture and language, and especially of theinstitutionalized linguistic practice of nominalization. In this way our epistemicgrasp of abstract intensional objects is explained in terms of the concepts thatunderlie our rule-following cognitive capacities in the use of language.

The natural kinds and natural properties and relations of conceptual naturalrealism, on the other hand, are not intensional objects; and in fact they are notobjects at all but are rather unsaturated causal structures that are complemen-tary to the structures of natural kinds of things. Unlike conceptual intensionalrealism, which is based on the logic of nominalized predicates, and hence isdirected upon an “object”-ification of predicable concepts, conceptual naturalrealism is directed upon the structure of reality and depends upon empiricalassumptions as to whether or not there are natural properties or relations cor-responding to particular predicable concepts, and similarly whether or not thereare natural kinds corresponding to particular sortal common-name concepts.

The difference between Plato’s and Aristotle’s ontologies has been one of thebasic issues of debate in the history of philosophy. The way both forms of realismare contained within the general framework of conceptual realism shows how amodern form of Aristotelian essentialism is compatible with an intensional logicthat is a counterpart to a modern form of Platonism. This is another reasonwhy conceptual realism is the formal ontology that is best to adopt.

There is an importance difference between conceptual intensional realism andlogical realism, however, if one extends and compares a metaphysical necessity

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and possibility in the latter with a conceptual necessity and possibility to theformer. The metaphysical modalities allow for a much stronger system in termsof what in chapter six are called intensional possible worlds, which can beexpressed either as world propositions or as world properties. The assumptionthat there are such intensional worlds is justifiable in modal logical realism but,as we explain in chapter six, not at all in conceptual intensional realism.

6. Predication in Conceptual Realism

Conceptual realism, we have noted, is based upon a cognitive theory of pred-ication, and referential and predicable concepts are in fact the rule-followingcognitive capacities that underlie our use of referential and predicable expres-sions. As explained in chapter seven, the nexus of predication in conceptualrealism is the result of jointly exercising a referential and predicable concept ascomplementary cognitive structures, and as such it is what accounts for bothpredication in language and the unity of thought. An important feature of thistheory is that it gives a unified account of both general and singular reference, afeature it has in common with the terminist logic of Ockham and other medievalphilosophers.

The theory also provides an account of complex predicate expressions thatcontain abstract noun phrases, such as infinitives and gerunds, and also complexpredicate expressions with quantifier phrases occurring as direct-object expres-sions of transitive verbs. Conceptually, as we explain in chapter seven, thecontent of such a quantifier phrase and the referential concept it stands for is“object”-ified through a doubly reflexive abstraction that by deactivation and“nominalization” of the quantifier phrase first generates a predicable conceptand then the intensional content of that predicable concept. All direct objectsof speech and thought are intensionalized in this way so that a parallel analysisis given for both ‘Sofia finds a unicorn’ and for ‘Sofia seeks a unicorn’. And yet,relations, such as Finds, that are extensional in their second argument positionscan still be distinguished from those that are not, such as Seeks, by appropriatemeaning postulates.

The same doubly reflexive abstraction explains the three different types ofexpressions that represent the natural number concepts, namely first, as numer-ical quantifier phrases, such as ‘three dogs’, ‘two cats, ‘five chairs’, etc., then,second, as the cardinal number predicates ‘has n instances’, or ‘has n mem-bers’, and the third as the numerals ‘1’, ‘2’, ‘3’, etc., i.e., as objectual termsthat purport to name the natural numbers as abstract objects.

Finally, the deactivation of referential expressions that is a part of this cog-nitive theory of predication is also involved in fictional discourse and in storiesin general. The objects of fiction, on this account, are none other than theintensional objects that deactivated referential expressions denote as abstractobjectual terms. This account of the ontology of fictional objects explains their“incompleteness” as well as their status as intensional content.

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7. Explaining Medieval Supposition Theoryand Defending General Reference

The unified account of both general and singular reference that is part of ourcognitive theory of predication is a feature, we have noted, that it has in commonwith medieval terminist logic. In chapter eight we explain how the medievalnotion of supposition, though different from reference, nevertheless can be ex-plained in terms of our cognitive theory. We also explain in that chapter how themedieval doctrine of the identity of the copula for categorical propositions canbe understood in terms of our theory of reference. These historically problem-atic notions, in other words, can be explained and accounted for in conceptualrealism, which is another reason why we think it is the best formal ontology toadopt.

It is well-known, of course, that Peter Geach, in his book Reference andGenerality, has argued against the possibility of a coherent account of gen-eral reference, including in particular the medieval theory of supposition. Inchapter nine, we give a detailed discussion of Geach’s arguments along witha refutation of those arguments. A key feature of our refutation is the notionof the deactivation of referential concepts, a notion that is basic to our analy-sis of the direct objects of intensional verbs. We also give an account in thatchapter of co-reference, as, e.g., in so-called donkey sentences, in terms of avariable-binding ‘that’-operator.

8. The Logic of Names in Lesniewski’s OntologyVersus in Conceptual Realism

The category of names in conceptual realism’s theory of reference contains bothproper and common names (common count nouns). It is through this broadernotion of a name that general as well as singular reference is part of this the-ory. A single category of names, both proper and common, is also a featureof Lesniewski’s ontology, which has been called a logic of names. In chapterten we formally describe both Lesniewski’s ontology, or logic of names, and thesimple logic of names that is part of our cognitive theory of reference. We thenshow how Lesniewski’s ontology is reducible to the simple logic of names of ourcognitive theory.

Names, whether proper or common, occur as parts of quantifier phrases inthe simple logic of names of our cognitive theory. In the broader theory ofreference of conceptual realism, however, names, whether proper or common,can also be “nominalized”, i.e., transformed into objectual terms that can occuras arguments of predicates. The result is a logic of “classes as many,” whereby a class as many we mean essentially the notion that was originally describedby Russell in his 1903 Principles of Mathematics. Nominalized common names,in this logic, provide both a semantic and an ontological ground for a logic of

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plurals within the more general framework of conceptual realism. We describesuch a logic in chapter eleven.

Semantically, such a logic is needed to account for irreducible forms of pluralreference and predication. The well-known Geach sentence ‘Some critics admireonly each other’, for example, which semantically says that there is a groupof critics who admire only other members of the group, cannot be analyzedin first-order logic alone; and it would be both semantically and ontologicallymisleading to analyze it in terms of set theory. Unlike a set, a group in thesense intended here is a plurality of individuals and not an abstract object.

A logic of plurals is needed not just as a semantical framework for pluralreference and predication in natural language and our commonsense framework,but, and perhaps more importantly, also for an ontological account of the prop-erties of groups of objects in our scientific theories. The temperature and pres-sure of a volume of gas, for example, are really properties of the group of atomsor molecules in that volume rather than properties of the individual atoms ormolecules that make it up. The visual, auditory, and other sensory properties ofdifferent modules of the brain are properties of the groups of neurons that makeup those modules rather than of the individual neurons in the group. Similarly,the dispersion and redistribution of different populations of species of plantsand animals are statistical properties of the groups of plants and animals andnot of the individuals in those groups. Groups, which are classes as many of twoor more objects, are plural objects, and as such they are values of the objectualvariables in this ontology.

9. Conceptual Realism and AristotelianEssentialism

Conceptualism and natural realism have a clear affinity for each other, eventhough they do not have the same overall logical structure. Conceptualism,for example, presupposes some form of natural realism as the causal ground ofour capacity for language and thought, and natural realism presupposes con-ceptualism as a framework by which it can be articulated as a formal ontology.Historically, in fact, the two ontologies have often been confused with one an-other, so that sometimes it was said that a universal “exists” in a double way,one being in the mind and the other in things in the world. Abelard, for ex-ample, held that a universal “exists” first as a common likeness in things, andthen as a concept that exists in the human intellect through the mind’s powerto abstract from our perception of things by attending to the likeness in them.Aristotle is sometimes also said to have held such a view. Aquinas, however,was clear about the distinction and maintained that a concept and a naturalproperty or natural kind are not really the same universal, and that in fact theydo not even have the same mode of being. The distinction is one of the issuesdiscussed in chapter twelve.

Natural realism is usually described as a moderate realism, where universals

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exist only in re, i.e., in things. As a scientifically acceptable ontology, however,this is much too restrictive a view. As we explain in chapter twelve, many of thenatural properties and relations that we now know to characterize atoms andcompounds as physical complexes did not have any instances at all at the timeof the Big Bang, i.e., when the universe began. Yet, these natural propertiesand relations must be acknowledged in any natural realism that is adequate forscience. What we propose instead in conceptual realism is a modal moderaterealism, where the modality in question is ontologically grounded in nature andits causal matrix.

A logic of natural kinds can also be developed within this framework, more-over, so that the result is a reconstruction of Aristotelian essentialism. Naturalkinds are general essences in this ontology, and not individual essences, whichare usually found in modal versions of logical realism. The question of whetherthere can be individual essences in a modern form of Aristotelian essentialismis briefly considered in chapter twelve.

10. Criteria of Adequacy

Criteria of adequacy for a formal ontology that we have indicated so far can besummarized as follows.

• A formal ontology must provide a logically perspicuous representationof our commonsense understanding of the world as well as our scientificunderstanding.

• A formal ontology must explain the distinction between being and exis-tence, i.e., give an ontological grounding of that distinction, or if it deniesthe distinction then explain why it does so and why the result is an ade-quate ontological framework.

• A formal ontology should provide an ontological, and not just a set-theoretical, account of modality.

• A formal ontology must explain the nature of predication in thought aswell as in language and indicate what theory of universals is part of thatexplanation.

It is because conceptual realism fulfills these criteria of adequacy, as well asothers indicated throughout this book, that it is the best formal ontology toadopt.

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Part I

Formal Ontology

1

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Chapter 1

Formal Ontology andConceptual Realism

Formal ontology is a discipline in which the formal methods of mathematicallogic are combined with the intuitive, philosophical analyses and principles ofontology.1 In this way formal ontology brings together the clarity, precisionand methodology of logical analysis on the one hand with the philosophicalsignificance of ontological analysis on the other. Father I.M. Bochenski has saidof ontology, for example, that it is a “sort of prolegomenon to logic” in thatwhereas ontology is an intuitive, informal inquiry into the categorial aspects ofreality in general, “logic is the systematic formal, axiomatic elaboration of thismaterial predigested by ontology.”2

Ontology, which is the study of being qua being (Aristotle, Meta. 1031a),has been a principal part of metaphysics since ancient times. Metaphysics itselfhas usually been divided into ontology and cosmology, where

• ontology = the study of being as such, and

• cosmology = the study of the physical universe at large; i.e., space, time,nature and causality.

Implicit in this division is a distinction between methodologies. The method-ology of cosmology, for example, is based on the analysis of such categories asspace, time, matter, and causality, where the goal is to discover by observationand experiment the laws connecting those categories and their constituents withone another, including in particular the natural kinds of things (beings) in na-ture. The methodology of ontology, on the other hand, is based on the analysisof ontological categories, i.e., categories of being, where the goal is to discoverthe laws connecting these categories and the entities in them with one another.

1This chapter is an extended version of my essay, “Formal Ontology,” in Burkhardt andSmith 1991.

2Bochenski 1974.

3

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The particular sciences that are part of cosmology and that are concernedwith particular natural kinds of beings may be prior to ontology in the order ofdiscovery—and even in the order of conception. But as an analysis of ontologicalcategories, ontology is a science that contains the ontological forms, if not thespecific content, of the ideas and principles of the different natural sciences, andin that sense it is a science that is prior to all of the others. Similarly, logiccontains the logical forms, but not the specific content, of the different scientifictheories that make up the content of our knowledge of nature, and in that senselogic is a science prior to all others. Thus, when the logico-grammatical formsand principles of logic are formulated with the idea of representing the differentcategories of being and the laws connecting them, i.e., when ontological andlogical categories are combined in a unified framework, then the result, whichis what we mean by formal ontology, is a comprehensive deductive science thatis prior to all others in both logical and ontological structure. In addition, byproving the consistency of the logical framework we can therefore also show thatthe intuitive ontological framework is consistent as well.

1.1 Formal Ontology as a CharacteristicaUniversalis

A system of logic can be constructed under two quite different aspects. Onthe one hand, it can be developed as a formal calculus and studied indepen-dently of whatever content it might be used to represent. Such a formal systemin that case is only a calculus ratiocinator. On the other hand, a system oflogic can be constructed somewhat along the lines of what Leibniz called acharacteristica universalis . Such a system, according to Leibniz, was to servethree main purposes. The first was that of an international auxiliary languagethat would enable the people of different countries to speak and communicatewith one another. Apparently, because Latin was no longer a “living” languageand new trade routes were opening up to lands with many different local lan-guages, the possibility of such an international auxiliary language was widelyconsidered and discussed in the 17th and 18th centuries.3 There were in fact anumber of proposals and partial constructions of such a language during thatperiod, but none of them succeeded in being used by more than a handful ofpeople. It was only towards the end of the 19th century when Esperanto wasconstructed that such a language came to be used by as many as eight millionpeople. At present, however, the question of whether even Esperanto will suc-ceed in fulfilling that purpose seems very much in doubt. Ido is another suchlanguage, which was constructed in 1907 by a committee of linguists, but it hasnot been used since about 1930.4 In any case, notwithstanding its visionary goal,the idea of an international auxiliary language is not the purpose of a formalontology.

3Cf. Cohen 1954 and Knowlson 1975.4Cf. Van Themaat 1962.

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1.1. FORMAL ONTOLOGY AS A CHARACTERISTICA UNIVERSALIS 5

The second and third purposes Leibniz set for his characteristica universalisare what distinguish it from its precursors and give his program its formal orlogistic methodology. The second purpose is that the universal character is to bebased upon an ars combinatoria, i.e., an ideography or system of symbolization,that would enable it to provide a logical analysis of all of the actual and possibleconcepts that might arise in science. Such an ars combinatoria would containboth a theory of logical form, i.e., a theory of all the possible forms that ameaningful expression might have in such a language, and a theory of definitionalforms, i.e., a theory of the operations whereby one could construct new conceptson the basis of already given concepts. The third purpose was that the universalcharacter must contain a calculus ratiocinator , and in particular a completesystem of deduction and valid argument forms, by which, through a study ofthe consequences, or implications, of what was already known, it could serve asan instrument of knowledge. These two purposes are central to the notion of aformal ontology.

With a universal character that could serve these purposes, Leibniz thoughtthat a unified encyclopedia of science could be developed about the world, andthat, by its means, the universal character would then also amount to a charac-teristica realis, i.e., a representational system that would enable us to see intothe inner nature of things and guide our reasoning about reality like an Ariadne’sthread.5 In other words, in Leibniz’s program for a characteristica universalis wehave an attempt to encompass the relationships between language and reality,language and thought, and language and knowledge, especially as representedin terms of scientific theories. In two fundamental parts of the program, namely,the construction of an ars combinatoria and a calculus ratiocinator, we also havetwo critical components that are necessary for a formal ontology.

The idea of a characteristica realis, i.e., a unified encyclopedia of science, isalso important for a formal ontology. That is because a formal ontology, as alogistic system, must be structurally rich enough so that in principle every sci-entific theory can be formulated within it so that the result would be a system ofmetaphysics containing both an ontology and a cosmology. Of course, this willbe possible only by adding to the general framework of a formal ontology ap-propriate nonlogical constants, axioms, and meaning postulates that representthe basic concepts and principles of a given science. In addition, though thisis not required from a strictly scientific point of view, a formal ontology shouldbe sufficiently structured so that with the addition of suitable nonlogical con-stants and meaning postulates a logical analysis of every meaningful declarativesentence of natural language can be given within it. That is, a formal ontologyshould be able to contain a semantics for natural language that captures theontology of our commonsense framework. In that case such a logistic system canbe taken not only as a characteristica realis, but also as a lingua philosophica.Of course, prior to the introduction of such constants and postulates, whetherfor science or natural language, a formal ontology is essentially just a shell con-taining the logico-ontological categorial forms and principles of science and of

5Cp. Cohen 1954, p. 50.

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our commonsense understanding of the world.There is still one important component of a formal ontology that Leibniz

did not include or discuss as part of a characteristica universalis. This is thecomponent that deals explicitly with the analysis of ontological categories andhow those categories are connected with the nexus of predication in language,thought and reality. It is this component that distinguishes formal ontologyfrom analytic metaphysics in the sense of formally constructed systems thatrepresent one or more aspects of reality. Analytic metaphysics is a part offormal ontology, to be sure, but it is not itself a formal ontology unless it dealsexplicitly with the formal analysis ontological categories and their connectionwith the nexus of predication in language, thought and reality.

1.2 Radical Empiricism and the LogicalConstruction of the World

Both Gottlob Frege and Bertrand Russell viewed the logical systems they con-structed as a framework for a universal characteristica, and each had a particu-lar analysis of the nexus of predication, which we will discuss in chapters three,four, and five. Russell, for example, described his theory of types as “a logicallyperfect language”, by which he meant a language that would show at a glancethe logical structure of the facts that are described by its means.6 According toRussell, such a language “would be one in which everything that we might wishto say in the way of propositions that are intelligible to us, could be said, andin which, further, structure would always be made explicit”.7 All that needsto be added to the theory of logical types to be such a language, Russell main-tained, is a vocabulary of (nonlogical) descriptive constants that correspond tothe meaningful words and phrases of natural science and ordinary language.The constants of pure mathematics, unlike those of the natural sciences, donot need to be added to the framework because they, according to Russell, areall definable in purely logical terms within the framework itself. Knowledge ofpure mathematics is explainable, in other words, as logical knowledge—a viewknown as logicism—and, in particular, as knowledge of the propositions thatare provable in the theory of logical types independently of any vocabulary ofdescriptive constants.

Despite his logicism regarding mathematics, Russell was a radical empiricistas far as our knowledge of physical or concrete existence was concerned. Allour empirical knowledge of the world, he maintained, must be reducible to ourknowledge of what is given in experience, by which he meant that it must beconstructible within the framework of the theory of logical types from the lowestlevel of objects, which he assumed to be events corresponding to our sensoryexperience. It was by means of logical constructions within this framework thatRussell proposed to bridge the gulf between the world of physical science and

6Cp. “The Philosophy of Logical Atomism” (1918) in Russell 1956.7Russell 1959, p. 165.

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1.2. RADICAL EMPIRICISM AND THE LOGICAL CONSTRUCTION 7

the world of sense, and he was guided in this regard by “the maxim whichinspires all scientific philosophizing, namely ‘Occam’s razor’: Entities are notto be multiplied without necessity. In other words, in dealing with any subject-matter, find out what entities are undeniably involved, and state everything interms of these entities”.8 Thus, because sense-data are the entities that are“undeniably involved” in all of our empirical knowledge according to Russell,“the only justification possible” for such knowledge “must be one which exhibitsmatter as a logical construction from sense-data”.9

Though Russell gave a number of examples of how such a construction mightbe given, it was Rudolf Carnap who, using nothing more than the frameworkof the simple theory of logical types (and a certain amount of empirical science,such as gestalt psychology, about the structure of experience), gave the mostdetailed analysis indicating how we might reconstruct all our knowledge of theworld in terms of what is given in experience.10This meant in particular thatall of the concepts of science could be analyzed and reduced to certain basicconcepts that apply to the content of what is given in experience. One ofthe important patterns for such an analysis is known today as definition byabstraction, whereby, relative to a given equivalence relation (i.e., a relation thatis reflexive, symmetric and transitive), certain concepts are identified with (orrepresented by) the equivalence classes that are generated by that equivalencerelation. This pattern was used by Frege and Russell in the analysis of thenatural numbers (as based on the equivalence relation of equinumerosity, or one-one correspondence), and was then used again in the analysis of the negativeand positive integers, the rational numbers, the real numbers, and even theimaginary numbers. Carnap, who took definition by abstraction as indicativeof the “proper analysis” of a concept, generalized the pattern into a form thathe called “quasi analysis” (but which, he acknowledged, really amounted toa form of synthesis), which could be based on relations of partial similarityinstead of full similarity, i.e., on relations that amount to something less thanan equivalence relation. The concepts definable in terms of a quasi-analysisspecify in what respect things (especially items of experience) that stand to oneanother in a relation of partial similarity agree, i.e., the respect in which theyare in part, but not fully, similar.11 In this way Carnap was able to define thevarious sense modalities (as classes of qualities that intuitively belong to thesame sense modality, where concepts for sense qualities are definable in termsof a partial similarity between elementary experiences), including in particularthe visual sense and color concepts (as determined by the three-dimensionalordering relation of the color solid).

Using the various sense modalities, Carnap went on to construct the four-dimensional space-time world of perceptual objects, which, with all its vari-ous sense qualities, “has only provisional validity”, and which, for that reason,“must give way to the strictly unambiguous but completely quality-free world

8Russell 1914, p.112.9Ibid., p. 106.

10Cp. Carnap 1967.11Cp. Carnap, op. cit., sections 71-74.

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of physics”.12 We will not go into the details of Carnap’s “constructional def-initions” here, but we should note that all that Carnap meant by a logicalanalysis by means of such definitions was translatability into his constructionallanguage—i.e., into an applied form of the simple theory of logical types as basedon a primitive descriptive constant for a certain relation of partial similaritybetween elementary experiences. Such a translation need not, and in generaldid not, preserve synonymy, nor did it in any sense amount to an ontologi-cal reduction of ordinary physical objects to the sensory objects of experience.What it did preserve, according to Carnap, was a material equivalence (i.e., anequivalence of truth-value) between the sentences of ordinary language, or of ascientific theory, and the sentences of the constructional language.13

Carnap’s project was not a formal ontology in the sense intended here, unlesswe view it as representing the radical empiricist doctrine that there is no more toreality than what we can construct in terms of sense data, or what Carnap calledelementary experiences (Elementarelebnisse). One extreme form of this positionis ontological solipsism, which is not at all the same as the methodologicalsolipsism that Carnap took it to be. In addition, the one important componentthat is missing is an analysis of the nexus of predication and an account of ourcommonsense understanding of the world.

1.3 Commonsense Versus ScientificUnderstanding

Our commonsense understanding of the world is sometimes said to be in conflictwith our scientific understanding, which, on this view, is taken as providing theonly proper criteria for truth. It is also claimed that the construction of a logisticsystem as the basis of a unified encyclopedia of science can represent only ourscientific understanding, because by its very nature such a system can operateonly with concepts and principles that have sharp and exact boundaries, suchas the concepts and principles we strive to formulate in our scientific theories.The same cannot be said, according to this so-called “eliminativist view,” ofthe concepts and principles of our commonsense understanding. That is, theconcepts underlying our use of natural language do not have sharp boundaries,and do not require the kind of precision of thought that is the goal of scientificknowledge, which alone can provide an adequate criterion of truth. Many of thewords and phrases of natural language by which we express our commonsenseunderstanding, for example, are vague or ambiguous, and as such are unsuitablefor the kind of logical representation involved in our methodology. Gottlob Fregeexpressed this view in comparing the difference between his logical system andordinary language with that between a microscope and the human eye. Eventhough the eye is superior to the microscope, Frege observed, “because of the

12Ibid., p. 207.13The same claim is made in Goodman 1951 for an alternative constructional language

based on nominalist principles that are opposed to the predicate quantifiers of type theory.

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1.3. COMMONSENSE VERSUS SCIENTIFIC UNDERSTANDING 9

range of its possible uses and the versatility with which it can adapt to the mostdiverse circumstances,” nevertheless, “as soon as scientific goals demand greatsharpness of resolution, the eye proves to be insufficient”.14

That only our scientific understanding can provide an adequate criterion oftruth about the natural world does not mean that our commonsense under-standing gives a false picture of the world, or a picture that, for the purposesof knowledge, ought to be eliminated.15 No doubt, many of our commonsensebeliefs and concepts about the natural world have been revised and correctedover the millennia, and probably many will be revised or corrected in times tocome. The concept water, for example, has been replaced by the concept H2Oin the scientific context of the atomic theory of matter, where the concept H2Ois systematically related to the concept H for hydrogen and the concept O foroxygen.16 This does not mean that the concept water is somehow misleadingand that the role it plays in our commonsense framework is to be eliminated.Indeed, not only has the concept continued to be functionally useful in everydaycontexts, but it also continues to serve in scientific contexts as well.

It is not just our commonsense concepts that are important for an under-standing of the world, however, but also how we structure our thought in ourcommonsense framework as well. How we reason and argue in this frameworkare preconditions of scientific knowledge and theorizing. Scientific understand-ing depends, in other words, both conceptually and pragmatically upon ourcommonsense understanding, including the way the world is categorially struc-tured, and the way we reason in terms of that structure. In this regard, therepresentation of our scientific knowledge involves more than the representationof a large number of facts or beliefs about the objects in a given domain ofscientific inquiry, regardless of whether those facts or beliefs are in conflict withwhat is believed by common sense. In particular, it involves the criteria forvalid reasoning that we bring to bear on our commonsense arguments, and theway those arguments are structured in terms of the categorial structure of ourcommonsense understanding.

It is precisely the formal representation of the categorial structure of ourcommonsense framework, as well as the criteria for valid reasoning within thatframework that is one of the goals of formal ontology and a criterion of adequacy.The arguments that we find in natural language and in terms of which wearticulate our reasoning can be evaluated as valid or invalid only with respectto a logical theory, and in particular one that provides an adequate formalrepresentation of the basic categories of natural language and the commonsenseframework expressed in its use. The adequacy of such a theory is judged on thebasis of how well it agrees with our commonsense intuition of which argumentsare valid and which invalid.

We are not claiming here that the ontology of our commonsense framework,based as it is on perceptible objects and their qualitative features, is also fun-

14Frege 1879, p, 6.15See, e.g., “Philosophy and the Scientific Image of man”, in Sellars 1963, for a account of

the eliminativist view.16Cf. Fodor 1993, p.86.

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damental to science. Certainly, our commonsense framework is prior in theorder of conception, but it is not necessarily prior in the order of being. Thisdistinction is especially important in regard to the mind-body problem and thenature of consciousness where the eliminativist view against our commonsenseunderstanding has been argued most forcefully. But some caution is necessaryhere because the mind-body problem really divides into at least two differentsub-problems:

(a) the study of the relations between physiological states and certain statesof consciousness, and

(b) the study of the emergence of consciousness, meaning, and the self andits relation to its body. consciousness, is a problem that is studied by experts

in neuropsychology and other neurosciences, and as such it is a proper part ofa characteristica realis. The second problem, i.e., the problem of the emergenceof consciousness, meaning and the self can be solved, on the other hand, only bytaking natural language, intentionality, and our commonsense framework intoaccount, which means the inclusion within formal ontology of an intensionallogic that can be used to represent our commonsense understanding and thecontents of our beliefs and theories, including the fables and stories that arepart of our culture and of our individual mental spaces. Such an intensionallogic will provide an account of the ontology of fictional objects in terms of thecontents of our concepts, and it will contain a logic of our various cognitivemodalities, including a logic of knowledge and belief.

1.4 The Nexus of Predication

Leibniz’s own ideography for his characteristica universalis was algebraic and,as we have noted, did not deal with the central feature of either a conceptualor ontological theory of logical form—namely, the nexus of predication. Howpredication is represented in a formal ontology depends on different theories ofuniversals, where by a universal we mean that which can be predicated of things(Aristotle, De Int. 17a39). Traditionally, there have been three main theoriesof universals: nominalism, conceptualism, and realism.

The difference between these three types of theories depends on what eachtakes to be predicable of things. In this regard, we will distinguish between:

• predication in language (nominalism),• predication in thought (conceptualism), and• predication in reality (realism).

All three types of theories agree that there is predication in language, and inparticular that predicates can be predicated of things in the sense of being trueor false of them. Nominalism goes further in maintaining that only predicates(or really prdicate tokens) can be predicated of things, that is, that there areno universals other than the predicate expressions of some language or other:

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Nominalism: only predicates are true or false of things; there are no uni-versals that predicates stand for.

Conceptualism opposes nominalism and maintains that predicates can betrue or false of things only because they stand for concepts, where predica-ble concepts are the cognitive capacities—intelligible universals—that underliepredication in thought and our rule-following abilities in the use of the predicateexpressions of natural language.

Conceptualism: predication in thought underlies predication in language;predicable concepts are rule-following cognitive capacities regarding the use ofpredicate expressions.

Realism also opposes nominalism in maintaining that there are real uni-versals, namely, properties and relations, that are the basis of predication inreality.

Realism: there are real properties and relations that are the basis of pred-ication in reality.

There are two distinct types of realism that should be distinguished; namely,various forms of logical realism as modern forms of Platonism, and various formsof natural realism, with at least one being a modern form of Aristotle’s theoryof natural kinds.

Realism↙ ↘

logical realism natural realism↙ ↘

with natural kinds without natural kinds↓

(Aristotelian essentialism)

Both forms of realism are compatible with conceptualism, but natural re-alism, especially Aristotle’s theory of natural kinds, is closely connected withthe kind of conceptualism we will describe in later chapters. That is because,natural realism as a formal ontology presupposes some form of conceptualismin order even to be articulated; and the kind of conceptualism that we will laterdefend depends in turn on some form of natural realism as its causal basis. Howconceptualism is compatible with logical realism, and how natural realism anda certain modern form of conceptualism are intimately connected are issues wewill take up later in our discussion of what we call conceptual realism.

Corresponding to these different theories of universals, there are differentformal ontologies containing different formal theories of predication, each rep-resenting some variant of one of these alternatives. That means that there willbe different comprehensive systems of formal ontology. Each formal ontology,of course, will view itself internally as the final arbiter of all logical and onto-logical distinctions. But the study of different possible formal ontologies, theirconsistency, adequacy, and relative strength with respect to one another, and,similarly, the study of the alternative subtheories that might be realized in the

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different branches of a comprehensive formal ontology, may together be calledcomparative formal ontology.

A comprehensive system of formal ontology will in general have differentbranches or subsystems within which different ontological tasks can be carriedout. One such branch, for example, might be a theory of parts and wholes, whichwould include a relation of foundation regarding how some parts are foundedor dependent upon other parts or wholes.17 There might also be a theory ofextensive and intensive magnitudes, i.e., a measurement theory, and a theory ofcontinuants and of the existence of the latter in space and time.18

1.5 Univocal Versus Multiple Senses of Being

One important distinction between different systems of formal ontology is whetherbeing is taken as univocal or as having different senses. It will have differentsenses when different types or categories of expressions are understood as rep-resenting different categories of being, in which case there will also be differenttypes of variables bound by quantifiers having the entities of those differentcategories as their values. Such is the case in both conceptualism and somevariants of realism. Where being is univocal, on the other hand, i.e., wherethere is just one ontological category of being (being simpliciter), only one typeof quantifiable variable will have semantic significance. This does not meanthat there are no different “kinds”, or sorts, of being, but only that in such aframework being is a genus, and that the different kinds of being all fall withinthat genus. In a formal ontology for nominalism, for example, there will beno ontological category corresponding to any grammatical category other thanthat of objectual terms (logical subjects), and in particular there will be no on-tological category or mode of being corresponding to the grammatical categoryof predicate expressions. Only objectual variables, i.e., the category of variableshaving objectual terms as their substituends, will have semantic-ontological sig-nificance in such a formal ontology. Predicate variables, and quantifiers bindingsuch, if admitted at all, must then be given only a substitutional and not asemantic interpretation, which means that certain constraints must be imposedon the logic of the predicate quantifiers in such a formal ontology.

Most nominalists in fact eschew even such a substitutional interpretationof predicate quantifiers and describe their ontology only in terms of first-orderlogic where there is but one type of boundable variable, i.e., where, as in W.V.O.Quine’s phrase:

to be = to be the value of a bound objectual variable.

It should be noted, however, that, unlike traditional nominalists, some con-temporary nominalists (e.g., Nelson Goodman), take abstract objects (e.g.,qualia) as well as concrete objects to fall under their supposedly univocal sense

17Cf. Husserl 1900, Volume 2, Investigation III, and Barry Smith 1982.18Cf. Brentano, 1933.

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of being.19 This means that although there is but one ontological category ofbeing in such an ontology, there may still be different “kinds” of being. Thatis, in such a system being is a genus, which is not at all the same as beingmultivalent.

Nominalism: being is univocal; i.e., being is a genus.

Being is also univocal in some forms of realism (regarding universals). Thiswould appear to be the case, for example, in the ultra-realism of certain earlyscholastic philosophers for whom the realm of being is the realm only of univer-sals (as in the teachings of John Scottus Eriugena and Remigius of Auxerre).It is certainly univocal in the case of certain contemporary forms of logical real-ism, where properties, relations, concrete objects, and perhaps states of affairsas well, are different kinds, as opposed to, categories of being. A formal ontologyfor such realists is developed today much as it is in nominalism, namely, as anaxiomatic first-order logic with primitive predicates standing for certain basicontological notions. Indeed, except perhaps for the distinction between an in-tensional and an extensional logic, there is little to distinguish realists who takebeing to be univocal from such nominalists as Goodman who include abstractobjects as values of their objectual variables and who describe such objectsaxiomatically (e.g., in terms of a mereological relation of overlap, or of part-to-whole).20 This is particularly true of those realists who, in effect, replace theextensional membership relation of an axiomatic set theory by an intensionalrelation of exemplification, and, dropping the axiom of extensionality, call theresult a theory of properties.21

Logical realism: being is univocal (i.e., being is a genus) if predication isbased on a relation of:

1. membership (set theory), or

2. exemplification (in first-order logic), or

3. part-to-whole (mereology).

Formal ontology, in other words, for both the nominalist and that kind ofrealist who takes being to be univocal and who has abstract as well as concreteobjects as values of their object variables, i.e., for whom being is a genus, is reallyno different from an applied theory of first-order logic. That is, it is no differentfrom a first-order logic to which primitive “nonlogical” (descriptive?) constantsand axioms are added and taken as describing certain basic ontological notions.In such a framework, it would seem, the dividing line between the logical and thenonlogical, or between pure formal ontology and its applications, has becomesomewhat blurred, if not entirely arbitrary.

19Cf. Goodman 1956, p. 17.20See also Goodman and Quine 1947.21Cf., Bealer 1982.

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1.6 Predication and Preeminent Being

Beginning with Aristotle, the standard assumption in the history of ontologyhas been that being is not a genus, i.e., that there are different senses of being,and that the principal method of ontology is categorial analysis. This raises theproblem of how the different categories of being fit together, and of whetherone of the senses or category of being is preeminent and the others somehowdependent on that sense or category of being. The different categorial analysesthat have been proposed as a resolution of this problem have all turned in oneway or another on a theory of predication, i.e., on how the different categoriesfit together in the nexus of predication, and they have differed from one anotherprimarily on whether the analysis of the fundamental forms of predication is tobe directed upon the structure of reality or the structure of thought. In formalontology, the resolution of this problem involves the construction of a formaltheory of predication.

Aristotle’s categorial analysis, for example, is directed upon the structure ofthe natural world and not upon the structure of thought, and the preeminentmode of being is that of concrete individual things, or primary substances.Aristotle’s realism regarding species, genera, and universals is a form of naturalrealism, it should be emphasized, and not of logical realism. Also, unlike logicalrealism, Aristotle’s realism is a moderate realism, though, as we indicate below,a modal moderate realism is better suited to a modern form of Aristotelianessentialism.

• Moderate realism = the ontological thesis that universals exist only inrebus, i.e., in things in the world.

• Modal moderate realism = the ontological thesis that universals existonly in things that, as a matter of a natural or causal possibility, could exist innature, even if in fact no such things actually do exist in nature.

Predication is explained in Aristotle’s realism in terms of two ontological con-figurations that together characterize the essence-accident distinction of Aris-totelian essentialism. These are the so-called essential predicative nexus be-tween an individual and the species or genera, i.e., the natural kinds, to whichit belongs, and the accidental, or nonessential, predicative nexus between anindividual and the universals that inhere in it. A formal theory of predicationconstructed as an Aristotelian formal ontology must respect this distinction be-tween essential and accidental predication, and it must do so in terms of anadequate representation of the two ontological configurations underlying predi-cation in an Aristotelian ontology.

Aristotle’s moderate natural realism has two types of predication:1. Predication of species, genera (natural kinds).2. Predication of properties and relations.

As a formal ontology, Aristotelian essentialism must contain a logic of naturalkinds. In addition, as a form of moderate realism it must impose the constraintthat every natural kind, property or relation is instantiated, because every nat-ural kind, property or relation exists only in rebus. This constraint leads to

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1.6. PREDICATION AND PREEMINENT BEING 15

Aristotle’s problem of the fixity of species, according to which members of aspecies cannot come to be except from earlier members of that species, and thattherefore there can be no evolution of new species.

The fixity of species:

Members of a species cannot come to be except from earlier members of thatspecies.

Therefore, there can be no evolution of new species.

This problem can be resolved, however, in a modified Aristotelian formalontology of modal natural realism, where the modal category of natural necessityand possibility is part of the framework of the formal ontology. On this modifiedaccount, instead of requiring that every natural property or relation actually beinstantiated at any given time, we require only that such an instantiation bewithin the realm of natural possibility, a possibility that might arise in time andchanging circumstances and not just in other possible worlds. Such a formalontology, needless to say, will contain a modal logic for natural necessity andpossibility, as well as a logic of natural kinds that is to be described in terms ofthat modal logic.22 Natural necessity, we will later argue, is a causal modalitybased on natural kinds and the laws of nature, and as such it is not the sameas logical necessity. As modalities, logical necessity and possibility, we willlater argue, can be made sense of only in an ontology of logical atomism, anontology in which there are no causal relations and no natural necessity as acausal modality.

Plato’s ontology is also directed upon the structure of reality, but the pre-eminent mode of being in this framework is not that of concrete or sensibleobjects but of the Ideas, or Forms. This leads to the problem of how and inwhat sense concrete objects participate in Ideas, and also to the problem ofhow and in what sense Ideas are “things” or abstract objects separate from theconcrete objects that participate in them. A Platonist theory of predication incontemporary formal ontology is the basis of logical realism, where it is assumedthat a property or relation exists corresponding to each well-formed predicateexpression (or open formula) of logical grammar, regardless of whether or notit is even logically possible that such a property or relation have an instance.When applied as a foundation for mathematics (as was Plato’s own original in-tent), logical realism is also called ontological logicism. The best-known form oflogical realism today is Bertrand Russell’s theory of logical types, which Russelldeveloped as a way to avoid his famous paradox of predication (upon which hisparadox of membership is based), a paradox not unrelated to Plato’s problemof the separate reality of Ideas. Whether and to what extent Russell’s theoryof logical types can satisfactorily resolve either of Plato’s problems and be thebasis of an adequate realist formal ontology is an issue that belongs to what wehave called comparative formal ontology.

22Cf. Cocchiarella 1976 and 1996.

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1.7 Categorial Analysis and TranscendentalLogic

Kant’s categorial analysis, unlike Aristotle’s, is directed upon the structure ofthought and experience rather than upon the structure of reality. The cate-gories function on this account to articulate the logical forms of judgments andnot as the general causes or grounds of concrete being. There is no preeminentmode of being identified in this analysis, accordingly, other than that of thetranscendental subject, whose synthetic unity of apperception is what unifiesthe categories that are the bases of the different possible judgments that canbe made. What categories there are and how they fit together to determinethe concept of an object in general is determined through a “transcendental de-duction” from Kant’s table of judgments, i.e., from the different possible formsthat judgments might have according to Kant. It is for this reason that the logicdetermined by this kind of categorial analysis is called transcendental logic.

The transcendental logic of Husserl in his later work is perhaps one of thebest-known versions of this type of approach to formal ontology. According toHusserl, logic, as formal ontology, is a universal theory of science, and as suchit is the justifying discipline for science. But even logic itself must be justified,Husserl insists, and it is that justification that is the task of transcendentallogic. This means that the grounds of the categorial structures that determinethe logical forms of pure logic are to be found in a transcendental subjectivity,and it is to a transcendental critique of such grounds that Husserl turns inhis later philosophical work. On the basis of such a critique, for example,Husserl gives subjective versions of the laws and rules of logic, such as the law ofcontradiction, the principle of excluded middle, and the rules of modus ponensand modus tollens, claiming that it is only in such subjective versions that therecan be found the a priori structures of the evidence for the objective versionsof those laws and rules.23 Husserl also claims on the basis of such grounds thatevery judgment can be decided24, and that a “multiplicity,” such as the systemof natural numbers, is to be “defined, not by just any formal axiom system, butby a ‘complete’ one”.25 That is, according to Husserl:

the axiom-system formally defining such a multiplicity is distin-guished by the circumstance that any proposition ... that can beconstructed, in accordance with the grammar of pure logic, out ofthe concepts ... occurring in that system is either true—that is tosay: an analytic (purely deducible) consequence of the axioms—or ‘false’—that is to say: an analytic contradiction—; tertium nondatur.26

Unfortunately, while such claims for transcendental logic are admirable ide-als, they are nevertheless in conflict with certain well-known results of

23Husserl 1929, §§75-8.24Ibid., §§79-80.25Ibid., §31, p. 96.26Ibid.. Cf. also Husserl 1913 §72, pp. 187f.

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1.8. THE COMPLETENESS PROBLEM 17

mathematical logic, such as Kurt Godel’s first incompleteness theorem.

1.8 The Completeness Problem

The transcendental approach to categorial analysis, as this last observation in-dicates, raises the important problem of the completeness of formal ontology. Itdoes this, moreover, not in just one but in at least two ways: first, as the prob-lem of the completeness of the categories; and, second, as the problem of thecompleteness of the laws of consequence regarding the logical forms generatedby those categories.

Two problems of the completeness of formal ontology:1. the completeness of the categories; and2. the completeness of the deductive laws with respect to those categories.

For Aristotle, for whom the categories are the most general “causes” orgrounds of concrete being, and for whom categorial analysis is directed upon thestructure of reality, the categories and their systematization must be discoveredby an inductive abstraction and reflection on the structure of reality as it isrevealed in the development of scientific knowledge, and therefore the questionof the completeness of the categories and of their systematization can never besettled as a matter of a priori knowledge. This is true of natural realism ingeneral.

Natural realism: the categories of nature and their lawsare not knowable a priori.

For Kant and the transcendental approach, however, the categories and theprinciples that flow from them have an a priori validity that is grounded inthe understanding and pure reason respectively—or, as on Husserl’s approach,in a transcendental phenomenology—and the question of the “unconditionedcompleteness” of both is said to be not only practical but also necessary. Thedifficulty with this position for Kant is that neither the system of categoriesnor the laws of logic described in terms of those categories can be viewed asproviding an adequate system of formal ontology as we have described it above.Kant’s description of logic, for example, restricts it to the valid forms of thesyllogism, which can in no sense account for the complexity of many intuitivelyvalid arguments of natural language, not to mention the complexity of proofsin mathematics. Husserl, unlike Kant, does not himself attempt to settle thematter of a complete system of categories, nor therefore of a complete system ofthe laws of logic or formal ontology; but he does maintain that such completenessis not only possible but necessary, and that the results achieved regarding thecategories and their systematization must ultimately be grounded on the a prioristructures of the evidence of a transcendental subjectivity.

Transcendental logic: the categories and their laws areknowable a priori.

The transcendental approach in general, in other words, or at least the apriori nature of its methodology as originally described, leaves no room for

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inductive methods or new developments in either logic or categorial analysis,especially in the way both are affected by new results in scientific theory (e.g.,the logic of quantum mechanics27 and the way that logic relates to the logicof macrophysical objects) or in theoretical linguistics, (e.g., universal grammarand the way that grammar is related to the pure logical grammar of a formalontology), or even in cognitive science (e.g., artificial intelligence and the waythat the computational theory of mind is related to the categorial and deductivestructure of logic).

Some categorial analyses not knowable a priori:1. The logic of quantum mechanics and how that logic relates to the logic

of macrophysical objects.2. Theoretical linguistics: is there a universal grammar underlying all nat-

ural languages? And, if so, how is that grammar related to the pure logicalgrammar of a formal ontology?

3. Cognitive science and artificial intelligence: are there categories and lawsof thought that can be represented in formal ontology? And, if so, how arethese categories and laws related to the categories of nature? And can they besimulated (duplicated?) in artificial intelligence?

Despite the difficulties with the problem of completeness of the a priorimethodology of the transcendental approach, it does not follow that we mustgive up the view that an analysis of the forms of predication is to be directedprimarily upon the structure of thought. There are alternatives other than thetranscendental idealism of either Kant or Husserl that such a view might adopt.Jean Piaget’s genetic epistemology with its “functional” (as opposed to abso-lute) a priori is such an alternative, for example, and so is Konrad Lorenz’sbiological Kantianism with its evolutionarily determined (and therefore non-transcendental) a priori.28

Some non-transcendental approaches:1. Jean Piaget’s genetic epistemology (a non-absolute “functional” a priori).2. Konrad Lorenz’s biological Kantianism (an evolutionarily determined a

priori).

Any version of a naturalized epistemology, in other words, where an a pos-teriori element would be allowed a role in the construction of a formal ontology,might serve as such an alternative; and in fact such a naturalized epistemol-ogy is presupposed by conceptual realism, which we will describe in more detaillater. The comparison of these alternatives, and a study of their adequacy (aswell as of the adequacy of a more complete and perhaps modified account oftranscendental apriority) as epistemological grounds for a categorial analysisthat is directed upon the structure of thought, are issues that properly belongto comparative formal ontology. The transcendental approach claims to be in-dependent of our status as biologically, culturally, and historically determined

27Cf. Putnam 1969 and Dummett 1976.28Cf. Piaget 1972 and Lorenz 1962.

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1.9. SET-THEORETIC SEMANTICS 19

beings, and therefore independent of the laws of nature and our evolutionaryhistory.

1.9 Set-Theoretic Semantics

The problem of the completeness of a formal ontology brings up a method-ological issue that is important to note here. This is the issue of how differentresearch programs can be carried out in restricted branches or subdomains ofa formal ontology without first deciding whether or not the categorial analysisof that formal ontology is to be directed upon the structure of thought or thestructure of reality. We do not always have to decide in advance whether or notthere must (or even ever can) be a final completeness to the categories or of thelaws of logic before undertaking such a research program. In particular, we cantry to establish restricted or relative notions of completeness for special areas ofa formal ontology, and we can then compare and evaluate those results in thecontext of comparative formal ontology. The construction of abstract formalsystems and model-theoretic semantics within set theory will be especially use-ful in carrying out and comparing such research programs. In other words, settheory is an ideal framework within which to carry out comparative analyses ofdifferent formal systems proposed either as a formal ontology or a subsystem ofsuch. Set theory is not itself a formal ontology, it should be noted, in that itdoes not contain a theory of predication.

We must be cautious in our use of set theory, however, and especially in howwe apply such well-known mathematical results as Kurt Godel’s incompletenesstheorems. Godel’s first incompleteness theorem, for example, does not show, asis commonly claimed, that every second-order predicate logic must be incom-plete, where by second-order predicate logic we mean an extension of first-orderpredicate logic in which quantifiers are allowed to reach into the positions thatpredicates occupy as well as of the subject or argument positions of those predi-cates. Rather, what Godel’s theorem shows is that second-order predicate logicis incomplete with respect to its so-called standard set-theoretic semantics. Inparticular, we must not confuse membership in a set with predication of a con-cept, property, or relation. Nor should we wrongly identify the logical conceptof a class, i.e., the concept of a class as the extension of a concept, property orrelation, with the mathematical concept of a set, i.e., a set in the sense of theiterative concept, which is based on Georg Cantor’s power-set theorem that theset of all subsets of any given set always has a greater cardinality than that set.Cantor’s theorem, for example, while essential to the iterative concept of set,will in fact fail in certain special cases of the logical concept of a class—such as,e.g., the universal class, which is the extension of the concept of self-identity.

For this reason we should note that1. a representation of concepts by sets in a set-theoretical semantics will not

always result in the same logical structure as a representation of those conceptsby the classes that are their extensions, and

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2. an incompleteness theorem based on the one kind of structure need notimply an incompleteness theorem based on the other.29

We should distinguish accordingly:• (a) The logical notion of a class (as one or as many) as the extension of

(and therefore having its being in) a concept.• (b) The mathematical iterative notion of a set (which has its being in its

members).

A set-theoretical semantics for a formal theory of predication must not beconfused, in other words, with a semantics for that theory based on its ownforms of predication taken primitively. For the latter is based on the very formsof predication that it is designed to interpret, and it is in that sense an internalsemantics for that theory, while the set-theoretical semantics, being based onthe membership relation of a framework not internal to the theory itself, isan external semantics for that theory. This means that in constructing a set-theoretical semantics for a formal theory of predication we must be cautious notto confuse and literally identify the internal content or mode of significance ofthe forms of predication of that theory with the external model-theoretic contentof the membership relation, or, as in the case of a set-theoretic possible-worldssemantics, with the external content of any function (e.g., on models as set-theoretic representatives of possible worlds) defined in terms of the semanticallyexternal membership relation. If we do not confuse predication with membershipin this way, then we will be able to see why the incompleteness of second-order predicate logic with respect to its standard set-theoretical semantics neednot automatically apply to any formal ontology designed to include second-order logic as part of its formal theory of predication. The careful separationand clarification of these issues is a topic that belongs to the methodology ofcomparative formal ontology.

Distinguish:1. Predication in a formal theory of predication corresponding to a given

theory of universals.2. Membership in a set based on the iterative concept

of set.

Godel’s first incompleteness theorem does show that any formal ontologythat includes arithmetic as part of its pure formal content must be deductivelyincomplete; that is, not every well-formed sentence of the pure logical grammarof such a formal ontology will be such that either it or its negation is provable inthat formal ontology. This does impose a limitation on what can be deductivelyachieved in such a formal ontology, and it requires a modification, if not acomplete revision, of any categorial analysis, such as Husserl’s, where the idealof deductive completeness even for an “infinite multiplicity” such as the systemof natural numbers is taken as an essential part of that analysis.

29See Cocchiarella 1988 and 1992 for a discussion and example of a framework in whichCantor’s theorem fails.

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The deductive incompleteness of an ontology that contains arithmetic is notthe same as the incompleteness of the categorial structure of that ontology, inother words, and in particular it does not show that the formal theory of predi-cation that is part of that structure is incomplete. What must be resolved in aformal ontology that is to contain arithmetic as part of its pure formal content isthe problem of how the possible completeness of its internal content as a formaltheory of predication is to be distinguished from its deductive incompleteness,and how within that pure formal content we are to characterize the content ofarithmetic (and perhaps, more generally, all of classical mathematics as well).30

Finally, in regard to Godel’s second incompleteness theorem what must also beresolved for such a formal ontology is the question of how, and with what sort ofsignificance or content, we are to prove its consistency, since such a proof is notavailable within that formal ontology itself. Again, these are issues that are tobe investigated not so much in a particular formal ontology as in comparativeformal ontology.

1.10 Conceptual Realism

Comparative formal ontology, as our remarks have indicated throughout, isthe proper domain of many issues and disputes in metaphysics, epistemology,and the methodology of the deductive sciences. Just as the construction of aparticular formal ontology lends clarity and precision to our informal categorialanalyses and serves as a guide to our intuitions, so too comparative formalontology can be developed so as to provide clear and precise criteria by whichto judge the adequacy of a particular system of formal ontology and by whichwe might be guided in our comparison and evaluation of different proposalsfor such systems. It is only by constructing and comparing different formalontologies that we can make a rational decision about which such system weshould ourselves ultimately adopt.

Since 1969 I have constructed and compared a number of such systems andhave come to the conclusion that the framework of conceptual realism is theformal ontology that we should adopt. Unlike the a priori approach of thetranscendental method, which claims to be independent of the laws of nature andour evolutionary history, i.e., of our status as biological beings with a culture andhistory that shapes our language and much of our thought, conceptual realismis framed within the context of a naturalistic epistemology and a naturalisticapproach to the relation between language and thought, thought and reality, andour scientific knowledge of the world. The following are some of the features ofconceptual realism that we will cover in this book.

As a conceptualist theory about the mental acts that underlie reference andpredication in language and thought, the categorial analyses of conceptual re-

30We should keep in mind in this context the distinction between a logical analysis of theconcept of a natural number on the one hand, which may be part of a pure formal theory ofpredication, and the axioms, such as infinity, that must be added to the logical backgroundin order to account for the standard laws of arithmetic.

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alism are primarily directed upon the structure of thought. But what guidesus in these analyses is the structure of natural language as a representationalsystem, and in particular as a representational system that is categorially struc-tured and logically oriented. Our methodology, in other words, is based on alinguistic and logical analysis of our speech and mental acts, and not, e.g., on aphenomenological reduction of those acts.

The realism part of conceptual realism, as we will see, contains both a naturalrealism and an intensional realism, each of which can be developed as separatesubsystems, one containing a modern form of Aristotelian essentialism, and theother containing a modern counterpart of Platonism based on the intensionalcontents of our speech and mental acts. We call these two subsystems conceptualnatural realism and conceptual intensional realism.

The realism of conceptual realism contains two subsystems:1. a conceptual natural realism (as a modern form of Aristotelian essential-

ism), and2. a conceptual intensional realism (as a modern counterpart of Platonism).

In addition to the categorial analyses that are directed upon our speech andmental acts, conceptual natural realism also contains a categorial analysis thatis directed upon the structure of reality, and in particular an analysis in whichnatural properties and relations are taken as corresponding to some, but notall, of our predicable concepts, and natural kinds are taken as correspondingto some, but not all, of our sortal common-name concepts. Natural kinds arenot properties in this framework. The category of natural kinds is the realistanalogue of a category of common-name concepts and not of predicable concepts.Common-name concepts are a fundamental part of conceptual realism’s theoryof reference, just as predicable concepts are a fundamental part of conceptualrealism’s theory of predication. Proper as well as common names are part ofthis theory of reference, and together both are described in a separate logic ofnames as another subsystem of conceptual realism. As we will explain later,S. Lesniewski’s ontology, which has also been described as a logic of names, isreducible to our conceptualist logic of names.31

Conceptual intensional realism, as we have developed it, is a logic of nom-inalized predicates and propositional forms as abstract singular terms, i.e., alogic of the abstract nouns and nominal phrases that we use in describing theintensional contents of our speech and mental acts. The intensional objectsthat are denoted by these abstract singular terms serve the same purposes inconceptual intensional realism that abstract objects serve in logical realism asa modern form of Platonism. The difference is that, unlike Platonic Forms, theintensional objects of conceptual realism do not exist independently of mindand the natural world, the way they do in logical realism, but are products ofthe evolution of culture and language, and especially of the institutionalizedlinguistic practice of nominalization.

31See Cocchiarella 2001 for a proof of this reduction. For a description of Lesniewski’sontology, see Slupecki, 1955.

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1.11. SUMMARY AND CONCLUDING REMARKS 23

The way both forms of realism are contained within the general frameworkof conceptual realism shows how a modern form of Aristotelian essentialism iscompatible with an intensional logic that is a counterpart to a modern form ofPlatonism.

1.11 Summary and Concluding Remarks

• Metaphysics consists of the separate disciplines of ontology and cosmology,each with their respective methodologies.

• Formal ontology connects logical categories—especially the categories in-volved in predication—with ontological categories.

• The goal of a formal ontology is the construction of a lingua philosophica,or characteristica universalis , as explicated in terms of an ars combinatoria anda calculus ratiocinator as part of a formal theory of predication.

• A formal ontology should serve as the framework of a characteristica realis ,and hence as the basis of a formal approach to science and cosmology. It shouldalso serve as a framework for our commonsense understanding of the world.

• The central feature of a formal ontology is how it represents the nexus ofpredication, which depends on what theory of universals it assumes.

• The three main theories of universals are nominalism, conceptualism, and(logical or natural) realism.

• The analysis of the fundamental forms of predication of a formal ontologymay be directed upon the structure of reality or upon the structure of thought.

• Natural realism, and in particular Aristotle’s ontology, is directed uponthe structure of the natural world, and the preeminent mode of being is that ofconcrete individual things, or primary substances. There are two major formsof natural realism, moderate realism and modal moderate realism.

• Aristotle’s moderate natural realism has two types of predication: predi-cation of species and genera (natural kinds), and predication of properties andrelations.

• Kant’s and Husserl’s categorial analyses, unlike Aristotle’s, are directedupon the structure of thought and experience rather than upon the structure ofreality. The categories function on this account to articulate the logical formsof judgments and not as the general causes or grounds of concrete being.

• Husserl’s formal ontology is based on a transcendental logic in which thelaws and rules of logic are justified in terms of subjective analyses of presumeda priori structures that provide the evidence for the objective versions of thoselaws and rules.

• There are two problems regarding the completeness of a formal ontology:first, the problem of the completeness of the categories of an ontology, andsecond, the problem of the completeness of the deductive laws that are basedon those categories.

• Set theory provides only an external semantics for a formal ontology, unlessthat ontology is set theory itself, which has no nexus of predication, and hence

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24 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM

strictly speaking is not a formal ontology. An incompleteness theorem for a for-mal ontology based a set-theoretic semantics need not show that the ontology isincomplete with respect to an internal semantics. In particular, sometimes gen-eral models are a better repesentation of a formal ontology’s internal semanticsthan are so-called “standard” models.

• Conceptual realism is a formal ontology framed within the context of anaturalistic epistemology and a naturalistic approach to the relations betweenlanguage, thought, and reality as based on our scientific knowledge of the world.

• Conceptual realism is based on a conceptualist account of the speech andmental acts that underlie reference and predication. It is directed in that regardprimarily upon the structure of thought. But, because its methodology is basedon a linguistic and logical analysis of our speech and mental acts, it is notcommitted to a phenomenological reduction of those acts. Nor does it precludesuch a reduction.

• Conceptual realism contains both a natural realism and an intensional real-ism, each of which can be developed as separate subsystems that are compatiblewithin the larger framework, one containing a modern form of Aristotelian es-sentialism, and the other containing a modern counterpart of Platonism basedon the intensional contents of our speech and mental acts.

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Chapter 2

Time, Being, and Existence

One criterion of adequacy for a formal ontology, we have said, is that it shouldprovide a logically perspicuous representation of our commonsense understand-ing of the world, and not just of our scientific understanding.1 Now a centralfeature of our commonsense understanding is how we are conceptually orientedin time with respect to the past, the present and the future, and the questionarises as to how we can best represent this orientation. It is inappropriate torepresent it in terms of a tenseless idiom of moments or intervals of time of acoordinate system, as is commonly done in scientific theories; for that amountsto replacing our commonsense understanding with a scientific view. A moreappropriate representation is one that respects the form and content of ourcommonsense speech and mental acts about the past, the present and the fu-ture. Formally, this can best be done in terms of a logic of tense operators, orin what is now called tense logic.2

The most natural formal ontology for tense logic is conceptual realism. Thatis because what tense operators represent in conceptual realism are certain cog-nitive schemata regarding our orientation in time and that are fundamental toboth the form and content of our conceptual activity. Thought and communi-cation are inextricably temporal phenomena, and it is the cognitive schemataunderlying our use of tense that structures that phenomena temporally in termsof the past, the present and the future.

A second criterion of adequacy for a formal ontology is that it must explainand provide an ontological ground for the distinction between being and exis-tence, or, if it rejects that distinction why it does so. Put simply, the problemis:

Can there be things that do not exist?Or is being the same as existence?

1This chapter is an extension and deveopment of my article “Quantification, Time, andNecessity,” in Lambert 1991.

2For an excellent book on tense logic and related philosophical issues, see Prior 1967. Othertexts are Gabbay 1976 and Benthem 1983.

25

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26 CHAPTER 2. TIME, BEING, AND EXISTENCE

Now the logic of time in conceptual realism provides the clearest ontologicalground for such a distinction in terms of the tense-logical distinction betweenpast, present and future objects, i.e., the distinction between things that didexist, do exist, or will exist, or what in the proposed book I call realia, asopposed to existentia, which is restricted to the things that exist at the timewe speak or think, i.e., the time we take to be the present of our commonsenseframework. The present, in other words, unlike the tenseless medium of ourscientific theories, is indexical, and refers at any moment of time to that momentitself.

Another criterion of adequacy for a formal ontology is that it must explainthe ontological grounds, or nature, of modality, i.e., of such modal notions asnecessity and possibility, as opposed to merely giving a set-theoretic semanticsfor modal logic. In this chapter we will deal not just with the fundamentalcategories of time, being and existence, but with modality as well. We explainbelow how some of the earliest views of necessity and possibility are groundedin such a framework as tense logic. As we explain below, even the temporal andmodal distinctions of the special theory of relativity theory can be understoodbest within the framework of conceptual realism.

In what follows we will develop separate logics for both possibilism andactualism, and then we will extend these logics to both possibilist and actualistversions of quantified tense logic, where by possible objects we mean only realia,i.e., the things that did, do, or will exist. These logics will serve not only asessential component parts of our larger framework of conceptual realism as aformal ontology, but also as paradigmatic examples of how different parts oraspects of a formal ontology can be developed independently of constructing acomprehensive system all at once. They also illustrate how the model-theoreticmethodology of set theory can be used to guide our intuitions in axiomaticallydeveloping the formal systems we construct as part of a formal ontology.

2.1 Possibilism versus Actualism

The two main parts of metaphysics, we have noted, consists of ontology andcosmology, where ontology is the study of being, and cosmology is the studyof the physical universe, i.e., the world of natural objects and the space-timemanifold in which they exist. If existence is the mode of being of the naturalobjects of the space-time manifold—i.e., of “actual” objects—then the questionarises as to whether or not being is the same as existence, and how this difference,or sameness, is to be represented in formal ontology. We will call the twopositions one can take on this issue possibilism and actualism, respectively.

In possibilism, there are objects that do not now exist but could exist in thephysical universe, and hence being is not the same as existence. In actualismbeing is the same as existence.

Possibilism: There are objects (i.e., objects that have being or) thatpossibly exist but that do not in fact exist.Therefore: Existence �= Being.

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2.1. POSSIBILISM VERSUS ACTUALISM 27

Actualism: Everything that is (has being) exists.Therefore: Existence = Being.

In formal ontology, possibilism is developed as a logic of actual and possibleobjects. Whatever exists in such a logic has being, but it is not necessary thatwhatever has being exists; that is, there can be things that do not exist. Muchdepends, of course, on what is meant here by ‘can’. Does it depend, for example,on the merely possible existence of objects that never in fact exist in the space-time manifold? Or is there a weaker, less committal sense of modality by whichwe can say that there can exist objects that do not now exist? The answeris there are a number of such senses, all explainable in terms of time or thespace-time manifold.

We can explain the difference between being and existence, first of all, interms of the notion of a local time (Eigenzeit) of a world-line of space-time.Within the framework of a possibilist tense logic, for example, being encom-passes past, present, and future objects with respect to such a local time, whileexistence encompasses only those objects that presently exist.3 No doctrine ofmerely possible existence is needed in such a framework to explain the distinc-tion between existence and being. We can interpret modality, in other words, sothat it can be true to say that some things do not exist, namely past and futurethings that do not now exist. In fact, there are potentially infinitely many differ-ent modal logics that can be interpreted within the framework of tense logic. Inthis respect, tense logic provides a paradigmatic framework within which possi-bilism can be given a logically perspicuous representation as a formal ontology.

Tense logic also provides a paradigmatic framework for actualism as well.Instead of possible objects, actualism assumes that there can be vacuous propernames, i.e., proper names that name nothing. Some names, for example, mayhave named something in the past, but now name nothing because those thingsno longer exist; and hence the statement that some things do not exist can betrue in a semantic, metalinguistic sense, i.e., as a statement about the denota-tions, or lack of denotations, of proper names. What is needed, according toactualism, is not that we should distinguish the concept of existence from theconcept of being, but only that we should modify the way that the concept ofexistence (being) is represented in standard first-order predicate logic with iden-tity. On this view, a first-order logic of existence should allow for the possibilitythat some of our singular terms might fail to denote an existent object, which,according to actualism, is only to say that those singular terms denote nothing,rather than that what they denote are objects (beings) that do not exist. Sucha logic for actualism amounts to what today is called a logic free of existentialpresuppositions, or simply free logic.4

The logic of actualism = free logic, i.e. logic free of existential pre-suppositions regarding the denotations of singular terms.

3For some philosophers, e.g., Arthur Prior, being encompasses only past and present ob-jects, apparently because, unlike the past and the present, the future is as yet undetermined.See Prior 1967, chapter viii.

4See Lambert 1991 for a collection of papers on free logic and its philosophical applications.

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28 CHAPTER 2. TIME, BEING, AND EXISTENCE

In what follows we shall first formulate a logic of actual and possible ob-jects in which existence and being are assumed to be distinct concepts that arerepresented by different universal quantifiers ∀e and ∀, respectively, with theexistential quantifiers ∃e and ∃ defined in terms of ∀e and ∀ and negation in theusual way. The free logic of actual objects, where existence is not distinguishedfrom being—but also where it is not assumed that all singular terms denote—isthen described as a certain subsystem of the logic of actual and possible objects.Of course, it is only from the perspective of possibilism that the logic of actualobjects is to be viewed as a proper subsystem of the logic of being, because,according to possibilism, the logic of being includes the logic of actual objects aswell. From the perspective of actualism, the logic of actual objects is all thereis to the logic of being.

Although both the free logic of actualism and the logic of possibilism havetheir most natural applications in tense and modal logic, we will first formulatethese logics without presupposing any such larger encompassing framework. Wewill then describe a framework for tense logic where we distinguish an appli-cation of the logic of actual and possible objects from an application of thefree logic of actual objects simpliciter. After that, we will explain how differentmodal logics can be interpreted in terms of tense logic, and how an applicationof the logic of actual and possible objects in modal logic can be distinguishedfrom an application of the free logic of actual objects simpliciter.5 Tense logic,as these developments will indicate, is a paradigmatic framework in which toformally represent the differences between actualism and possibilism.

2.2 Logics of Actual and Possible Objects

We will initially consider only the first-order logic of actual and possible objects.Later, after we have considered different formal theories of predication, we willextend the logic into a fuller account of being and existence. We turn first tothe syntax of the logic.

As logical constants, we have the following:

1. The negation sign: ¬

2. The (material) conditional sign: →

3. The conjunction sign: ∧

4. The disjunction sign: ∨

5. The biconditional sign: ↔5For an account of the kinds of qualifications that are required in the statement of the laws

involving the interplay of quantifiers, tenses, and modal operators, or what are called de remodalities, see Appendix 1 of this chapter. The tense-logical frameworks for which these lawsare stated provide logically perspicuous representations of the differences between actualismand possibilism, including a restricted version of temporal possibilism where determinate beingincludes only what did or does exist, leaving the future as an indeterminate realm of nonbeing.

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2.2. LOGICS OF ACTUAL AND POSSIBLE OBJECTS 29

6. The identity sign: =

7. The possibilist universal and existential quantifiers: ∀, ∃

8. The actualist universal and existential quantifiers: ∀e, ∃e

When stating axioms, we will assume that ¬, →, =, ∀, and ∀e are the onlyprimitive logical constants, and that the others are defined in terms of these inthe usual way. We do this only for the convenience of not having to deal withtoo many axioms when proving metatheorems.

We take a formal language L to be a set of objectual constants and pred-icates of arbitrary (finite) degrees. We define a formal language in this waybecause we want every formal language to have the same logical grammar andtherefore differ from other formal languages only in the objectual and predicateconstants in that language. Objectual constants are the symbolic counterpartsof proper names in this sort of logic, and n-place predicate constants are thesymbolic counterparts of n-ary relation expressions, with one-place predicateconstants the counterparts of monadic predicates. Whether or not predicateconstants stand for concepts or properties and intensional relations depends onwhat formal theory of predication is assumed in the larger framework, i.e., thesort of framework that we will turn to later in our discussion of formal theoriesof predication. Also, whether the use of objectual constants is the best wayto represent proper names and singular reference is a matter we will turn tolater as well. For now, we note only that this is the standard, modern way torepresent proper names.

• Objectual constants6: symbolic counterparts of proper names.

• n-place predicate constants: symbolic counterparts of n-place predi-cate expressions of natural language, for some natural number n.

• A formal language L: a set of objectual and predicate constants.

The objectual terms, or for brevity, terms, of a formal language L are theobjectual variables and the objectual constants in that language. Atomic for-mulas of L are the identity formulas of L, i.e., formulas of the form a = b, or theresult of concatenating an n-place predicate constant of L with n many singularterms of L.

• The (objectual) terms of L =df {a : a is either an objectual constantin L or an objectual variable}.

• The atomic formulas of L=df {a = b : a, b are terms of L}∪{F (a1, ..., an) :for some natural number n, F is an n-place predicate constant in L anda1, ..., an are terms of L}.

6We use the phrase ‘objectual constant’ and ‘objectual variable’ instead of the more usual‘individual constant’ and ‘individual variable’ so as to accommodate our account in a laterchapter of plural expressions that name plural objects, i.e., objects that are not individualsin the sense of single entities.

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30 CHAPTER 2. TIME, BEING, AND EXISTENCE

We use two quantifiers—though only one style of objectual variable—one forquantification over possible objects, or possibilia, and the other for quantifica-tion over actual objects. The formulas of a language L are those objects thatbelong to every set K containing the atomic formulas of L and such that ¬ϕ,(ϕ→ ψ), (∀xϕ), (∀exϕ) ∈ K whenever ϕ, ψ ∈ K and x is an objectual variable.As indicated, we use Greek letters as variables for expressions of the syntacticalmetalanguage (set theory).

• χ is a (first-order) formula of L if, and only if, for all sets K, if (1)every atomic formulas of L is in K and (2) for all ϕ, ψ ∈ K and objectualvariables x, ¬ϕ, (ϕ→ ψ), (∀xϕ), and (∀exϕ) ∈ K, then χ ∈ K.

Note: The induction principle for formulas follows from this definition.

Induction Principle: If L is a (formal) language, then if:(1) every atomic formula of L ∈ K, and(2) for all ϕ, ψ ∈ K and all objectual variables x, ¬ϕ, (ϕ → ψ), (∀xϕ),and (∀exϕ) ∈ K,then every formula of L ∈ K.

2.3 Set-theoretic Semantics

A set-theoretic semantics for the logic of actual and possible objects is onlya mathematical tool. It does not explain the difference between being andexistence but merely models it mathematically. In this respect it guides ourintuitions about how validity and logical consequence are to be determined inthe logic.

A model for a formal language L is characterized in terms a universe U ofactual objects, a nonempty domain of discourse of possible objects D containingthe universe of actual objects, and an assignment R of extensions drawn fromthe domain of discourse to the objectual and predicate constants in L. Theextension of an objectual constant is what is denoted by that constant, and theextension of an n-place predicate constant F is the set of n-tuples of objects inthe domain that F is understood to be true of in the model.

Definition: A is a model for a formal language L if, and only if, for someU,D,R,(1) A = 〈U,D,R〉,(2) D is a nonempty set,(3) U ⊆ D, and(4) R is a function on L such that for each objectual constant in L, R(a) ∈D, and for each natural number n and each n-place predicate constant Fin L, R(F ) ⊆ Dn, i.e., R(F ) is a set of n-tuples drawn from D.

The assumption that the domain of discourse of possible objects is not emptycan be dropped, and later we will have reasons to do just that; but, insofar as

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2.4. AXIOMS IN POSSIBILIST LOGIC 31

the domain is restricted to concrete objects—i.e., insofar as it does not includeany abstract objects—then, as applied to time, the assumption says only thatsome concrete object exists at some time or other, which seems appropriate;and as applied to possible worlds of concreta, it says only that some concreteobject exist in some world or other, which again seems appropriate.

The notions of satisfaction and truth of a formula of a language L in amodel for L are defined in the usual Tarski manner, except that the satisfactionclause for the actual quantifier applies only to the universe of the model inquestion, whereas the satisfaction clause for the possible quantifier covers theentire domain of discourse, i.e., the set of possibilia of the model. We will notgo into those details here. A formula ϕ is said to be logically true if for somelanguage L of which ϕ is a formula, ϕ is true in every model suited to L.

Definition: ϕ is logically true if, and only if, for some language L, ϕ is aformula of L and ϕ is true in every model A suited to L.

2.4 Axioms in Possibilist Logic

We turn now to an axiomatization of the logic of actual and possible objectsas our first-order description of possibilism. We note as well that a first-orderlogic of actualism is properly contained in this version of possibilism.

Where ϕ, ψ, χ are formulas, x, y are variables, and a, b are (objectual) terms(variables or constants), we take all instances of the following schemas to beaxioms of the logic of actual and possible objects.

(A1) ϕ→ (ψ → ϕ)

(A2) [ϕ→ (ψ → χ)] → [(ϕ→ ψ) → (ϕ→ χ)]

(A3) (¬ϕ→ ¬ψ) → (ψ → ϕ)

(A4) (∀x)[ϕ → ψ] → [(∀x)ϕ → (∀x)ψ]

(A5) (∀ex)[ϕ→ ψ] → [(∀ex)ϕ→ (∀ex)ψ]

(A6) ϕ→ (∀x)ϕ, where x is not free in ϕ

(A7) (∀x)ϕ → (∀ex)ϕ

(A8) (∃x)(a = x), where x is not a

(A9) (∀ex)(∃ey)(x = y) where x, y are distinct variables

(A10) a = b→ (ϕ→ ψ), where ϕ, ψ are atomic formulas and ψis obtained from ϕ by replacing anoccurrence of b by a

As inference rules we assume only modus ponens and the rule of universalgeneralization The turnstile, �, is read as ‘is a theorem of the logic of actualand possible objects’.

MP: If � ϕ→ ψ and � ϕ, then � ψ.

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32 CHAPTER 2. TIME, BEING, AND EXISTENCE

UG: If � ϕ, then � (∀x)ϕ.

By axiom (A7) and (UG), we also have as a derived rule:

UGe: If � ϕ, then � (∀ex)ϕ.

Axioms (1)–(A3), as is well-known, suffice to validate all tautologous formu-las, which we will assume hereafter. Note that where a, b, c are terms (variablesor constants), the transitivity of identity law,

� a = b→ (b = c→ a = c),

is an immediate consequence of axiom (A10). The reflexive law of self-identity,

� a = a,

is a consequence of axioms (A10), (UG), (A4), (A6) (A8) and tautologous trans-formations.7 The symmetry of identity law for identity then follows from (A10),transitivity and reflexivity.8 Finally, note that by a simple induction on formu-las, Leibniz’s law, (LL),

� a = b→ (ϕ↔ ψ), (LL)

where ψ is obtained from ϕ by replacing one or more free occurrences of a byfree occurrences of b, is provable.9 In other words, the full logic of identity,which we will assume hereafter, is contained in this system.

Note that the principle of universal instantiation,

(∀x)ϕ → ϕ(a/x), (UI)

where a can be properly substituted for x in ϕ, is not one of our axiom schemas.10

This is convenient because when extending the logic by adding tense and modaloperators we generally have to revise this principle when taken as an axiom so asto cover the cases when x has de re occurrences in ϕ, i.e., when x occurs within

7That is, where a and y are distinct terms, then by (A10), � a = y → (a = y → a = a),and hence, � a = y → a = a, and therefore � a �= a → a �= y. Thus, by (UG) and(A4), � (∀y)(a �= a) → (∀y)(a �= y),. But by (A6), � a �= a → (∀y)(a �= a), and hence� a �= a → (∀y)(a �= y). That is, by tautology, � ¬(∀y)(a �= y) → a = a. But, by (A8) andthe definition of ∃, � ¬(∀y)(a �= y), from which it follows that � a = a.

8THat is, by (A10), � a = b → (a = a → b = a). But � a = a, so therefore by tautologyand modus ponens, � a = b→ b = a.

9The case for atomic formulas is a consequence of (A10), the symmetry of identity law, andtautologous transformations. The other cases follow by the inductive hypothesis, tautologoustransformations, and finally (A4) and (A5).

10The notion of proper substitution of a term for a variable that is needed for (UI) isalso not involved in any of the axioms; and even the notion of bondage and freedom in (A6)can be replaced by the notion of an occurrence simpliciter. This is very convenient becausethese notions are complex and difficult for students to grasp at first. Also, it is convenientto avoid these notions when using Godel’s arithmetization technique, because they add somuch complexity to that technique. A yet further, and perhaps even more important reasonis noted above.

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2.4. AXIOMS IN POSSIBILIST LOGIC 33

the scope of a tense or modal operator—or any other intensional operator. Therestrictions needed in each case can be determined by seeing what is needed inthe proof by induction on Leibniz’s law, (LL), from which this principle follows.That is, by (LL),

� a = x→ [ϕ→ ϕ(a/x)],

where ϕ(a/x) is the result of replacing all free occurrences of x by a (which weassume to be distinct from x). Then, by tautologous transformations, (UG),(A4) and (A6), we have

� (∃x)(a = x) → [(∀x)ϕ→ ϕ(a/x)],

from which, by axiom (A8) and modus ponens, (UI) follows. The law forexistential generalization

,ϕ(a/x) → (∃x)ϕ, (EG)

is of course the converse of (UI) and therefore provable on its basis.Axiom (A8) is the important axiom here. One should not think that it is

redundant because it is provable by (EG) from the law of self-identity. Ofcourse,

a = a→ (∃x)(a = x)

is an an instance of (EG); but (EG) is provable from (UI), and, as noted,in the proof of (UI) we need (A8), i.e., (∃x)(a = x). So it would be circularreasoning to try to prove the latter in terms of (EG).

What axiom (A8) says in effect is that every objectual term denotes a pos-sible object (as a value of the bound objectual variables), even if that objectdoes not exist (as a value of the variables bound by the actualist quantifier,∀e). That is not a problem if the logic does not introduce complex objectualterms, i.e., objectual terms that might contain formulas that describe impos-sible situations (such as ‘the round square’), or if the logic is not extended toa situation where certain complex objectual terms must fail to denote on painotherwise of resulting in a contradiction. (This is what in fact happens, as wewill see in the next chapter when we extend the logic to second-order predicatelogic with nominalized predicates as abstract objectual terms.) It is this lattersituation that we will later be concerned with, in which case we will then haveto replace axiom (A8) by (∀x)(∃y)(x = y), which is the possibilist counterpartof the actualist axiom (A9).

Finally, we note that If we restrict ourselves to formulas in which the actu-alist quantifier does not occur, then axioms (A1)-(A4), (A6), (A8), and (A10)yield all and only the standard logical truths of first-order predicate logic.11

The standard logical truths, moreover, are none other than the logical truthsas defined above when formulas are restricted to those in which the actualistquantifier does not occur. What these results show, in other words, is that the

11The completeness of the system as described above is due to D. Kalish and R. Montague1965, their result being obtained by a modification of an original formulation by A. Tarski.

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34 CHAPTER 2. TIME, BEING, AND EXISTENCE

logic of possible objects is none other than standard first-order predicate logicwith identity. Hereafter we will assume all the known results about this logic.

The main result of this section is that a formula is a theorem of the logic ofactual and possible objects if, and only if, it is logically true.

Metatheorem: For all formulas ϕ, � ϕ if, and only if, ϕ is logically true.

2.5 A First-order Actualist Logic

Essentially the same proof that we described above for (UI) applies to theprinciple of universal instantiation for the actualist quantifier, except that nowthe antecedent condition in one step of that proof is not an axiom. That is, byLeibniz’s law, (UG), (A5), the counterpart of (A6) for ∀e (which is derivablefrom (A6) and (A7)), we have

(∃ey)(a = y) → [(∀ex)ϕ→ ϕ(a/x)], (UIe)

which is the actualist version of universal instantiation (where a and y aredistinct terms). We cannot assume the antecedent here unless we are given asa separate premise that the objectual term a denotes an actual, existent, object(as a value of the variable bound by ∃e). Note that in addition to the quantifierconcept of existence represented by ∀e (and its dual ∃e), the predicable conceptof existence can be defined in this first-order logic as follows (where a is distinctfrom the variable y):

E!(a) =df (∃ey)(a = x).

Now by an E-formula, let us understand a formula in which the possiblequantifier, ∀, does not occur.

Definition: ϕ is an E-formula =df ϕ is a formula in which the possibilistquantifier, ∀, does not occur.

Note that if we restrict ourselves to E-formulas, then neither

(A6e) ϕ→ (∀ex)ϕ, where x is not free in ϕ,

nor

(A8e) a = a, where a is a objectual term

are provable. That is, the proofs of these valid formulas depend on severalpossibilist axioms, namely, (A6), (A7) and (A8). That means that if we wantto consider only actualism, and not the full logic of possible and actual objects,then we need to take both of these schemas as axioms of the logic of actualism.Indeed, it can be shown that axioms (Al)-(A3), (A5), (A9), (A10), together withthese schemas, (A6e) and (A8e), yield all and only those logical truths that areE-formulas.12

12See Cocchiarella 1966.

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2.6. TENSE LOGIC 35

Metatheorem: All logical truths that are E-formulas are derivable from ax-iom schemas (Al)-(A3), (A5), (A9), (A10), (A6e) and (A8e) with modusponens and (UGe) as the only inference rules.

Thus, whereas

1. the standard formulas that are logically true constitute the logic of possibleobjects simpliciter,

2. the E-formulas that are logically true constitute the logic of actual objectssimpliciter.

3. All of the formulas together—i.e., the standard formulas, the E-formulas,and the formulas that contain both the possible and the actual quantifiers—that are logically true constitute the logic of actual and possible objects,which can be shown to be complete for the semantics described above.

2.6 Tense Logic

As already indicated, one of the most natural applications of the logic of actualand possible objects is in tense logic, where existence applies only to the thingsthat presently exist, and possible objects are none other than past, present, orfuture objects, i.e., objects that either did exist, do exist, or will exist. The mostnatural formal ontology for tense logic is conceptual realism. This is becauseas forms of conceptual activity, thought and communication are inextricablytemporal phenomena, and to ignore this fact in the construction of a formalontology is to court possible confusion of the Platonic with the conceptual viewof intensionality.

Propositions on the conceptualist view, for example, are not abstract entitiesexisting in a platonic realm independently of all conceptual activity. Rather,according to conceptual realism, they are conceptual constructs correspondingto a projection on the level of objects of the truth-conditions of our temporallylocated assertions. However, on our present level of analysis, where propositionalattitudes are not being considered, their status as constructs can be temporarilyignored.

What is also a construction, but which, should not be ignored in a concep-tualist framework, are certain cognitive schemata characterizing our conceptualorientation in time and implicit in the form and content of our assertions asmental acts. These schemata, whether explicitly recognized as such or not, areusually represented or modelled in terms of a tenseless idiom (such as our set-theoretic metalanguage) in which reference can be made to moments or intervalsof time (as objects of a special type). Of course, for most scientific purposessuch a representation is quite in order. But to represent them only in this wayin a context where our concern is with a perspicuous representation of the formof our assertions as speech or mental acts might well mislead us into thinkingthat the schemata in question are not essential to the form and content of an

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36 CHAPTER 2. TIME, BEING, AND EXISTENCE

assertion after all—the way they are not essential to the form and content of aproposition on the Platonic view.

Now it is important to note that even though the cognitive schemata inquestion can be modelled in terms of a tenseless idiom of moments or intervalsof time, as in fact they are in our set-theoretic metalanguage, they are reallythemselves the conceptually prior conditions that lead to the construction ofour referential concepts for moments or intervals of time. In other words, interms of conceptual priority, these cognitive schemata are implicitly presup-posed by the same tenseless idiom in which they are set-theoretically modelled.In this regard, despite the use of moments or intervals of time in the semanticclauses of the metalanguage, there is no need for conceptualism to assume thatmoments or intervals of time are independently existing objects as opposed tobeing merely constructions out of the different events that actually occur innature, constructions that can be given within a tense-logically based language.

Now because the temporal schemata implicit in our assertions enable us toorientate ourselves in time in terms of the distinction between the past, thepresent, and the future, a more appropriate or perspicuous representation ofthese schemata is one based upon a system of quantified tense logic in whichtenses are represented by tense operators. As applied in thought and commu-nication, what these operators correspond to is our ability to refer to what wasthe case, what is the case, and what will be the case—and to do so, moreover,without having first to construct referential concepts for moments or intervalsof time. In the simplest case we have operators only for the past and the future.

P it was the case that ...F it will be the case that ...

We do not need an operator for the simple present tense, ‘it is the case that’,because it is already represented in the simple indicative mood of our predicates.With negation applied both before and after a tense operator, we can shortenthe long reading of ¬P¬ , namely, ‘it was not the case that it was the case thatit was not the case’ to simply ‘it was always the case’. A similar shorter readingapplies to ¬F¬ as well. In other words,. we also have the following readings:

¬P¬ it always was the case that ...¬F¬ it always will be the case that ...

Tensed formulas are defined inductively as follows.

Definition: ϕ is a tensed formula of a language L if, and only if, ϕ isin every set K such that (1) every atomic formula of L is in K, and (2)whenever ϕ, ψ ∈ K and x is a variable, then ¬ϕ, (ϕ→ ψ), Pϕ, Fϕ, (∀x)ϕ,(∀ex)ϕ ∈ K.

We will avoid going into all of the details here of a set-theoretical semanticsfor tense logic.13 Briefly, the idea is that we consider the earlier-than relationof a local time (Eigenzeit) of a world-line in space-time. Though it is natural to

13For such details see Cochiarella 1966 or 1974.

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2.6. TENSE LOGIC 37

assume that this relation is a serial ordering, i.e., that it is transitive, asymmetricand connected, we initially impose no other constraints at all upon it, and thenconsider validity with respect to the different kinds of structures that it canhave, e.g., that it is discrete, dense, or continuous, has a beginning and end,or neither, etc. What is important is that we distinguish the objects that existat each moment of a local time from the objects that exist at any of the othermoments of that local time. In this respect a local time determines a history ofthe world from its unique point of point . Formally, this can stated as follows.

Definition: If L is a (formal) language and A is a model (as defined earlier)for a language L, then(1) UA = the universe (of existing objects) of A, and(2) DA = the set of possibilia of A.

Definition: If L is a language and R is a serial relation, then B is an R-history with respect to L if there are a nonempty index set I includedin the field of R and an I-termed sequence A of models suited to L suchthat(i) B = 〈R,Ai∈I〉;(ii) I is identical with the field of R if I has more than one element; and(iii) ∪j∈IUAj ⊆ DAi , for all i ∈ I; and(iv) DAi = DAj , for all i, j ∈ I.

Where 〈R,Ai∈I〉 is such a history, we take the members of the set I to be themoments of the local time with respect to which the history is determined andR to be the earlier-than relation ordering those moments. The structure of Ris the temporal structure of that history. It may, for example, have a beginning,or an end, both, or neither, and it may be discrete, dense, or continuous, and soon. Condition (iii) stipulates that whatever is actual at one time or another in ahistory is a possible object of that history. Condition (iv) states the requirementthat whatever is a possible object at one moment of a history is a possible objectat any other moment of that history. A complete description of the world relativeto the language L and a moment i of a history 〈R,Ai∈I〉 is given by the modelAi that is associated with i in that history.

Except where tense operators are involved, satisfaction and truth in a history〈R,Ai∈I〉 at a given moment i of the history is understood as satisfaction andtruth in the model Ai. The satisfaction clauses for the tense operators havethe obvious references to the models associated with the moments before andafter the moment i. Validity in a history is defined as truth at all times in thathistory. If R is a relation, then ϕ is said to be R-valid if ϕ is a tensed formulaof some language L such that for each R-history B with respect to L, ϕ is validin B. A tensed schematic formula ϕ is understood to characterize a class K ofrelations if for each relation R, ϕ is R-valid if, and only if, R ∈ K.14

14See Cocchiarella 1966 and 1974 for the details of this semantics.

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38 CHAPTER 2. TIME, BEING, AND EXISTENCE

Definition: If ϕ is a tensed formula of a language L and B =〈R,Ai∈I〉 is anhistory, then:(1) ϕ is valid in B if, and only if, for all i ∈ I, ϕ is true at i in B;(2) ϕ is R-valid if, and only if, for each R-history B′, ϕ is valid in B′;and(3) ϕ characterizes a class K of relations if, and only if, for eachrelation R, ϕ is R-valid if, and only if, R ∈ K.

Special schematic formulas can be shown to characterize various classes ofrelations. For example, the tensed formulas

PPϕ→ Pϕ

FFϕ→ Fϕ

characterize the class of transitive relations, whereas the converse schemas

Pϕ→ PPϕ

Fϕ→ FFϕ

characterize the class of dense relations, i.e., where between any two momentsof time there is always another moment of time.15

The most natural assumption, we have said, is that the earlier-than relationof a local time (Eigenzeit) of a world-line is a serial ordering, i.e., that therelation is transitive, asymmetric and connected. This is not just a matter of“logical purity,” as Arthur Prior has suggested, but of how we conceive thestructure of the earlier-than relation of a local time.16 The “logical purity”assumed in this regard is an abstraction from all features of the structure ofthe earlier-than relation of a local time other than its seriality. That is, thestructure is assumed to be invariant for all serial orderings regardless whethertime has a beginning, end, or neither, and whether time is discrete, dense andcontinuous or not. Prior suggested, however, that “if we really want to be safe,it’s odd to begin by insisting on linearity [i.e., seriality], and it might be better... to confine one’s ‘basic’ laws to those which put no special assumptions on theearlier-than relation at all,” including, e.g., the condition of transitivity as wellas that of connectedness.17 This kind of purity abstracts beyond the fact thatit is the structure of the earlier-than relation of a local time and characterizesinstead the structure of a binary relation simpliciter. That much abstractionmight be an appropriate characterization of the accessibility relation betweenpossible worlds in modal logic, but it abstracts too much insofar as it is thestructure of the earlier-than relation of a local time that is in question.

15The formulas¬F¬ϕ → Fϕ¬P¬ϕ → Pϕ

characterize time as having no end, or beginning, respectively.16See Prior 1967, p. 51, for a discussion of this issue.17Ibid.

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2.6. TENSE LOGIC 39

Time, as well as causality, is a cosmological category, which, for some, mightindicate that “tense logic is not really logic but physics, or that it has a gooddeal of physics ‘built into it’,” as Prior has noted.18 That in fact may be trueto some extent, because insofar as such cosmological categories as time andcausality can be said to have a logic of their own they are to that extent alsopart of formal ontology. In any case, it is clear that we are not dealing with thestructure of time if the earlier-than relation is not assumed to be transitive.

The issue of the connectedness of the earlier-than relation of a local timemight be thought to be another matter, however, especially in the context ofEinstein’s theory of special relativity. But, as we will argue in section 11 below,the claim that Einstein’s theory of special relativity shows that connectednessdoes not apply is based on a confusion of:

1. the causal signal relation, which is not connected, between the differentmomentary states of different world lines with

2. the earlier-than relation of a local time (Eigenzeit) of a given worldline, which is connected.

In any case, in regard to the characterization of logical truth as extended toall tensed formulas, we restrict our considerations—in deference to this funda-mental feature of (local) time—to serial histories, i.e., histories whose temporalordering is a series.

Definition: ϕ is tense-logically true if for some language L of which ϕ is atensed formula, ϕ is valid in every serial history suited to L.

Metatheorem: ϕ is tense-logically true if, and only if, for every serial orderingR, ϕ is R-valid.

Given modus ponens and universal generalization for ∀, and the followingas inference rules

(i) if �t ϕ, then �t ¬P¬ϕ(ii) if �t ϕ, then �t ¬F¬ϕ

then these rules together with all instances of (A1)-(A10) of the logic of actualand possible objects (applied now to tensed formulas) and all instances of thefollowing axiom schemas as well yield all and only the tense-logical truths:

(A11) ¬P¬(ϕ→ ψ) → (Pϕ→ Pψ)(A12) ¬F¬(ϕ→ ψ) → (Fϕ→ Fψ)(A13) ϕ→ ¬P¬Fϕ(A14) ϕ→ ¬F¬Pϕ(A15) PPϕ→ Pϕ(A16) FFϕ→ Fϕ

18Ibid.

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40 CHAPTER 2. TIME, BEING, AND EXISTENCE

(A17) Pϕ ∧ Pψ → P(ϕ ∧ ψ) ∨ P(ϕ ∧ Pψ) ∨ P(ψ ∧ Pϕ)

(A18) Fϕ ∧ Fψ → F(ϕ ∧ ψ) ∨ F(ϕ ∧ Fψ) ∨ F(ψ ∧ Fϕ)

(A19) x = y → ¬P¬(x=y)∧¬F¬(x=y) where x, y are variables

Metatheorem: For all tensed formulas ϕ, �t ϕ if, and only if, ϕ is tense-logically true.19

A completeness theorem for actualist tensed logic is also forthcoming if werestrict ourselves to tensed E-formulas, i.e., those formulas in which the possiblequantifier does not occur. Restricted to tensed E-formulas, the axioms foractualism are as follows:

Tensed Actualist axioms: (A1)-(A3), (A5), (A6e), (A8e), (A9), and (A10)-(A19).

We will also need the inference rules listed above—but with universal general-ization for ∀e instead of ∀—as well as the following (somewhat complex) rules:

(iii) If �t ¬P¬(ϕ1 → ¬P¬(ϕ1 → ...→ ¬P¬(ϕn−1 → ¬P¬ϕn)...)),

and x is not free in ϕ1, ..., ϕn−1, then

�t ¬P¬(ϕ1 → ¬P¬(ϕ1 → ...→ ¬P¬(ϕn−1 → ¬P¬(∀ex)ϕn)...)).

(iv) If �t ¬F¬(ϕ1 → ¬F¬(ϕ1 → ...→ ¬F¬(ϕn−1 → ¬F¬ϕn)...)),

and x is not free in ϕ1, ..., ϕn−1, then

�t ¬F¬(ϕ1 → ¬F¬(ϕ1 → ...→ ¬F¬(ϕn−1 → ¬F¬(∀ex)ϕn)...)).

The above axioms and rules yield all and only those tense-logical truths thatare tensed E-formulas.20

2.7 Temporal Modes of Being

In assuming that being and existence are not the same concept, possibilism doesnot also assume that whatever is (i.e., whatever has being) either did exist, doesexist, or will exist, a thesis we shall call temporal possibilism. We will call theobjects of temporal possibilism realia. Formally, this thesis is stated as follows:

Temporal Possibilism: (∀x)[PE!(x) ∨ E!(x) ∨ FE!(x)].

Realia: What did, does, or will exist.

If we add this formula as a new axiom, then to render it tense-logically true weneed only require that the condition stated in clause (iii) of the definition of anR-history be an identity rather than just an inclusion.

19See Cocchiarella 1966 for a proof of this metatheorem.20See Cocchiarella 1966.

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2.7. TEMPORAL MODES OF BEING 41

It should be noted that Aristotle seems to have held such a view in that hethought that whatever is possible is realizable in time, which, for Aristotle, hasno beginning or end.

A more restrictive view than temporal possibilism—but one that still fallsshort of actualism—is that only the past and the present are metaphysically de-terminate, and for that reason only objects that either do exist or did exist havebeing. The future, being indeterminate metaphysically as well as epistemicallyhas no being. Being, on this account, covers only past or present existence.Future objects have no being but only come into being in the present when theyexist, and then continue to have being in the past. We can characterize this po-sition by first defining quantification over past objects, and then quantificationover past and present objects, as follows (where ∃p and ∃p

p are defined in theusual way as the duals of ∀p and ∀p

p, respectively):

(∀px)ϕ =df (∀x)[PE!(x) → ϕ]

(∀ppx)ϕ =df (∀x)[PE!(x) ∨ E!(x) → ϕ]

The metaphysical thesis that being comprises only what either did exist ordoes exist can now be expressed as follows:

(∀x)(∃ppy)(x = y).

Alternatively, instead of having the concept of being in such a framework repre-sented by the possibilist quantifier ∀, we can take it to be represented directlyby ∀p

p as a primitive quantifier together with ∀e for the concept of existence.A sound and complete axiom set for this system is then given by (A1)-(A3),

(A5), (A9), (A10), together with the schemas:

(∀ppx)(ϕ → ψ) → [(∀p

px)ϕ→ (∀ppx)ψ],

ϕ→ (∀ppx)ϕ, where x is not free in ϕ,

(∀ppx)ϕ → (∀ex)ϕ,

(∀ppx)(∃p

py)(x = y), where x, y are distinct variables,

(∀ppx)[(∃p

py)(x = y) ∧ ¬F¬(∃ppy)(x = y)],

(∀ppx)¬P¬ϕ → ¬P¬(∀p

px)ϕ,

¬F¬(∀ppx)ϕ→ (∀p

px)¬F¬ϕ,a = a, where a is an arbitrary term.

The inference rules for this system are modus ponens, universal gener-alization for ∀p

p, rules (i), (ii) as described earlier, and the counterpart of rule(iv) using ∀p

p in place of ∀e.21

21The counterpart of rule (iii) with ∀pp in place of ∀e is provable, and it does not need to be

taken as a primitive rule in this system.

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42 CHAPTER 2. TIME, BEING, AND EXISTENCE

2.8 Past and Future Objects

Actualists claim that quantificational reference to either past or future objectsis possible only indirectly (de dicto)—i.e., through the occurrence of an actualistquantifier within the scope of a tense operator.22 This is true of some of ourreferences to past objects, as for example in an assertion of,

Someone did exist who was a King of France.

In this case, the apparent reference to a past object can be accounted for asfollows:

P(∃ex)King-of -France(x),

where the reference to a past object is not direct but indirect, i.e., within thescope of a past-tense operator. Here, by a direct quantificational reference to apast object that no longer exists (or a future one that has yet to exist) we meanone in which the quantifier is outside the scope of a tense operator.

Note: What is apparently is not possible on this account about a direct quan-tificational reference to past objects that no longer exist is our presentinability to actually confront and apply the relevant identity criteria toobjects that do not now exist.

A present ability to identify past or future objects, however, is not the sameas the ability to actually confront and identify those objects in the present; thatis, our existential inability to do the latter is not the same as, and should notbe confused with, what is only presumed to be our inability to directly referto past or future objects. Indeed, the fact is that we can and do make directreference to realia, and to past and future objects in particular, and that we doso not only in ordinary discourse but also, and especially, in most if not all ofour scientific theories. The real problem is not that we cannot directly refer topast and future objects, but rather how it is that conceptually we come to doso.

One explanation of how this comes to be can be seen in the analysis of thefollowing English sentences:

1. There did exist someone who is an ancestor of everyone now existing.

2. There will exist someone who will have everyone now existing as an an-cestor.

Assuming, for simplicity, that we are quantifying only over persons, it isclear that (1) and (2) cannot be represented by:

3. P(∃ex)(∀y)Ancestor-of(x, y)

4. F(∃ex)(∀y)Ancestor-of(y, x).22Cf. Prior, 1967, Chapter 8.

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2.8. PAST AND FUTURE OBJECTS 43

What (3) and (4) represent are the different sentences:

5. There did exist someone who was an ancestor of everyone then existing.

6. There will exist someone who will have everyone then existing as an an-cestor.

Of course, in temporal possibilism, referential concepts are available thatenable us to refer directly to past and future objects. Thus, for quantificationover past objects we have the quantifier ∃p and for quantification over futureobjects we have ∃f as the future-counterpart of ∃p. Using these quantifiers, theobvious representation of (1) and (2) is:

7. (∃px)(∀y)Ancestor-of(x, y)

8. (∃fx)(∀y)FAncestor-of(y, x).

We should note here that the relational ancestor concept is such that:

x is an ancestor of y only at those times when either y exists andx did exist, though x need not still exist at the time in question, orwhen x has continued to exist even though y has ceased to exist.

When y no longer exists as well as x, we say that x was anancestor of y; and where y has yet to exist, we say that x will be anancestor of y.

Now although these last analyses are not available in actualist tense logic,nevertheless semantical equivalences for them are available once we allow us theuse of the now-operator , N .

N : It is now the case that ...The now-operator is unlike the simple present tense in that it always brings

us back to the present even when it occurs within the scope of either the past-or future-tense operators. Thus, although the indirect references to past andfuture objects in (3) and (4) fail to provide adequate representations of (1)and (2), the same indirect references followed by the now-operator succeed incapturing the direct references given in (7) and (8):

9. P(∃x)N (∀y)Ancestor-of(x, y)

10. F(∃x)N (∀y)FAncestor-of(y, x).

In other words, at least relative to any present-tense context, we can in generalaccount for direct reference to past and future objects, and hence to all of theobjects of temporal possibilism, as follows:

(∀px)ϕ↔ ¬P¬(∀ex)Nϕ

(∀fx)ϕ↔ ¬F¬(∀ex)Nϕ

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44 CHAPTER 2. TIME, BEING, AND EXISTENCE

(∀x)ϕ ↔ (∀px)ϕ ∧ ϕ ∧ (∀fx)ϕ.

These equivalences, it should be noted, cannot be used other than in apresent tense context; that is, the above use of the now-operator would beinappropriate when the equivalences are stated within the scope of a past- orfuture-tense operator, because in that case the direct reference to past or futureobjects would be from a point of time other than the present. Formally, whatis needed in such a case is the introduction of a so-called “backwards-looking”operator, such as the then-operator, which can be correlated with occurrences ofpast or future tense operators within whose scope they lie and that semanticallyevaluate the formulas to which they are themselves applied in terms of the pastor future times already referred to by the tense operators they are correlatedwith23. Backwards-looking operators, in other words, enable us to conceptuallyreturn to a past or future time already referred to in a given context in the sameway that the now-operator enables us to return to the present. In that regard,their role in the cognitive schemata characterizing our conceptual orientation intime and implicit in each of our assertions is essentially a projection of the roleof the now-operator.

We will not formulate the semantics of these backwards-looking operatorshere. But we do want to note that by means of such operators we can accountfor the development of referential concepts by which we can refer directly topast or future objects. Such an account is already implicit in the fact thatsuch direct references to past or future objects can be made with respect tothe present alone. This shows that whereas the reference is direct at least ineffect, nevertheless the application of any identity criteria associated with suchreference will itself be indirect, and in particular, not such as to require a presentconfrontation, even if only in principle, with a past or future object.

2.9 Modality Within Tense Logic

It is significant that the first modal concepts to be discussed and analyzed in thehistory of philosophy are concepts based on the distinction between the past,the present, and the future, that is, concepts that can be analyzed in terms ofthe temporal modalities that are represented by the standard tense operators.The Megaric logician Diodorus, for example, is reported as having argued thatthe possible is that which either is or will be the case, and that the necessary isthat which is and always will be the case.24 Formally, the Diodorean modalitiescan be defined as follows:

♦fϕ =df (ϕ ∨ Fϕ)

�fϕ =df ϕ ∧ ¬F¬ϕ

∴ �fϕ↔ ¬♦f¬ϕ23Cf. Vlach,1973 and Saarinen 1976.24See Prior 1967, chapter 2, for a discussion of Diodorus’s argument.

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2.9. MODALITY WITHIN TENSE LOGIC 45

Aristotle, on the other hand, included the past as part of what is possible;that is, for Aristotle the possible is that which either was, is, or will be the casein what he assumed to be the infinity of time, and therefore the necessary iswhat is always the case25:

♦tϕ =df Pϕ ∨ ϕ ∨ Fϕ

�tϕ =df ¬P¬ϕ ∧ ϕ ∧ ¬F¬ϕ

∴ �tϕ↔ ¬♦t¬ϕ

Both Aristotle and Diodorus assumed that time is real and not ideal. Inother words, the Diodorean and Aristotelian temporal modalities are under-stood to be real modalities based on the nature of time. In fact they provide aparadigm by which we might understand what is meant by a real, as opposedto a merely formal, modality such as logical necessity. These temporally-basedmodalities contain an explanatory, concrete interpretation of what is called theaccessibility relation between possible worlds in modal logic, except that worldsare now construed as momentary states of the universe as described by the mod-els associated with the moments of a local time. That is, where possible worldsare momentary descriptive states (models) of the universe with respect to thelocal time (Eigenzeit) of a given world-line, then the relation of accessibilitybetween worlds is ontologically grounded in terms of the earlier-than relation ofthat local time.

The Aristotelian modalities are stronger than the Diodorean, of course, andin fact they provide a complete semantics for the quantified modal logic knownas S5.

Definition: If L is a language, then ϕ is an S5t-formula of L if, and only if,ϕ belongs to every set K containing the atomic formulas of L and suchthat ¬ϕ, �tϕ, ♦tϕ, (ϕ→ ψ), (∀x)ϕ, (∀ex)ϕ ∈ K whenever ϕ, ψ ∈ K andx is a variable.

Definition: ϕ is S5-valid if, and only if, ϕ is an S5t-formula that is tense-logically true.

We obtain the system we call S5t if to the axioms (A1)-(A10) of the logicof actual and possible objects we add all instances of schemas of the followingforms:

(S5t-1) �tϕ→ ϕ

(S5t-2) �t(ϕ→ ψ) → (�tϕ→ �tψ)(S5t-3) ♦tϕ→ �t♦tϕ

(S5t-4) (x = y) → �t(x = y), where x, y are variables.

25See Hintikka, 1973, Chapters V and IX. Aristotle may have intended his notion of possibleto apply to individuals as well, a position that is validated in the quantified modal logicdescribed in this section.

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46 CHAPTER 2. TIME, BEING, AND EXISTENCE

As inference rules we take in addition to modus ponens and universal general-ization for ∀ the following:

(MG) if �S5t ϕ, then �S5t �tϕ .

Note that by (S5t-4), (A8) and (MG),

(∃y)�t(x = y)

is provable, and from this it can be shown that the Carnap-Barcan formula,

(∀x)�tϕ↔ �t(∀x)ϕ

is also provable. A completeness theorem for this version of quantified S5 modallogic can be proved in the usual way.

Metatheorem: For each S5t-formula ϕ, �S5t ϕ if, and only if, ϕ is S5-valid.

For an actualist S5 modal logic, we need only restrict the S5-formulas tothose that are tensed E-formulas—i.e., tensed formulas in which the possibilistquantifier does not occur—and use only the logic of actual objects as describedearlier together with the axiom schemas (S5t-1)-(S5t-4) and one new inferencerule added to those of S5t. That is, where S5t

e is that subsystem of S5t that isthe result of replacing (Al)-(A10) of the logic of actual and possible objects bythe axioms for the logic of actual objects simpliciter and adding to the inferencerules of S5t the following:

If �S5te

�t(ϕ1 → �t(ϕ2 → ...→ �t(ϕn−1 → �tϕn)...)),and x is not free in ϕ1, ..., ϕn−1, then�S5t

e�t(ϕ1 → �t(ϕ2 → ...→ �t(ϕn−1 → �t(∀ex)ϕn)...)).

A completeness theorem can be shown for the actualist modal logic S5te.

Metatheorem: For each S5t-formula ϕ that is also an E-formula, �S5teϕ if,

and only if, ϕ is S5-valid.

The Diodorean modalities, we have noted, are weaker than the Aristotelianmodalities, and the corresponding quantified modal logic is not S5 but theweaker system known as S4.3.

Definition: If L is a language, then ϕ is an S4.3t-formula of L if, and onlyif, ϕ belongs to every set K containing the atomic formulas of L and suchthat ¬ϕ, ♦fϕ, (ϕ → ψ), (∀x)ϕ, (∀ex)ϕ ∈ K whenever ϕ, ψ ∈ K and x isa variable.

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2.9. MODALITY WITHIN TENSE LOGIC 47

Definition: ϕ is S4.3-valid if, and only if, ϕ is an S4.3t-formula that is tense-logically true.

We obtain the system we call S4.3t if we add to the axioms (A1)-(A10) ofthe logic of actual and possible objects all instances of schemas of the followingforms:

(S4.3t-1) �fϕ→ ϕ

(S4.3t-2) �f(ϕ→ ψ) → (�fϕ→ �fψ)(S4.3t-3) �fϕ→ �f�fϕ

(S4.3t-4) ♦fϕ ∧ ♦fψ → ♦f (ϕ ∧ ψ) ∨ ♦f (ϕ ∧ ♦fψ) ∨ ♦f (ψ ∧ ♦fϕ)(S4.3t-5) ♦f (x = y) → �f (x = y)(S4.3t-6) (∀x)�fϕ→ �f(∀x)ϕ

As inference rules for S4.3t we have the same inference as those for S5t, exceptexcept for having �f where �t occurs in those rules. It can be shown that foreach S4.3t-formula ϕ, ϕ is a theorem of S4.3t if, and only if, ϕ is S4.3-valid,which is our completeness theorem for S4.3t.

Metatheorem: For each S4.3t-formula ϕ, �S4.3t ϕ if, and only if, ϕ is S4.3-valid.

For an actualist S4.3t modal logic we need first to restrict the tensed S4.3t-formulas to tensed E-formulas. Then, to obtain the subsystem S4.3t

e of S4.3t

when the latter is restricted to E-formulas, we must first delete the axiomschema (S4.3t-6), which is not an E-formula, and replace (A1)-(A10) of thelogic of actual and possible objects by the axioms of the logic of actual objectssimpliciter. We then adopt the same modal inference rules as already describedfor S5t

e, except for using �f instead of �t in those rules. Then, it can beshown that for each S4.3-formula ϕ that is also an E-formula, ϕ is a theoremof S4.3t

e if, and only if, ϕ is S4.3-valid, which is our completeness theorem forS4.3t-formulas when the latter are restricted to E-formulas.

Metatheorem: For each S4.3t-formula ϕ that is also an E-formula, �S4.3teϕ

if, and only if, ϕ is S4.3-valid.

Infinitely many other modal logics can be generated in ways similar to theabove by various combination of tenses—e.g., merely iterating new occurrencesof F in the definition of the Diodorean modalities will lead to new modalities.In addition to these temporal notions of modality, the semantics for yet anothercan be given corresponding roughly to the idea that a formula is conditionallynecessary (in a given history at a given moment of that history) because of theway the past has been. The semantics for this notion also yields a completenesstheorem for an S5 type modal structure, and it may be used for a partial or fullexplication of the notions of causal modality and counterfactuals.

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48 CHAPTER 2. TIME, BEING, AND EXISTENCE

2.10 Causal Tenses in Relativity Theory

One of the defects of Aristotle’s notion of necessity as a paradigm of a temporally-based modality is its exclusion of certain situations that are possible in specialrelativity theory as a result of the finite limiting velocity of causal influences,such as a light signal moving from one point of space-time to another. For ex-ample, relative to the present of a given local time T0, a state of affairs can cometo have been the case, according to special relativity, without its ever actuallybeing the case.26 That is, where FPϕ represents ϕ’s coming (in the future) tohave been the case (in the past of that future), and ¬♦tϕ represents ϕ’s neveractually being the case, the situation envisaged in special relativity might bethought to be represented by:

FPϕ ∧ ¬♦tϕ. (Rel)

This conjunction is incompatible with the connectedness assumption of thelocal time T0 in question; for on the basis of the assumption of connectedness,

FPϕ→ Pϕ ∨ ϕ ∨ Fϕ

is tense-logically true, and therefore, by definition of ♦t,

FPϕ→ ♦tϕ

is also tense-logically true. That is, FPϕ, the first conjunct of (Rel), implies♦tϕ, which contradicts the second conjunct of (Rel), ¬♦tϕ. The connectednessassumption cannot be given up, moreover, without violating the notion of alocal time or of a world-line as an inertial reference frame upon which that localtime is based. The notion of a local time is a fundamental construct not only ofconceptualism and our common-sense framework but of natural science as well,as in the assumption of an Eigenzeit in relativity theory.

In conceptualism the connectedness of a local time is part of the notion ofthe self as a center of conceptual activity, and in fact it is one of the principlesupon which the tense-logical cognitive schemata characterizing our conceptualorientation in time are constructed.

This is not to say that in the development of the concept of a self as a centerof conceptual activity we do not ever come to conceive of the ordering of eventsfrom perspectives other than our own. Indeed, by a process that Jean Piagetcalls decentering, children at the stage of concrete operational thought (7–11years) develop the ability to conceive of projections from their own positions tothat of others in their environment; and subsequently, by means of that ability,they are able to form operational concepts of space and time whose systematiccoordination results essentially in the structure of projective geometry.27

Spatial considerations aside, however, and with respect to time alone, thecognitive schemata implicit in the ability to conceive of such projections can be

26Cf. Putnam, 1967.27Cf. Piaget 1972.

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2.10. CAUSAL TENSES IN RELATIVITY THEORY 49

represented in part by means of tense operators corresponding to those alreadyrepresenting the past and the future as viewed from one’s own local time. Thatis, because the projections in question are to be based on actual causal connec-tions between the momentary descriptive states of inertial frames, or world-lines,we can represent the cognitive schemata implicit in such projections by whatwe will here call causal-tense operators.

Pc : it causally was the case that ...

Fc : it causally will be the case that ...

Semantically, the causal-tense operators go beyond the standard tenses byrequiring us to consider not just a single local time but a causally-connectedsystem of local times. The causal connections are between the momentarystates of the different inertial reference frames upon which such local times arebased; and, given the finite limiting velocity of light, these causal connectionscan be represented by a signal relation between the moments of the local timesthemselves—so long as we assume that the sets of moments of different localtimes are disjoint. (This assumption is harmless if we think of a moment of alocal time as an ordered pair one constituent of which is the inertial frame uponwhich that local time is based.) The only constraint that should be imposed onsuch a signal relation is that it be a strict partial ordering, i.e., transitive andasymmetric.28

Thus, by the causal past, as represented by Pc, we mean not just thepast with respect to the here-now of our own local time, but also the pastwith respect to any momentary state of any other world-line that can send asignal to our here-now; and by the causal future, as represented by Fc, wemean not just the future with respect to our here-now, but the future of anymomentary state of any world-line to which we can send a signal from here-now.The geometric structure at a given momentary state of a world-line of a causallyconnected system is that of a Minkowski light-cone. That is, at each momentarystate X of a world-line there is both a prior light cone (the causal past)consisting of all the momentary states (or space-time points) of world-lines thatcan send a signal to X and a posterior light cone (causal future) of allthe momentary states (or space-time points) of world-lines that can receive asignal from X . Momentary states are then said to be simultaneous if no signalrelation can be sent from one to the other.

The causal past (prior light-cone) of the here-now momentary stateX of a world-line = the momentary states of world-lines that cansend a signal to X .

28See Cocchiarella 1984, section 15, for the details of this semantics. The signal relation,incidentally, provides yet another example of a concrete interpretation of an accessibilityrelation between possible worlds, reconstrued now as the momentary states of the universe atdifferent space-time points.

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50 CHAPTER 2. TIME, BEING, AND EXISTENCE

The causal future (posterior light-cone) of the here-now momentarystate X of a world line = the momentary states of world-lines towhich a signal can be sent from X.

A momentary state X of a world-line W1 is simultaneous with amomentary state Y of a world-line W2 if, and only if, no signal canbe sent from X to Y , nor from Y to X .

Now because the signal relation has a finite limiting velocity, simultaneitywill not be a transitive relation. As a result any one of a number of momentarystates of one world-line can be simultaneous with the same momentary stateof another world-line. This is what leads to the type of situation described byPutnam:

Let Oscar be a person whose whole world-line is outside of thelight-cone of me-now. Let me-future be a future ‘stage’ of me suchthat Oscar is in the lower half of the light cone of me-future [i.e.,the prior cone of me-future]. Then, when that future becomes thepresent, it will be true to say that Oscar existed, although it willnever have had such a truth value to say in the present tense ‘Oscarexists now’. Things could come to have been, without its ever havingbeen true that they are!29

What all this indicates is that the possibility according to special relativ-ity theory of a state of affairs coming to have been the case without its everactually being the case is a possibility that should be represented in terms ofthe causal tense-operators Fcϕ and Pcϕ—i.e., in terms of the causal past andcausal future—and not in terms of the simple past- and future-tense operatorsP and F , i.e., the past and future according to the ordering of events within asingle local time.

Note that because there is a causal connection from the earlier to the latermomentary states of the same local time, the signal relation is assumed tocontain as a proper part the connected temporal ordering of the moments ofeach of the local times in such a causally connected system.30 The following, inother words, are valid theses of such a causally connected system:

Pϕ→ Pcϕ

Fϕ→ Fcϕ

But note also that, because the signal relation has a finite limiting velocity, theconverses of these theses will not also be valid in such a system. Were we toreject the assumption of relativity theory that there is a finite limit to causalinfluences, namely, the speed of light—as was implicit in classical physics and isstill implicit in our commonsense framework where simultaneity is assumed to be

29Putnam 1967, p. 204.30See Carnap, 1958, Sections 49–50, for such an analysis of the notion of a causally connected

system of local times.

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2.10. CAUSAL TENSES IN RELATIVITY THEORY 51

absolute across space-time—then we would validate the converses of the abovetheses, in which case the causal tense operators would be completely redundant,which explains why they have no counterparts in natural language, which, priorto the special theory of relativity, allowed for unlimited causal influences.

A related point is that unlike the cognitive schemata of the standard tenseoperators whose semantics is based on a single local time, those represented bythe causal tense-operators are not such as must be present in one form or anotherin every speech or mental act. They are derived schemata, in other words,constructed on the basis of those decentering abilities whereby we are able toconceive of the ordering of events from a perspective other than our own. Theimportance and real significance of these derived schemata was unappreciateduntil the advent of special relativity.

One important consequence of the divergence of the causal from the standardtense operators is the invalidity of

FcPcϕ→ Pcϕ ∨ ϕ ∨ Fcϕ

and therefore the consistency of

FcPcϕ ∧ ¬♦tϕ.

Unlike its earlier counterpart in terms of the standard tenses, this last formulais the appropriate representation of the possibility in special relativity of a stateof affairs coming (in the causal future) to have been the case (in the causal past)without its ever actually being the case (in a given local time). Indeed, not onlycan this formula be true at some moment of a local time of a causally connectedsystem, but so can the following formula31:

[Pc♦tϕ ∨ Fc♦tϕ] ∧ ¬♦tϕ.

Quantification over realia, which now includes things that exist in space-timewith respect to any local time and not just with respect to a given local time,also finds justification in special relativity. For just as some states of affairscan come to have been the case in the causal past of the causal future withouttheir actually ever being the case, so too there can be things that do not existin the past, present or future of our own local time, but which neverthelessmight exist in a causally connected local time at a moment that is simultaneouswith our present. In this regard, reference to such objects as real even if notpresently existing would seem hardly controversial—or at least not at that stageof conceptual development where our decentering abilities enable us to constructreferential concepts that respect other points of view causally connected withour own. Realia encompass all the objects of temporal possibilism and possiblymore as well.

31This formula would be true at a given moment t of a local time X if in either the priorcone or posterior cone of that moment there is a space-time point t′ of a world-line Y such thatϕ is always true in Y , even though ϕ is never true in X. Putnam’s Oscar example indicateshow this is possible in relativity theory.

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52 CHAPTER 2. TIME, BEING, AND EXISTENCE

Finally, it should be noted that there is also a causal counterpart to Diodorus’snotion of possibility as what either is or will be the case, namely, possibility aswhat either is or causally will be the case:

♦cfϕ =df ϕ ∨ Fcϕ.

Instead of the modal logic S4.3, this causal Diodorean notion of possibilityresults in the modal logic S4. Moreover, if we also assume, as is usual in specialrelativity, that the causal futures of any two moments t, t′ of two local times ofa causally connected system eventually intersect, i.e., that there is a moment wof a local time such that both t and t′ can send a signal to w, then the thesis

Fc¬Fc¬ϕ→ ¬Fc¬Fcϕ

will be validated, and the causal Diodorean notion of possibility will then resultin the modal system S4.2,32 i.e., the system S4 plus the thesis

♦fc�fcϕ→ �fc♦fcϕ.

Many other modal concepts can also be characterized in terms of a causallyconnected system of local times, including, e.g., the notion of something beingnecessary because of the way the past has been. What is distinctive aboutthem all is the unproblematic sense in which they can be taken as material ormetaphysical modalities.

Tense logic is not the only framework in which both the logic of actual andpossible objects and the logic of actual objects simpliciter have natural appli-cations and in which the differences between possibilism and actualism can bemade perspicuous. There is also, for example, the logic of belief and knowledgeand the differences between the possible quantifier and the actual quantifierbinding variables occurring free within the scope of operators for propositionalattitudes. Still, even these other frameworks must presuppose some account ofthe logic of tenses, in which case the differences between possibilism and actu-alism within tense logic becomes paradigmatic. Indeed, as we have indicated,this is certainly the case for the differences between possibilism and actualismin modal logic, since some of the very first modal concepts ever to be discussedin the history of philosophy have been modal concepts that can be analyzed inthe framework of tense logic.

2.11 Summary and Concluding Remarks

• A formal ontology must provide an ontological ground for the distinc-tion between being and existence, or, if the distinction is rejected, an adequateaccount of why it is rejected. This is a criterion of adequacy for any formalontology.

32Cf. Prior, 1967, p. 203.

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2.12. APPENDIX 53

• A first-order logic of actual and possible objects is a component part ofconceptual realism. A fuller ontological account of the distinction between ac-tualism and possibilism is given in chapter six, section one.

• Our commonsense understanding includes a representation of how we areconceptually oriented in time with respect to the past, the present and thefuture.

• Tense logic provides a formal representation that respects the form andcontent of our commonsense speech and mental acts about the past, the presentand the future.

• Because tense operators represent cognitive schemata regarding our ori-entation in time, conceptual realism is the most natural formal ontology for atense-logical representation of our commonsense understanding of time.

• Conceptual realism provides the clearest ontological ground for the dis-tinction between being and existence in terms of the tense-logical distinctionbetween past, present and future objects, i.e., the distinction between thingsthat did exist, do exist, or will exist.

• Another criterion of adequacy for a formal ontology is that it must explainthe ontological grounds, or nature, of modality, i.e., of such modal notions asnecessity and possibility, as opposed to merely giving a set-theoretic semanticsfor modal logic.

• Some of the earliest ontological views of modality, going back as far asAristotle and Diodorus, are temporal notions. Different notions of necessityand possibility can be grounded in the tense-logical part of conceptual realism.

• The analysis of necessity and possibility within tense logic provides a clearand unproblematic paradigm by which to understand the notion of a possibleworld and the accessibility relation between possible worlds.

• Different temporal modes of being can be represented within the tense-logical framework of conceptual realism.

• Special relativity is not incompatible with the connectedness of a localtime (Eigenzeit) as represented by the axioms of standard tense logic.

• The cognitive schemata implicit in our ability to conceive of projectionsto other reference frames, and hence to other local times, can be representedin terms of causal tense operators based on a signal relation between differentspace-time points.

• The causal past and the causal future that are represented by the causaltense operators are not the same as the simple past and the simple future. It iswith respect to the causal tense operators that the connectedness thesis fails tobe valid, not with respect to the standard tense operators.

2.12 Appendix

Once the logic of actual and possible objects is extended by introduction oftense and modal operators, there are certain complications that arise in theapplication of Leibniz’s law and the law of universal instantiation (and its dual,existential generalization) in tense and modal contexts. We describe some of

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54 CHAPTER 2. TIME, BEING, AND EXISTENCE

these features here as well as certain laws regarding the commutation of tenseand modal operators with quantifier phrases.

In describing some of the theorem schemas involving quantifiers, tenses andmodal operators in these different logics, we shall use the following notation:

(t) �t : is a theorem of tense logic (of local time) with quantification overactual andpossible objects.

(te) �te : is a theorem of tense logic (of local time) with quantification overjust actual objects.

(tp) �tp : is a theorem of tense logic (of local time) with quantification overjust past andpresent objects.

Note that what is provable in (te) is provable in (tp), and what is provablein (tp) is provable in (t). That is,

{ϕ : �teϕ} ⊆ {ϕ : �tpϕ} ⊆ {ϕ : �tϕ}

We may use �te , accordingly, to state what is provable in all three systems,and �tp for what is provable in (tp) and (t). We will also use the followingcounterparts of notions already defined:

(∀fx)ϕ =df (∀x)[FE!(x) → ϕ]

♦pϕ =df (ϕ ∨ Pϕ)

�pϕ =df ϕ ∧ ¬P¬ϕ

Leibniz’s Law: Assume that a, b are objectual constants, ϕ is a tensedformula, and ψ is obtained from ϕ by replacing one or more occurrences of a byoccurrences of b. Then, we have the following theses about Leibniz’s law:

(1) �te �t(a = b) → (ϕ↔ ψ)

(2) �te (x = y) → (ϕ↔ ψ), where x, y are variables.

(3) �te (a = b) → (ϕ ↔ ψ), if a does not occur in ϕ within the scope of atense operator.

(4) �te �p(a = b) → (ϕ ↔ ψ), if a does not occur in ϕ within the scope ofa future tense operator.

(5) �te �f (a = b) → (ϕ↔ ψ), if a does not occur in ϕ within the scope ofa past tense operator.

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2.12. APPENDIX 55

Identity and Non-identity: Although identity and non-identity asexpressed in terms of objectual variables is always necessary, that is,

�te (x = y) → �t(x = y)

�te (x �= y) → �t(x �= y),

The same theses are not true for objectual constants—unless they are “rigiddesignators,” i.e., denote the same object at all times, which is symbolizedas (∃x)�t(a = x). Without assuming that, the relevant qualifications are asfollows:

(6) �t (∃x)�t(a = x) ∧ (∃y)�t(b = y) → [a = b↔ �t(a = b)]∧

[a �= b↔ �t(a �= b)]

�te (∃ex)�t(a = x) ∧ (∃ey)�t(b = y) → [a = b↔ �t(a = b)]∧

[a �= b↔ �t(a �= b)]

�tp (∃ppx)�t(a = x) ∧ (∃p

py)�t(b = y) → [a = b↔ �t(a = b)]∧

[a �= b↔ �t(a �= b)]

Similar theorems hold when �t is uniformly replaced throughout (6) by �f

and �p, respectively.

Universal Instantiation: The law of universal instantiation does nothold in general in these logics without qualification. The different qualificationsare as follows, where x and y are variables, a is a term distinct from y, and a isfree for x in ϕ:

(7) �t (∃y)�t(a = y) → [(∀x)ϕ→ ϕ(a/y)]

�te (∃ey)�t(a = y) → [(∀ex)ϕ→ ϕ(a/y)]

�tp (∃ppy)�t(a = y) → [(∀p

px)ϕ→ ϕ(a/y)]

(8) If either a is a variable or x does not occur in ϕ within the scope of a pastor future tense operator, then:

�t (∀x)ϕ→ ϕ(a/y)

�te (∃ey)(a = y) → [(∀ex)ϕ→ ϕ(a/y)]

�tp (∃ppy)(a = y) → [(∀p

px)ϕ→ ϕ(a/y)]

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56 CHAPTER 2. TIME, BEING, AND EXISTENCE

(9) If x does not occur in ϕ within the scope of a future tense operator, then:

�tp (∃y)�p(a = y) → [(∀x)ϕ→ ϕ(a/y)]

�te (∃ey)�p(a = y) → [(∀ex)ϕ→ ϕ(a/y)]

�tp (∃ppy)�p(a = y) → [(∀p

px)ϕ→ ϕ(a/y)]

(10) If x does not occur in ϕ within the scope of a past tense operator, then:

�t (∃y)�f (a = y) → [(∀x)ϕ→ ϕ(a/y)]

�te (∃ey)�f (a = y) → [(∀ex)ϕ→ ϕ(a/y)]

�tp (∃ppy)�f (a = y) → [(∀p

px)ϕ→ ϕ(a/y)]

Laws of Commutation: The possible quantifier ∃ commutes withboth the past and future tense operators and therefore with ♦f , ♦p, and ♦t aswell. Dually, ∀ commutes with ¬P¬ and ¬F¬ and therefore with �f , �p, and�t as well:

(11) �t P(∃x)ϕ ↔ (∃x)Pϕ �t ¬P¬(∀x)ϕ↔ (∀x)¬P¬ϕ

�t F(∃x)ϕ↔ (∃x)Fϕ �t ¬F¬(∀x)ϕ ↔ (∀x)¬F¬ϕ

�t ♦f (∃x)ϕ ↔ (∃x)♦fϕ �t �f(∀x)ϕ ↔ (∀x)�fϕ

�t ♦p(∃x)ϕ ↔ (∃x)♦pϕ �t �p(∀x)ϕ ↔ (∀x)�pϕ

�t ♦t(∃x)ϕ ↔ (∃x)♦tϕ �t �t(∀x)ϕ↔ (∀x)�tϕ

The actual quantifier ∃e does not commute with the past or future tenseoperators except under special conditions, and even then different conditionsare required for each direction—unless it is assumed that nothing ever comes toexist or ceases to exist, in symbols:

�t(∀ex)�tE!(x) Nothing ever comes to exist or ceases to exist,

in which case ∃e commutes with ♦f , ♦p, and ♦t (and therefore, by duality, ∀e

commutes with ¬P¬, ¬F¬, �f , �p, and �t).

(12) �te (∀ex)¬P¬E!(x) → [(∃ex)Pϕ→ P(∃ex)ϕ]

�te ¬P¬(∀ex)¬F¬E!(x) → [P(∃ex)ϕ→ (∃ex)Pϕ]

�te (∀ex)¬F¬E!(x) → [(∃ex)Fϕ→ F(∃ex)ϕ]

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2.12. APPENDIX 57

�te ¬F¬(∀ex)¬P¬E!(x) → [F(∃ex)ϕ→ (∃ex)Fϕ]

�te �t(∀ex)�tE!(x) → [(∃ex)Pϕ↔ (∃ex)Pϕ]

�te �t(∀ex)�tE!(x) → [(∃ex)Fϕ↔ F(∃ex)ϕ]

�te �t(∀ex)�tE!(x) → [(∃ex)♦fϕ↔ ♦f (∃ex)ϕ]

�te �t(∀ex)�tE!(x) → [(∃ex)♦pϕ↔ ♦p(∃ex)ϕ]

�te �t(∀ex)�tE!(x) → [(∃ex)♦tϕ↔ ♦t(∃ex)ϕ]

Assumptions weaker than the condition that nothing ever comes into or goesout of existence—such as that everything presently existing always has existedand always will exist, or that everything now existing will never cease to exist,or that everything now existing always has existed yield commutations in onlyone direction.

(∀ex)�pE!(x) Everything presently existing always has existed

(∀ex)�fE!(x) Everything presently existing always will exist.

(∀ex)�tE!(x) Everything presently existing always has and always will ex-ist.

�te (∀ex)�tE!(x) → [�t(∀ex)ϕ→ (∀ex)�tϕ]

�te (∀ex)�fE!(x) → [�f(∀ex)ϕ→ (∀ex)�fϕ]

�te (∀ex)�pE!(x) → [�p(∀ex)ϕ→ (∀ex)�pϕ]

The quantifier ∃pp commutes with the past and future tense operators in only

one direction, each the converse to the other, and therefore it commutes with♦p and ♦f in only one direction as well. Similarly, ∀p

p commutes with ¬P¬ and¬F¬, and therefore with �p and �f in only one direction:

(13) �tp P(∃ppx)ϕ→ (∃p

px)Pϕ �tp (∀ppx)¬P¬ϕ→ ¬P¬(∀p

px)ϕ

�tp (∃ppx)Fϕ→ F(∃p

px)ϕ �tp ¬F¬(∀ppx)ϕ→ (∀p

px)¬F¬ϕ

�tp (∃ppx)♦fϕ→ ♦f (∃p

px)ϕ �tp �f (∀ppx)ϕ→ (∀p

px)�fϕ

�tp ♦p(∃ppx)ϕ→ (∃p

px)♦pϕ �tp (∀ppx)�pϕ→ �p(∀p

px)ϕ

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58 CHAPTER 2. TIME, BEING, AND EXISTENCE

∀pp commutes with �p in both directions if every past and present object

always was a past or present object:

�tp (∀ppx)�p[E!(x) ∨ PE!(x)] → [�p(∀p

px)ϕ↔ (∀ppx)�pϕ]

Strong conditions are needed in order to commute ∀pp with �t, and in fact

only a very strong condition suffices for commutation in both directions:

�tp (∀ppx)�t[E!(x) ∨ PE!(x)] → [�t(∀p

px)ϕ→ (∀ppx)�tϕ]

�tp �t(∀ppx)�t[E!(x) ∨ PE!(x)] → [�t(∀p

px)ϕ→ (∀ppx)�tϕ].

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Chapter 3

Logical Necessity andLogical Atomism

As indicated in the previous chapter, the analysis of necessity and possibilitywithin tense logic provides a clear and unproblematic paradigm by which tounderstand the notion of a possible world and the accessibility relation betweenpossible worlds. It also shows the unproblematic nature of quantifying intomodal contexts so long as possible objects are restricted to realia. The situationis more problematic, however, once we turn to the different notions of logicalnecessity and possibility, or a metaphysical necessity and possibility that goesbeyond causal modalities and the space-time continuum.

The problem with logical and metaphysical necessity and possibility, accord-ing to its critics, is quantification into modal contexts, which is traditionallyknown as de re modality. By allowing such quantification, these critics argue,we become committed to essentialism, and perhaps a bloated universe of logi-cally possible objects as well. The essentialism is avoidable, it is claimed, butonly by turning to a Platonic realm of individual concepts whose existence isno less dubious or problematic than logically possible objects. Moreover, bas-ing one’s semantics on individual concepts would in effect render all identitystatements containing only proper names either necessarily true or necessarilyfalse—i.e., there would then be no contingent identity statements containingonly proper names.1

These claims are not true independently of what formal ontology we adopt.The claim that identity statements containing only proper names would beeither necessarily true or necessarily false does not depend, for example, on acommitment to individual concepts, but on whether or not proper names are“rigid designators,” i.e., on whether a proper name is assumed to denote thesame object in every possible world, or at any time, at which that object exists.

The commitment to essentialism that these critics have in mind, moreover,is not Aristotelian essentialism, even though Quine, who has been the most

1Cf. Quine 1943, 1947, 1953.

59

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60 CHAPTER 3. LOGICAL NECESSITY AND LOGICAL ATOMISM

vocal of the critics, explicitly condemns it as such.2 The examples Quine givesof essentialism, such as,

the number nine is necessarily, i.e., essentially, greater than seven,but nine is only contingently and not essentially the number of plan-ets, and everything is necessarily self-identical,

is not based on an ontology of natural kinds and have nothing to do with Aris-totelian essentialism. Apparently, what Quine and the other critics of modallogic have in mind is either a logical essentialism based on logical modalities ora metaphysical essentialismbased on some form of realism other than naturalrealism and perhaps even some sort of logical modalities broadly understood.

Now there is an ontology in which all of these objections to modal logic,including especially those directed against de re modalities, can be shown tobe completely false. This is the ontology of logical atomism, where there areno individual concepts and no possibilia other than the simple objects thatexist in the actual world. In addition, it is probably only in logical atomismthat logical necessity and possibility find their clearest, and perhaps their only,adequate explication. And yet, not only is logical essentialism false in logicalatomism, but it is refutable as well. In other words, with respect to the logicalmodalities, a modal thesis of anti-essentialism is valid in logical atomism, andone consequence of this is that all de re logical modalities are reducible to dedicto logical modalities, and hence that there is no problem of de re logicalmodalities.

Logical atomism:(1) There are no individual concepts and no possibilia, i.e., existence= being.(2) Logical essentialism is refuted because the modal thesis of anti-essentialism is logically true in this framework.(3) All de re logical modalities are reducible to de dicto logicalmodalities.

These results do not mean that logical atomism provides the kind of formalontology we should adopt, and in fact there are good reasons why just theopposite is the case. Nevertheless, logical atomism does provide the paradigmframework by which to understand logical necessity and possibility, and it showswhy a logical essentialism based on this kind of necessity—as opposed, e.g., toa natural necessity—not only does not follow but is actually refuted.

Now opposed to logical atomism, but on a par with it in its referential inter-pretation of quantifiers and proper names, is Saul Kripke’s semantics for whathe calls metaphysical necessity.3 There are no individual concepts in Kripke’ssemantics, and yet proper names are “rigid designators,” which means that therecan be no contingent identity statements containing only proper names. But

2Cf. Quine 1953, p. 173–174.3Cf. Kripke, 1971, p. 150.

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3.1. THE ONTOLOGY OF LOGICAL ATOMISM 61

unlike what is meant by the clear and primary meaning of the phrase “all pos-sible worlds” in logical atomism, and with respect to which logical necessity isinterpreted, Kripke’s semantics allows for a “cut-down” on the totality of possi-ble worlds, so that the notion of “all possible worlds,” and hence necessity, has asecondary meaning in his semantics.4 Such a secondary meaning or “cut-down”on the notion of “all possible worlds,” and therefore on necessity, amounts toan initial, but incomplete, step toward something like Aristotelian essentialism.The problem with this initial step, and why it is incomplete, is its failure toprovide any ontological content—as opposed to a merely formal, set-theoreticstructure—to what is meant by necessity and possibility. This problem of thesecondary meaning of necessity, or “cut-down,” on the notion of “all possibleworlds,” is only compounded, moreover, by adding to this semantics a relationof accessibility between possible worlds. What must then be explained, in otherwords, is not just the philosophical significance of the “cut-down” on the no-tion of “all possible worlds,” but also the ontological content of the accessibilityrelation between possible worlds.

The real problem of quantified modal logic for an ontology otherthan logical atomism, in other words, is to give an ontological ac-count of the “cut-down” on the notion of “all possible worlds” andof the accessibility relation between possible worlds. The questionis: can this really be done other than in tense logic or Aristotelianessentialism, both of which are contained in conceptual realism?

3.1 The Ontology of Logical Atomism

Reality, according to logical atomism, consists of the existence and nonexistenceof atomic states of affairs, where the existence of a state of affairs is “a positivefact” and its nonexistence “a negative fact”.5 The actual world, in other words,consists of all that is the case, namely the totality of facts, whether positiveor negative.6 Every other possible world consists of the same atomic states ofaffairs that make up reality, except that what are positive facts in one world canbe negative facts in another, with every possible combination of atomic states ofaffairs being realized in some possible world or other.7 The totality of possibleworlds, in other words, is completely determined by all the combinations of theexistence or nonexistence of the atomic states of affairs that make up reality.

4A secondary semantics for necessity stands to the primary semantics in essentially thesame way that nonstandard models for second-order logic stand to standard models. SeeCocchiarella 1975.

5Wittgenstein 1961, 2.06. For a fuller discussion of logical atomism as a formal ontology,see chapters 6 and 7 of Cocchiarella 1987.

6Wittgenstein 1961, 1. It is an issue as to whether the Tractatus allowed for negative facts.There are negative facts in Russell’s version of logical atomism.

7If we did not include negative facts, then a world that contains none of the states of affairsthat “exist” in the actual world—i.e., that would contain as positive facts all of the negativefacts of the actual world—would be an empty world, a world devoid of all facts, and hence ofall objects as well, and therefore not a possible world at all.

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62 CHAPTER 3. LOGICAL NECESSITY AND LOGICAL ATOMISM

The direction of this determination is important. Atomic states of affairs do nothave being (the-case-or-not-the-case) because they exist (are the case) in somepossible worlds; rather, possible worlds are possible because they are resolvableinto the atomic states of affairs that make up reality.

The determination of the totality of possible worlds in terms of theatomic states of affairs that make up reality is what makes logicalatomism a paradigm framework for the semantics of logical necessity.

Every atomic state of affairs is a configuration of objects, and thereforebecause every state of affairs is a positive or negative fact in each possible world,each possible world consists of the same totality of objects as every other possibleworld. There is no distinction, accordingly, between the existence and being ofobjects. In the ontological grammar of logical atomism, in other words, thereis no distinction between the possibilist quantifiers ∀ and ∃ and the actualistquantifiers ∀e and ∃e, and for that reason we will not include the latter in theformal ontology of logical atomism.

Ordinary proper names of natural language are not “logically proper names”inthe framework of logical atomism. The things they name, if they name anythingat all, are not the simple objects that are the constituents of atomic states ofaffairs. The names of ordinary language have a sense (Sinn), moreover, in sofar as they are introduced into discourse with identity criteria, usually providedby a sortal common noun with which they are associated.8 The logically propernames of logical atomism have no sense other than what they denote. In otherwords, in logical atomism, “a name means (bedeutet) an object. The object isits meaning (Bedeutung).”9 Different identity criteria have no bearing on thesimple objects of logical atomism, and (pseudo) identity propositions, strictlyspeaking, have no sense (Sinn), i.e., they do not represent an atomic state ofaffairs.

Semantically, what this comes to is that logically proper names, or objectualconstants, are rigid designators; that is, their introduction into formal languagesrequires that the formula

(∃x)�(a = x)

be logically true in the primary semantics for each objectual constant a.Kripke also claims that proper names are rigid designators, but his proper

names are those of ordinary language, and his necessity is metaphysical and notlogical necessity. Nevertheless, in agreement with logical atomism the functionof a proper name, according to Kripke, is simply to refer, and not to describethe object named10; and this applies even when we fix the reference of a proper

8Cf. Geach, 1980, p. 63f. Individual constants cannot be vacuous in logical atomism,moreover, which means that the free logic of the quantifiers ∀e and ∃e is to be excluded, asopposed to the logic of the possibilist quantifiers ∀ and ∃, which, as already noted is standardpredicate logic.

9Wittgenstein 1961, 3.203.10Kripke, 1971, p. 140.

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3.1. THE ONTOLOGY OF LOGICAL ATOMISM 63

name by means of a definite description—for the relation between a proper nameand a description used to fix the reference of the name is not that of synonymy.11

Because logically proper names are rigid designators, there are no contingentidentity statements involving only proper names. Accordingly, where a, b areobjectual constants,

(a = b) → �(a = b)

(a �= b) → �(a �= b)

are both understood to be logically true in logical atomism, and in Kripke’sframework of metaphysical necessity as well. The fact that there can be nocontingent identities or non-identities in logical atomism is reflected, moreover,in the logical truth of both of the formulas

(∀x)(∀y)[(x = y) → �(x = y)],

(∀x)(∀y)[(x �= y) → �(x �= y)]

in the semantics of logical atomism.12 But then even in the framework ofKripke’s metaphysical necessity (where quantifiers also refer directly to objects),an object cannot but be the object that it is, nor can one object be identicalwith another—a metaphysical fact which is reflected in the above formulas beingvalid in that framework as well.

Another observation about the ontology of logical atomism is that the num-ber of objects in the world is part of its logical scaffolding.13 That is, for eachpositive integer n, it is either logically necessary or impossible that there areexactly n objects in the world; and if the number of objects is infinite, then, foreach positive integer n, it is logically necessary that there are at least n objectsin the world.14 This is true in logical atomism because every possible worldconsists of the same totality of objects.

One important consequence of the fact that every possible world (of a givenlogical space) consists of the same totality of objects is the logical truth of theCarnap-Barcan formula (and its converse)

(∀x)�ϕ↔ �(∀x)ϕ.

Carnap, it should be noted, was the first to actually give a semantic argumentjustifying the logical truth of this principle.15 The idea, in effect, is that anyuniversally quantified sentence (∀x)ϕ, no matter whether ϕ contains occurrencesof modal operators or not, “is to be interpreted as a joint assertion for all valuesof the variable.”16

11Ibid., pp. 156f.12See Carnap 1946 for the first clear recognition of the validity of these noncontingent

identity theses.13This observation was first made by Ramsey in his adoption of logical atomism. Cf. Ramsey

1960.14Cf. Cocchiarella, 1975, Section 5.15Cf. Carnap, 1946, p.37 and 1947, Section 40. Unlike Carnap, Barcan assumed the formula

as an axiom, and gave no explanation of why it should be accepted.16Carnap 1947, p. 37.

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64 CHAPTER 3. LOGICAL NECESSITY AND LOGICAL ATOMISM

3.2 The Primary Semantics of Logical Necessity

Let us now turn to what we take to be the primary semantics of logical ne-cessity.17 Our terminology will proceed as a natural extension of the syntaxand semantics of standard first-order logic with identity. A formal languageconsists then just of predicate constants (of various finite degree) and objectualconstants.

As primitive logical constants we take:

→ the material conditional sign,

¬ the negation sign,

∀ the universal quantifier,

= the identity sign, and

� the logical necessity sign.

The conjunction, disjunction, biconditional, existential quantifier and possi-bility signs—∧, ∨, ↔, ∃ and ♦, respectively—are understood to be defined in theusual way as metalinguistic abbreviatory devices. The formulas of a languageare defined as in the logic of actual and possible objects, except that now thequantifier ∀e for existent objects is excluded and the logical necessity operator� is included.

Because all possible objects are actual objects in logical atomism, we canrestrict the notion of an model suited for a formal language L as follows.

Definition: If L is a language, then a model A is an L-model if, and onlyif, for some nonempty set D and function R on L, A = 〈D,R〉, and foreach objectual constant a ∈ L, R(a) ∈ D and for each positive integer nand each n-place predicate Fn ∈ L, R(Fn) ⊆ Dn, i.e., R(Fn) is a set ofn-tuples of members of D.

Note: In the context of logical atomism, a model 〈D,R〉 for a language Lrepresents a possible world of a logical space based upon D as the universeof objects of that space and L as the predicates characterizing the atomicstates of affairs of that space. The possible worlds of this logical space arethe L-models having D as their domain and that assign to each objectualconstant a in L the same denotation that R assigns to a, because a is a“rigid designator”.

Definition: If A = 〈D,R〉 is a model for a language L representing the actualworld, then the logical space determined by A = the totality of possibleworlds based on A, in symbols Wlds(A), is defined as follows:Wlds(A) = {〈D,R′〉 : 〈D,R′〉 is an L-model and for all objectual constantsa in L, R′(a) = R(a)}.

17One or another version of this primary semantics for logical necessity, it should be noted,occurs in Carnap, 1946; Kanger, 1957; Beth, 1960 and Montague, 1960. Only Carnap, how-ever, was clear about the association of his semantics with logical atomism.

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3.2. THE PRIMARY SEMANTICS OF LOGICAL NECESSITY 65

Definition: A is an assignment (of values to variables) in a domain D if, andonly if, A is a function with the set of objectual variables as domain andsuch that for each variable x, A(x) ∈ D.

Definition: If A is an assignment in a domain D, x is an objectual variable andd ∈ D, then A(d/x) is an assignment in D that is exactly like A exceptfor its assigning d to x.

Definition: If L is a language, A = 〈D,R〉 is an L-model, A is an assignment inD, and a is a variable or an individual constant in L, then (the denotationof a in A):

denA,A ={

A(a) if a is a variableR(a) if a is an objectual constant .

The satisfaction in A of a formula ϕ of L by an assignment A in D, insymbols A, A |= ϕ, is recursively defined as follows:

1. A, A |= (a = b) iff denA,A(a) = denA,A(b);

2. A, A |= Fn(denA,A(a1, ..., an) iff 〈denA,A(a1), ..., denA,A(an)〉 ∈ R(Fn);

3. A, A |= ¬ϕ iff A, A � ϕ;

4. A, A |= (ϕ→ ψ) iff either A, A � ϕ or A, A |= ψ;

5. A, A |= (∀x)ϕ iff for all d ∈ D, A, A(d/x) |= ϕ; and

6. A, A |= �ϕ iff for all B ∈Wlds(A), B, A � ϕ.

The truth of a formula in a model (indexed by a language suitable to thatformula) is as usual the satisfaction of the formula by every assignment in theuniverse of the model. Logical truth is then truth in every model (indexed byany appropriate language).

Definition: If L is a language, ϕ is a standard formula of L, A = 〈D,R〉, andA is an L-model, then ϕ is true in A if, and only if, for each assignmentA in D, A, A |= ϕ.

Definition: ϕ is logically true if, and only if, for some language L, ϕ is aformula of L, and ϕ is true in every L-model.

These definitions are the natural extensions of the same semantical conceptsas defined for the modal free formulas of standard first-order predicate logicwith identity.

Note that every model, because it specifies both a domain and a lan-guage, determines both a unique logical space and a possible worldof that space. In this regard, the clause for the necessity operatorin the above definition of satisfaction is the natural extension of thestandard definition of satisfaction and interprets the necessity oper-ator as ranging over all the possible worlds (models) of the logicalspace to which the given one belongs.

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66 CHAPTER 3. LOGICAL NECESSITY AND LOGICAL ATOMISM

3.3 The Modal Thesis of Anti-Essentialism

Now it may be objected that logical atomism is an inappropriate frameworkupon which to base a system of quantified modal logic; for if any frameworkis a paradigm of anti-essentialism, it is logical atomism with its ontologicallysimple objects and independent atomic states of affairs. The objection is voidand begs the question, because it assumes that any system of quantified modallogic is committed to essentialism insofar as it allows quantifiers to reach intomodal contexts, i.e., insofar as it allows de re modalities.

Now essentialism means different things in different ontologies. For Aristotleand Aquinas the doctrine of essentialism is a form of natural realism, only inaddition to natural properties and relations there are also natural kinds hierar-chically ordered in terms of species and genera. On this doctrine, if an objectbelongs to a natural kind, then it necessarily belongs to that natural kind, i.e.,it belongs to that kind in every possible world in which it exists. But a naturalkind of object will also have natural properties that are not essential to it, i.e.,properties that it has in some but not in all possible worlds in which it exists.This doctrine is similar to Quine’s characterization of what he calls “Aristotelianessentialism,” namely, “the doctrine that some of the attributes of a thing (quiteindependently of the language in which that thing is referred to, if at all) maybe essential to the thing and others accidental.”18 But Quine’s characterizationfails to distinguish natural kinds from attributes in general, and in fact, as wewill see in our later development of the logic of natural kinds, natural kinds arenot “attributes” at all, at least not in the sense of the natural properties that anobject might have. Instead, natural kinds are certain types of causal structuresthat have a complementary relationship with natural properties and relations,and as such they are general essences as opposed to the individual essences of,e.g.,Alvin Plantinga’s ontology.19

Now, according to Quine “what Aristotelian essentialism says” is that youcan have open sentences, e.g., ϕx and ψx such that ϕ is necessary to x but ψ isnot necessary to x even though ψ is true of x, which formally can be symbolizedas follows:20

(∃x)[�ϕx ∧ ¬�ψx ∧ ψx].

As an example of this so-called “Aristotelian essentialism”, Quine gives thefollowing,

(∃x)[�(x > 5) ∧ there are just x planets ∧ ¬there are just x planets].

Note that here (x > 5) represents the “attribute” of being greater than 5, acondition that in no sense can be understood as a natural kind in Aristotle’snatural realism. According to Quine, “something yet stronger can be shown”,namely

18Cp, e.g., Quine 1966, pp.173f.19See Plantinga 1974, chapter 5, §2. Also, see our discussion of the difference between

general and individual essence in chapter ??, §5.20Ibid.

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3.3. THE MODAL THESIS OF ANTI-ESSENTIALISM 67

(∀x)[�(x = x) ∧ (∀x)(x = x ∧ P ) ∧ (∀x)¬�(x = x ∧ P ).

Here, again, (x = x), the condition of being self-identical, does not representa natural kind, nor even a natural property in Aristotle natural realism. Self-identity, unlike belonging to a natural kind, is a condition that is necessary toall objects and in no way marks off what is essential to some objects as opposedto others.

If we ignore Aristotelian essentialism and the issue of natural kinds, thenQuine’s characterization can be slightly reformulated as the more interestingdoctrine that there are some conditions (predicates, properties, or concepts)that are necessarily true of some objects, but, unlike self-identity, not of others.It is this more interesting version of essentialism that is actually invalidated inlogical atomism where the opposite thesis of anti-essentialism is logically true.This anti-essentialist thesis was first formulated and validated by Rudolf Carnapin his state-description semantics, which, in the case of an infinite domain, isequivalent to the above semantics for logical atomism.21

The general idea of the modal thesis of anti-essentialism is that if a predicateexpression or open formula ϕ in which no objectual constants occur can be true ofsome objects in a given universe (satisfying a given identity-difference conditionwith respect to the variables free in ϕ), then ϕ can be true of any objects inthat universe (satisfying the same identity-difference conditions).

In other words, no condition is essential to some objects that is notessential to all, which is as it should be if necessity means logicalnecessity.22

The restriction to identity-difference conditions mentioned above can bedropped, it should be noted, if nested quantifiers are interpreted exclusivelyand not, as we have done, inclusively where, e.g., it is allowed that the valueof y in (∀x)(∃y)ϕ(x, y) can be the same as the value of x, as for example in(∀x)(∃y)(x = y).23 Now our point is that when nested quantifiers are inter-preted exclusively, then identity and difference formulas are superfluous—whichis especially appropriate in logical atomism where an identity formula does notrepresent an atomic state of affairs.24

Retaining the inclusive interpretation and identity as primitive, however, anidentity-difference condition is defined as follows.

Definition: If x1, ..., xn are distinct objectual variables, then an identity-differencecondition for x1, ..., xn is a conjunction of one each but not both of theformulas (xi = xj) or (xi �= xj), for all i, j such that 1 ≤ i < j ≤ n.

21See Carnap, 1946, T10-3.c, p.56, for the first formulation of this thesis ever to be given,and also Parsons, 1969 for a much later formulation. Note, however, that whereas in Carnap’ssemantics the thesis is valid, in Parson’s semantics the thesis is merely consistent.

22If objectual constants do occur in a formula, they can be replaced uniformly by distinctnew objectual variables not already occurring in the formula.

23See Hintikka, 1956 for a development of the exclusive interpretation.24Cf. Wittgenstein, 1961, and Cocchiarella, 1987, chapter V1.

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68 CHAPTER 3. LOGICAL NECESSITY AND LOGICAL ATOMISM

Note: Because there are only a finite number of nonequivalent identity-differenceconditions for x1, ..., xn, we can assume an ordering, ID1(x1, ..., xn),...,IDj(x1, ..., xn), is given of all of these nonequivalent conditions.

The modal thesis of anti-essentialism may now be stated for all formulas ϕin which no objectual constants occur as follows: for all positive integers j suchthat 1 ≤ j ≤ n, every formula of the form,

(∃x1)...(∃xn)[IDj(x1, ..., xn) ∧ �ϕ] → (∀x1)...(∀xn)[IDj(x1, ..., xn) → �ϕ]

is to be logically true, where x1, ..., xn are all the distinct objectual variablesoccurring free in ϕ. We can also phrase this thesis in terms of its equivalentcontrapositive form:

(∃x1)...(∃xn)[IDj(x1, ..., xn) ∧ ♦ϕ] → (∀x1)...(∀xn)[IDj(x1, ..., xn) → ♦ϕ]

Note: Where n = 0, the above formula is understood to be just (�ϕ → �ϕ);and where n = 1, it is understood to be just (∃x)�ϕ → (∀x)�ϕ, orequivalently (∃x)♦ϕ→ (∀x)♦ϕ. The first of these last formulas state thatif something is essentially ϕ, then everything essentially ϕ.

The validation of the thesis in our present semantics is easily seen to be aconsequence of the following lemma (whose proof is by a simple induction onthe formulas of L). That is, given that some objects satisfy ϕ in some L-model,then, by the following lemma, any permutation of those objects with any othersin a domain D of the same size will also satisfy ϕ in some other L-model withthat domain D, i.e., there will be an isomorphism between the two L-models.

LEMMA: If L is a language, A,B are L-models, and h is an isomorphism of Awith B, then for all formulas ϕ of L and all assignments A in the universeof A, A, A � ϕ if, and only if, B, A/h |= ϕ.25

As already noted, one of the consequences of the modal thesis of anti-essentialism is the reduction of all de re formulas to de dicto formulas. Such aconsequence indicates the correctness of our association of the present semanticswith logical atomism.

25We understand h to be an isomorphism of A with B if (1) h is a 1-1 mapping of thedomain of A onto the domain of B, (2) for each individual constant a ∈ L, denB,A/h(a) =h(denA,A(a)), and (3)for all positive intergers n and n-place predicate constants F ∈ L,the extension F is assigned in B is {〈h(d1) , ..., h(dn)〉 : for 〈d1, ...dn〉 in the extension thatF is assigned in A}. Also, we take the relative product A/h to be that assignement inthe domain of B such that for all variables x, A/h(x) = h(A(x)). In the case of anatomic formula in the inductive argument for this lemma, we have A, A |= F (a1, ..., an) iff〈denA,A(a1), ..., denA,A(an)〉 ∈ R(F ) iff 〈h(denA,A(a1)), ..., h(denA,A(an))〉 ∈ R′(F ), whereR,R′ are the assignments to predicate constants in A and B, respectively; and thereforeA, A |= F (a1, ..., an) iff B, A/h |= F (a1, ..., an). The remaining cases follow in each case bythe inductive hypothesis.

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3.4. AN INCOMPLETENESS THEOREM 69

Note: A de re formula ϕ is one in which some objectual variable has a freeoccurrence in a subformula of ϕ of the form �ψ, and hence a variable thatcan be bound by a quantifier applied to ϕ. A de dicto formula is a formulathat is not de re.

De Re Elimination Theorem: For each de re formula ϕ, there is a de dictoformula ψ such that (ϕ↔ ψ) is logically true.26

3.4 An Incompleteness Theorem

Another result of the semantics for logical atomism is its essential incomplete-ness with respect to any language containing at least one relational predicate.This result depends on the conditional possibility of there being infinitely manyobjects, and a relational predicate is needed in order to state such an infinitarycondition. In other words, an essential incompleteness theorem results if thereare relational states of affairs in logical atomism.

If there are only properties, i.e., monadic states of affairs, then theformal ontology is not only complete but decidable as well. Theabove semantics yields both a completeness theorem and a decisionprocedure for logical truth, in other words, for any language con-taining only monadic predicates.

The incompleteness theorem for languages containing a relational predicateis easily seen to follow from the following lemma and the well-known fact thatthe modal-free nonlogical truths of a first-order language containing at least onerelational predicate is not recursively enumerable.27

LEMMA: If ψ is a sentence that is satisfiable, but only in an infinite model,and ϕ is a modal-free and identity-free sentence and ϕ, ψ contain no ob-jectual constants, then (ψ → ¬�ϕ) is logically true iff ϕ is not logicallytrue.28

26A proof of this theorem can be found in McKay, 1975. Briefly, where x1, ..., xn are all thedistinct individual variables occurring free in ϕ and ID1(x1, ..., xn), ...,Dk(x1, ..., xn) are allthe nonequivalent identity-difference conditions for x1, ..., xn, then the equivalence in questioncan be shown if ψ is obtained from ϕ by replacing each subwff �χ of ϕ by:

[ID1(x1, ...xn) ∧ �∀x1...∀xn(ID1(x1, ..., xn) → χ)] ∨ ...∨[IDk(x1, ..., xn) ∧ �∀x1...∀xn(IDk(x1, ..., xn) → χ)].

27Cf. Cocchiarella, 1975.28Proof. Assume the antecedent and, for the left-to-right direction that (ψ → ¬�ϕ) is

logically true. We note that if ϕ were logically true, then it would be true in every L-modelfor any language L of which ϕ is a formula; but then ϕ would be true in an infinite L-modelA in which ψ is satisfiable, in which case, by assumption, ¬�ϕ would be true A as well; butthat is impossible because ϕ would then be false in some L-model when by assumption ϕ islogically true, and therefore true in every L-model. For the right-to-left direction, suppose

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THEOREM: If L is a language containing at least one relational predicate,then the set of formulas of L that are logically true is not recursivelyenumerable.

Proof. Because the modal and identity free formulas of L that are not logicallytrue are not recursively enumerable, it follows by the above lemma that thelogically true formulas of L of the form (ψ → ¬�ϕ) are also not recursivelyenumerable, and hence that the set of formulas of L that are logically true isnot recursively enumerable.

This last result does not affect the association we have made of the primarysemantics with logical atomism. Indeed, given the Lowenheim-Skolem theorem,what the above lemma shows is that

there is a complete concurrence between logical necessity as an in-ternal condition of modal free propositions, or of their correspondingstates of affairs, and logical truth as a semantical condition of themodal free sentences expressing those propositions, or representingtheir corresponding states of affairs. And that of course is as itshould be if the operator for logical necessity is to have only formaland no material content.

Finally, it should be noted that the above incompleteness theorem explainswhy Carnap was not able to prove the completeness of the system of quantifiedmodal logic formulated in Carnap 1946. For on the assumption that the numberof objects in the universe is denumerably infinite, Carnap’s state descriptionsemantics is essentially that of the primary semantics restricted to denumerablyinfinite models; and, of course, precisely because the models are denumerablyinfinite, the above incompleteness theorem applies to Carnap’s formulation aswell. Thus, the reason why Carnap was unable to carry though his proof ofcompleteness is finally answered.

3.5 The Semantics of Metaphysical Necessity

Like the situation in standard second-order logic, the incompleteness of theprimary semantics can be avoided by allowing the quantification over possibleworlds in the interpretation of necessity to refer not to all of the possible worlds(models) of a given logical space but only to those in a given non-empty set ofsuch.

ϕ is not logically true. Let A be an arbitrary L-model for any language L of which ϕ andψ are formulas. It suffices to show that (ψ → ¬�ϕ) is true in A. If ψ is not satisfiablein A, then (ψ → ¬�ϕ) is vacuously true in A. Suppose then that ψ is satisfiable in A.Then, by hypothesis, D, the domain of A, is infinite. Now because ϕ is modal and identityfree and not logically true, then, by Lowenheim-Skolem theorem, ϕ must be false in someL-model B having D as its domain, and hence, because no objectual constants occur in ϕ orψ, B∈ Wlds(A). But then, by the semantic clause for �, ¬�ϕ is true in A, and therefore sois (ψ → ¬�ϕ).

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That is, by allowing a “cut-down” on the notion of “all possibleworlds” in the interpretation of necessity, we can obtain a complete-ness theorem instead of an incompleteness theorem. Of course, theworld in question must be in the “cut-down” as part of the definitionof satisfaction.

Accordingly, where L is a language and D is a non-empty set, we understanda model structure based on D and L to be a pair 〈A,K〉, where A ∈ K, Kis a set of L-models all having D as their domain of discourse and all of theobjectual constants in L are assigned the same denotation in each L-model inK.

Definition: If L is a language and D is a nonempty set, then 〈A,K〉 is a modelstructure based on D and L if, and only if, A ∈ K and K is a set ofL-models all having D as their domain of discourse and all agreeing onthe assignment of members of D to the objectual constants in L, i.e., theassignment of a members of D to objectual constants in L is the same foreach member of K.

The satisfaction of a formula ϕ of L in such a model structure by an assign-ment A in D, in symbols 〈A,K), A |= ϕ, is recursively defined exactly as in theprimary semantics except for clause (6), which is now defined as follows:

6. 〈A,K〉, A |= �ϕ iff for all B ∈ K, 〈B,K〉, A |= ϕ.

Instead of logical truth, a formula is understood to be universally valid ifit is satisfied by every assignment in every model structure based on a languageto which the formula belongs.

Definition: ϕ is universally valid if, and only if, for every language L, everynonempty domain D, every model structure 〈A,K〉 based on D and L,and every assignment A in A, if ϕ is a formula of L, then 〈A,K〉, A |= ϕ.

Where QS5 is standard first-order logic with identity supplemented withthe axioms of S5 propositional modal logic, a completeness theorem for thesecondary semantics of logical necessity was proved by Kripke in 1959.

Completeness Theorem: A set Γ of formulas is consistent in QS5 if, andonly if, all the members of Γ are simultaneously satisfiable in a modelstructure; and (therefore) a formula ϕ is a theorem of QS5 if, and only if,ϕ is universally valid.

Despite the above completeness theorem, the secondary semantics has toohigh a price to pay as far as logical atomism is concerned.

Unlike the situation in the primary semantics, the secondary seman-tics does not validate the modal thesis of anti-essentialism—i.e., itis false that every instance of the thesis is universally valid.

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Example 1 As an example of a false instance of the thesis, consider a modelstructure 〈A,K〉, where K = {A,B},D = {d1, d2}, d1 �= d2, and F is a monadicpredicate that is assigned only d1 in B and nothing in A. Then (∃x)♦F (x) istrue in 〈A,K〉, whereas (∀x)♦F (x) is false in 〈A,K〉, which invalidates themodal thesis of anti-essentialism (in the case of a monadic formula).

The reason why the modal thesis of anti-essentialism can be invalidated inthe secondary semantics is because necessity no longer represents an invariancethrough all the possible worlds of a given logical space but only through thosein a nonempty set of such worlds.

In this way, necessity is no longer a purely formal concept having nomaterial content the way it is in logical atomism. Instead, necessityis now allowed to represent an internal condition of states of affairs—i.e., a condition that has material and not merely formal content—forwhat is invariant through all of the members of such a nonempty setneed not be invariant though all the possible worlds (models) of thelogical space to which those in the set belong.

One example of how such material content affects the implicit metaphysicalbackground can be found in monadic modal predicate logic. First let us note awell-known fact about modal-free monadic predicate logic.

Note: Modal-free monadic predicate logic is decidable and no modal-free monadicformula can be true in an infinite model unless it is true in a finite modelas well. Therefore, any substitution instance of a modal-free monadicformula for a relational predicate in an axiom of infinity—i.e., a formulathat is true only in an infinite domain of discourse—is not only false butlogically false; and hence, contrary to certain metaphysical views therecan be no modal-free analysis or reduction of all relational predicates (oropen formulas with two or more free variables) in terms only of monadicpredicates, i.e., in terms only of modal-free monadic formulas.

Now the same result also holds for quantified monadic modal logic withrespect to the primary semantics, where “all possible worlds” means all possibleworlds.

Theorem: Modal monadic predicate logic is also decidable with respect to theprimary semantics of logical atomism; and therefore no monadic formula,modal-free or otherwise, can be true in an infinite model unless it is alsotrue in a finite model.29 That is, there can be no reduction of all relationalpredicates or open formulas in terms only of monadic formulas, modal freeor otherwise.

29Cf. Cocchiarella 1975. This is proved by interpreting � as a string of universal quantifierson the predicates occurring within the scope of �, and thereby translating modal formulas intomodal-free formulas of second-order monadic predicate logic, which is known to be decidable.

It is also shown in Cocchiarella 1975 that if second-order monadic predicate logic is given asecondary, “cut-down” semantics, then, unlike its primary semantics, the resulting logic is nolonger decidable.

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The following formula, e.g., is true in some models based on an infinitedomain but false in all models based on a finite domain. This formula, in otherwords, can be taken as an axiom of infinity.

(∀x)¬R(x, x) ∧ (∀x)(∃y)R(x, y) ∧ (∀x)(∀y)(∀z)[R(x, y) ∧R(y, z) → R(x, z)]

Consider substituting the open formula (with two free variables) ♦[F (x)∧G(y)],with two monadic predicate constants F and G, for the two-place predicate R inthis formula. The first conjunct then yields the following substitution instance.

(∀x)¬♦[F (x) ∧G(x)],

which is equivalent to¬♦(∃x)[F (x) ∧G(x)].

But the formula ♦(∃x)[F (x) ∧ G(x)] is logically true in the primary semanticsof logical atomism, and hence its negation is logically false, which shows thatthat the conjunction that is an instance of the above infinity formula is logicallyfalse.

In other words, where L is any language having the monadic predicates Fand G as members, then given any nonempty domain D there will be an L-model A in which [F (x) ∧ G(x)] is satisfied by some member of D, and hence(∃x)[F (x) ∧G(x)] will be true in A, which means that ♦(∃x)[F (x) ∧G(x)] willbe true in any world (L-model) in the logical space determined by A, i.e., truein any world in Wlds(A), and hence that ♦(∃x)[F (x) ∧ G(x)] is logically truewith respect to the primary semantics of logical atomism. Thus, substituting♦[F (x) ∧ G(y)] for R in the above formula representing an axiom of infinityresults in a logically false sentence in the primary semantics.

With respect to the secondary semantics, however, the situation is quitedifferent, because all we need do is exclude all of the L-models in the logicalspace based on L and D in which (∃x)[F (x)∧G(x)] is true. The “cut-down” orresulting model structure 〈B,K〉 will be such that ¬♦(∃x)[F (x) ∧G(x)] is truein all of the models in K. Equivalently, the formula

�(∀x)[F (x) → ¬G(x)],

which clearly represents a necessary, i.e., an internal, relation between beingan F and not-being a G, will be true in 〈B,K〉 with respect to the secondarysemantics.

Instead of modal monadic predicate logic being decidable the way it is inthe primary semantics, modal monadic predicate logic is undecidable in thesecondary semantics, as Kripke has shown. Moreover, on the basis of that se-mantics a modal analysis of relational predicates in terms of monadic predicatescan in general be given.30 Indeed, substituting ♦[F (x) ∧ G(y)] for the binarypredicate R in the above infinity axiom results in a modal monadic sentence

30Cf. Kripke 1962.

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that is true in some model structure based on an infinite universe and false inall model structures based on a finite domain.31

Somehow, in other words, by means of a “cut-down” on the notion of“all possible worlds”, relational content has been incorporated intothe semantics for necessity, and thereby of possibility as well.

In this respect, the secondary semantics is not the semantics of a merely formal,or logical, necessity but of a necessity having material, nonlogical content as well.

3.6 Metaphysical Versus Natural Necessity

Kripke does not describe the necessity of his semantics as a formal, or logical,necessity, but as a metaphysical necessity. He has also argued that not everynecessary proposition is a priori, and that not every a posteriori proposition iscontingent.32 It is because Kripke is concerned with a metaphysical necessityand not a logical, or formal, necessity that not every necessary proposition needsto be a priori, nor every a posteriori proposition contingent. But such a positioncannot be taken as a refutation of the claim in logical atomism that everylogically necessary proposition is a priori and that every a posteriori propositionis logically contingent. These are two different metaphysical frameworks, eachwith its own notion of necessity and thereby of contingency as well. The ontologyof logical atomism’s framework is very clear and precise, moreover, whereas itis not at all clear just what ontological framework Kripke has in mind with hisnotion of “metaphysical necessity”.

Now we can extend the notion of a model structure 〈B,K〉 based on a domainD so that instead of having D represent the same universe of existing objectsin all of the worlds in K, D would represent only the same domain of possibleobjects of the structure 〈B,K〉, i.e., the union, or sum, of all of the objects thatexist in some world or other in K. We would then distinguish a universe ofexisting objects for each world A ∈ K (i.e., the objects that exist in A and notnecessarily in the other worlds in K) from the possible objects made up of theobjects that exist in some world or other in K, just as we did in the semanticsfor the logic of actual and possible objects in tense logic where instead of worldswe had moments of time. We would then reintroduce the actualist quantifiers ∀e

and ∃e and the free logic of actualism to represent the restricted quantificationover existing objects. We could either retain the full logic of actual and possibleobjects in that case, or we could restrict ourselves to just the actualist modallogic, depending on whether we want to represent modal possibilism or modalactualism.33

31As shown in Cocchiarella 1975, a parallel result holds for second-order monadic predicatelogic. That is, although second-order monadic predicate logic is decidable with respect to itsstandard set-theoretic semantics, it is not decidable with respect to a secondary semanticsthat allows for a “cut-down” on the notion of all values of the monadic predicate variables.

32Cf. Kripke 1971, p. 150.33See Cocchiarella 1984, section 6.

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We can also deepen the material content of the secondary semantics byadding a relation R of accessibility between possible worlds, i.e., between themodels in K, where 〈B,K〉 is a model structure, so that necessity in a world A ∈K is an invariance condition with respect to the worlds in K that are accessiblefrom A.34 If R is an equivalence relation, then the modal logic S5, whether asactualist or possibilist, characterizes the class of model structures that result.Other quantified modal logics characterize classes of model structures in whichthe accessibility relation between models is weaker than an equivalence relation.The modal logic S4, for example, characterizes the class of model structures inwhich the accessibility relation is transitive and reflexive.

The question remains, however, as to just what ontology is represented bythese kinds of set-theoretic semantics, and in particular what notion of necessityother than logical necessity is in question. Calling the result a metaphysicalmodality is not an adequate answer. We need a philosophical, and in particularan ontological, account of what principles determine the “cut-down” on possibleworlds (models), and how the accessibility relation between worlds is to beexplained in terms of such principles. A set-theoretic structure with respect towhich a completeness theorem can be proved is not itself such an ontological orphilosophical account.

In the ontology of conceptual realism, for example, the “cut-down” on pos-sible worlds can be accounted for either in terms of time, as in tense logic, orin terms of the network of natural laws that are part of nature’s causal matrix.Not all logically possible states of affairs will be realized in time, so that timeitself provides a metaphysical ground for such a “cut-down”. That is why theAristotelian and Diodorean modalities definable in terms of a local time areunproblematic. The earlier-than relation of a local time, or the signal relationof a causally connected system of local times also provide unproblematic meta-physical bases for different accessibility relations, as well as ontological groundsfor different modal logics.

Similarly, an ontological ground for such a “cut-down” on all logically pos-sible worlds can be given in terms of the set L of laws of nature. Thus, forexample, where K is the set of possible worlds of a multiverse35 in which all ofthe laws in L are true, and B ∈ K, the model structure 〈B,K〉 would character-ize an invariance condition based on the laws in L. Different sets of laws wouldthen determine different model structures.36 But because each model structure〈B,K〉 would be determined by the same set of laws, then all of the worlds inK would have the same laws of nature, and hence each would be accessible fromevery other member of K. The modal logic that results would then be S5.

34Ibid., section 7.35See Kaku 2005 for account of the multiverse, or megaverse, of parallel worlds as described

in modern cosmology, and especially as based on string theory and the inflationary universe.36According to the Astronomer Royal of Great Britain, Sir Martin Rees, “what’s conven-

tionally called ‘the universe’ could be just one member of an ensemble. Countless other waysmay exist in which the laws are different. The universe in which we’ve emerged belongs tothe unusual subset that permits complexity and consciousness to develop.” (Quoted in Kaku2005. p. 15.)

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3.6.1 The Concordance Model of a Multiverse

One example of a multiverse is what is known as the concordance model of cos-mology. The universe, as is now well-known, is expanding, and because of thatexpansion the most distant objects we can see now with the most powerful tele-scopes are about 40 billion lightyears (4× 1026 meters) away.37 A sphere of thisradius, and hence our universe as well, is called a Hubble volume; and, accordingto some cosmologists, beyond the Hubble horizon there are other universes withthe same laws of nature as ours, but with possibly different initial conditions.Space is assumed to be infinite on this model and because the distribution ofmatter is relatively uniform and represented by an ergodic random field it is alsoassumed that “there are infinitely many other regions the size of our observableuniverse, where every possible cosmic history is played out.”38 On this view thecurvature of space is flat, which excludes a curved, bounded, and hence finite,space-time such as Einstein once favored. Of course the universe could be hy-perspherical, and hence finite after all, but because it is so large it appears flatand Euclidean to us, just as a small part of the Earth’s surface appears flat tous. Nevertheless, on this model, it is assumed both that space is infinite andthat there are infinitely many physical objects in the multiverse; that is, eventhough there are only finitely many objects in each region, the total numberof objects is infinite because the number of regions is infinite. This multiverseis essentially just the cosmos, however, because if the cosmic expansion wereto decelerate then it would be physically possible to travel to regions beyondthe Hubble horizon. On the other hand, if the acceleration of the universe’sexpansion continues indefinitely, as most cosmologists today assume, then therate of expansion will exceed the speed of light and the different regions willamount essentially to different possible worlds, all with the same laws of natureas ours. The objects in these regions can then be considered as physically possi-ble objects, and quantifying over them would be different from quantifying overthe “actual” objects of our region. The logic of a formal ontology representingthis situation would then be an S5 modal logic with actualist and possibilistquantifiers.

3.6.2 The Multiverse of the Many-Worlds Model

Another example of a multiverse are the parallel worlds in the omnium ofthe many-worlds interpretation of quantum mechanics. In quantum mechan-ics (QM) each particle in the universe is associated with a probability wavethat specifies the different probabilities of where that particle might be locatedanywhere in the universe at each moment. Whether a particle is the same as itswave function, or whether the wave function is merely a mathematical construct

37See Tegmark [2003]. Although the universe is only 13.7 billion years old and the lightthat is now reaching us from the most distant stars took that many years to reach us, thosestars, because of the expansion of the universe, are now more than 13.7 billion lightyears away.They are in fact now 40 billion lightyears away.

38Ibid.

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that describes the particle’s motion is one of the issues that distinguishes dif-ferent versions of QM. In standard quantum mechanics, when a measurementis made and a particle is observed at a given location, then the probability offinding it at that location becomes 100 percent while the probability of finding itat any other location at that time drops to zero. This is what is meant in sayingthat the wave function “collapses”. The many-worlds interpretation, (MWI),denies that a particle’s wave function ever collapses.

Instead of a collapse of the wave function, what happens according to theMWI is that every potential outcome described in the particle’s probabilityfunction is realized in a separate parallel world, so that anything that couldhappen in the sense of being physically possible according to QM in fact doeshappen in some parallel world.39 All of the worlds accessible in this way from agiven world when a measurement is made at a given moment have the same pastup to that moment, but, except for the laws of nature, they differ thereafterin some way. An infinite number of parallel worlds populated by copies ofourselves is assumed in this way, where all of the worlds “co-exist” in a quantumsuperposition.40 Although the objects in those worlds are not “real” in the samesense in which the objects of our universe are real, nevertheless, they have anontological status as objects of the multiverse, or what following Roger Penrosemight preferably be called the omnium.41 This type of situation is representedin a formal ontology in terms of an S4 modal logic in which necessity andpossibility are based on what is physically possible in QM.42 In other words,we distinguish between quantifying over the real objects of our universe fromquantifying over the objects in other possible (parallel) worlds. There wouldthen seem to be infinitely many possible objects, i.e., objects that exist in somepossible world of the multiverse, even though in our universe there are onlyfinitely many objects.43

It is by these kinds of interpretations, or models of cosmology, that we cangive content to what is meant by a “cut-down” on the notion of all possibleworlds, and thereby on the notions of necessity and possibility.

The real problem of quantified modal logic for an ontology otherthan logical atomism is to give an ontological account of the “cut-down” on the notion of “all possible worlds” and of the accessibilityrelation between possible worlds.

39In Everett’s original version of the axioms of MWI no account was given of how thebranching into different parallel worlds takes place. Later proposals by Graham and DeWittintroduce the complicated notion of a measuring device that results in observations (by humansor automata) upon which the splitting into parallel worlds is based. See De Witt & Graham[1973].

40Penrose [2004], p. 784.41Penrose [2004], p.784.42The modal logic is S4 because the accessibility relation between possible worlds is a

partial ordering determined by the wavefunctions that split each universe into its relatedparallel universes. The result in effect is a branched-tree model of the universe something likethe semantics for S4 in terms of the causal-tense operators described in chapter 2, §10.

43For a fuller account of MWI see De Witt & Graham [1973].

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We have already indicated how such an account can be given for conceptualrealism in terms of time and also, as described above, in terms of a cosmologicalmodel of the multiverse. In a later chapter we will base a modal logic of anatural or causal necessity on the laws of nature or the causal mechanisms of ahierarchically structured universe of the natural kinds that make up the causalnexus of the world.

The question is: can this really be done other than in the modalitiesconstructible within tense logic on the basis of time or the causal ornatural modalities of a cosmological model of the multiverse?

3.7 Summary and Concluding Remarks

• Logical atomism provides a paradigmatic example of a formal ontology inwhich being is the same as existence.

• Logical atomism is also the paradigm of a formal ontology in which astrictly formal interpretation of logical necessity and possibility can be given.This is because it is only in an ontology of simple objects and simple propertiesand relations as the bases of logically independent atomic states of affairs thatan absolute totality of possible worlds is uniquely determined; and it is onlywith respect to this totality, as opposed to arbitrary sets of possible worlds,that logical necessity and possibility can be made sense of as modalities.

• The modal thesis of anti-essentialism is logically true in the formal ontologyof logical atomism.

• All de re logical modalities are reducible to de dicto logical modalities inthe formal ontology of logical atomism.

• Every logically necessary proposition is logically true and therefore a prioriin logical atomism, and every a posteriori proposition is logically contingent.

• If there are only properties, and hence only monadic states of affairs, thenthe formal ontology of logical atomism is both complete and decidable.

• But if there is at least one relation and infinitely many objects, then logicalatomism is essentially incomplete.

• A completeness theorem is forthcoming but only by allowing for arbitrary“cut-downs” on the totality of possible worlds.

• Allowing for arbitrary “cut-downs” on the totality of possible worlds in-troduces material content into the modalities of a formal ontology, which isantithetical to logical atomism, and results in something other than logical ne-cessity and possibility.

• The modal thesis of anti-essentialism is consistent, but not universallyvalid, in a semantics based on a “cut-down” to arbitrary non-empty sets ofpossible worlds.

• A “cut-down” on the set of possible worlds can be accounted for in termsof time, as in tense logic, where an ontological account can also be given of theaccessibility relation beween worlds as momentary states of a local time.

• A “cut-down” on the set of possible worlds can also be accounted for interms of the network of natural laws that determine nature’s causal matrix.

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But in that case we are no longer dealing with logical necessity and possibilitybut a causal necessity and possibility; and of course not every proposition thatis causally necessary is a priori, nor is every a posteriori proposition causallycontingent.

• There are alternative theories of a multiverse of possible worlds in whicha “cut-down” on the totality of possible worlds can be ontologically grounded:one, e.g., is the concordance model, which validates the laws of S5 modal logic,and another is the many-world interpretation of quantum mechanics, whichvalidates the laws of S4 modal logic.

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Chapter 4

Formal Theoriesof Predication

A formal ontology, we have said, is based on a theory of predication and noton set theory as a theory of membership. Set theory, of course, might be usedas a model-theoretic guide in the construction of a theory of predication; butsuch a guide can be misleading if we confuse membership with predication.Unlike set theory, a theory of predication depends on what theory of universalsit is designed to represent, where, by a universal we mean that which can bepredicated of things. A universal is not just an abstract entity such as a set,in other words, but something that has a predicative nature, which sets do nothave.

Our methodology in representing a theory of predication is to reconstructit as a second-order predicate logic that includes the salient features of thattheory. Now by a second-order predicate logic we mean an extension of first-order predicate logic in which quantifiers are allowed to reach into the positionsthat predicates occupy in formulas (sentence forms), as well as into the subject orargument positions of those predicates.1 This means that just as the quantifiersof first-order logic are indexed by object variables, which are said to be boundby those quantifiers, so too the quantifiers of second-order logic are indexed bypredicate variables, which are said to be bound by those quantifiers. In thisrespect we need only add n-place predicate variables, for each natural numbern, to our syntax for first-order logic. We will use for this purpose the capitalletters Fn, Gn, Hn, with or without numerical subscripts, as n-place predicatevariables; but we will generally drop the superscript when the context makesclear the degree of the predicate variable. Sometimes, for relational predicates,i.e., where n > 1, we will use R and S as relational predicate variables as well.

We still understand a formal language L to be a set of object and predicateconstants. The atomic formulas of such a language L are defined in the same way

1Here, by first-order predicate logic we mean the logic of possible objects described inchapter 2.

81

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as in first-order logic, except that now the predicate of an atomic formula mightbe a predicate variable instead of a predicate constant. Thus, the (second-order)formulas of a language L are understood to be defined as follows.

Definition: χ is a (second-order) formula of a language L if, andonly if, for all sets K, if (1) every atomic formula of L is in K, and (2) for allϕ, ψ ∈ K, all object variables x, and, for each natural number n, all n-placepredicate variables Fn, ¬ϕ, (ϕ→ ψ), (∀x)ϕ, and (∀Fn)ϕ ∈ K.

It is formally convenient, incidentally, to take propositional variables to be 0-place predicate variables, and hence to allow for atomic formulas of the form F 0.In other words, a propositional variable is a predicate variable that takes zeromany terms as arguments to result on an atomic formula. We will generally usethe capital letters P , Q, with or without numerical subscripts as propositionalvariables.

A principle of induction over second-order formulas follows of course just asit did in first-order logic. By way of axioms, we add the following to the ten wealready gave for standard first-order logic. These axioms are the distributionlaw for predicate quantifiers, and the law of vacuous quantification:

(A11) (∀Fn)[ϕ→ ψ] → [(∀x)ϕ → (∀x)ψ]

(A12) ϕ→ (∀Fn)ϕ, where Fn is not free in ϕ

We also add the inference rule of universal generalization for predicate quanti-fiers:

UG2: If � ϕ, then � (∀Fn)ϕ.

4.1 Logical Realism

The axioms described so far apply to nominalism and conceptualism, as well asto logical realism. What distinguishes logical realism is an axiom schema thatwe call an impredicative comprehension principle, (CP). This principle is statedas follows:

(∃Fn)(∀x1)...(∀xn)[F (x1, ..., xn) ↔ ϕ], (CP)

where ϕ is a (second-order) formula in which Fn does not occur free, andx1, ..., xn are pairwise distinct object variables occurring free in ϕ.

What the comprehension principle does in logical realism is posit the ex-istence of a universal corresponding to every second-order formula ϕ. This isso even when ϕ is a contradictory formula, e.g., ¬[G(x) → G(x)]. This is be-cause, contrary to what has sometimes been held in the history of philosophy,there are properties and relations, that on logical grounds alone cannot have anyinstances. Such properties and relations cannot be excluded without seriouslyaffecting the logic of logical realism. In particular, because we cannot effectivelydecide even when a first-order formula is contradictory, the logic would then notbe recursively axiomatizable.

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The comprehension principle, (CP), is said to be impredicative because itposits properties and relations in terms of totalities to which they belong. For-mally, this is a consequence of the fact that even formulas with bound predicatequantifiers are allowed to specify, i.e., “comprehend”, properties and relations.

Note that by a simple inductive argument on the structure of the formulaϕ, a second-order analogue of Leibniz’s law is provable independently of (CP):

(∀x1)...(∀xn)[F (x1, ..., xn) ↔ ϕ] → (ψ ↔ ψ[ϕ/F (x1, ..., xn)]),

where ψ[ϕ/F (x1, ..., xn)] is the result of properly substituting ϕ for the freeoccurrences of F in ψ with respect to the object variables x1, ..., xn.2 Fromthis, by UG2, axioms (A11), (A12) and elementary transformations, whatfollows as a theorem schema is the law of universal instantiation of formulas forpredicate variables:

(∀F )ψ → ψ[ϕ/F (x1, ..., xn)], (UI2)

where ϕ can be properly substituted for F in ψ. The contrapositive of (UI2)is of course the second-order law for existential generalization,

ψ[ϕ/F (x1, ..., xn)] → (∃F )ψ. (EG2)

The second-order predicate logic described so far, where the comprehen-sion principle (CP) is valid for any formula ϕ, is sometimes called “standard”second-order logic. The use of “standard” in this context should not be con-fused with the notion of a “standard” set-theoretic semantics for second-orderlogic, i.e., a semantics based on confusing predication with membership in aset D, where all sets of n-tuples drawn from the power-set of Dn are takenas the values of the n-place predicate variables.3 Second-order predicate logic,it is well-known, is essentially incomplete with respect to this so-called “stan-dard set-theoretic semantics”. But, as we have already noted (in chapter one),whether or not that incompleteness applies to the theory of universals that isthe basis of a second-order predicate logic is another matter altogether.

Now it is noteworthy that had we assumed (UI2) as an axiom schema insteadof (CP), then, by (EG2), the comprehension principle (CP) would be derivableas a theorem schema instead. This raises the question of whether or not thereare any reasons to prefer (CP) over (UI2), or (UI2) over (CP).

2If x1, ...xn are pairwise distinct object variables, then ψ[ϕ/F (x1, ..., xn)] is just ψ unlessthe following two conditions are satisfied: (1) no free occurrence of Fn in ψ occurs withina subformula of ψ of the form (∀a)χ, where a is a predicate or object variable other thanx1, ...xn that occurs free in ϕ; and (2) for all terms a1, ..., an, if F (a1, ..., an) occurs in ψ insuch a way that the occurrence of Fn in question is a free occurrence, then for each i such that1 ≤ i ≤ n, if ai is a variable, then there is no subformula of ϕ of the form (∀ai)χ in which thevariable xi has a free occurrence. If these two conditions are satisfied, then ψ[ϕ/F (x1, ..., xn)]is the result of replacing, for all terms a1, ..., an, each occurrence of F (a1, ..., an) in ψ at whichFn is free by an occurrence of ϕ(a1/x1, ..., an/xn). When conditions (1) and (2) hold, we saythat ϕ can be properly substituted for Fn in ψ.

3By Dn we understand the set of n-tuples drawn from D. In the so-called “standard”set-theoretic semantics for second-order logic, the power-set of Dn is taken as the range ofvalues assigned to n-place predicate variables. The set theory involved here is assumed to bebased on the iterative concept of set and Cantor’s power-set theorem.

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One reason to prefer (UI2) over (CP) is that whereas (CP) is existentialin form (UI2) is universal. Gottlob Frege, who was the first to formulate aversion of second-order logic, argued that the laws of logic should be universalin form, which is why he had a version of (UI2) as one of his basic laws.Bertrand Russell assumed a version of (EG2), the contrapositve of (UI2), asa “primitive proposition” of his higher-order logic. This principle, accordingto Russell, “gives the only method of proving ‘existence theorems’”4, whichsuggests that Russell did not think that “existence theorems” of the form (CP)were provable in his logic.

It is elegant, perhaps, to have the basic laws of logic all be universal in form,but there is something to be said for putting our existential posits up front, andthat is precisely what (CP) does—just as the related axiom (A8), (∃x)(a = x),for first-order logic puts our our existential presuppositions up front for objectualterms. The latter, i.e., axiom (A8), may in fact be rejected, as we will see, oncepredicates are allowed to be transformed into objectual terms as counterparts ofabstract nouns. In other words, in a somewhat larger ontological context thatwe will consider later, where complex objectual terms are allowed, certain ofthese complex objectual terms will lead to a contradiction if axiom (A8), whichis the first-order counterpart of (CP), is not modified along the lines of “freelogic”.

Can (CP) also be rejected in that larger ontological context? No, at least notin logical realism. In the ontology of natural realism, however, the assumptionthat a natural property or relation corresponds to any given predicate or formulais at best an empirical hypothesis. That is, just as whether or not an objectualterm denotes an object in free logic is not a question that can be settled bylogical considerations alone, so too in natural realism the question whether ornot a given predicate or open formula stands for a natural property or relationcannot be settled by logic alone. The comprehension principle (CP), in otherwords, is not a valid thesis in natural realism. We will forego giving a fulleranalysis of natural realism at this point, however, until a later lecture when weconsider conceptual natural realism.

The status of the comprehension principle, (CP), as these remarks indicate,is an important part of the question of what metaphysical theory of universalsis being assumed as the basis of our logic as a formal theory of predication.

4.2 Nominalism

In nominalism, the basic thesis is that there are no universals, and that thereis only predication in language. This suggests that the comprehension principle(CP) must be false in nominalism, which is why the formal theory of predicationthat is commonly associated with nominalism is standard first order predicatelogic with identity. In fact, however, the situation is a bit more complicatedthan that.

4Russell & Whitehead 1910, p. 131.

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It is true that according to nominalism first-order predicate logic gives alogically perspicuous representation of the predicative nature of the predicateexpressions of language. According to nominalism it is the logico-grammaticalroles that predicates have in the logical forms of first-order predicate logic thatexplains their predicative nature.5

Nominalism: The logico-grammatical roles that predicate expres-sions have in the logical forms of first-order predicate logic explainstheir predicative nature.

Predicate constants are of course assigned the paradigmatic roles in thisexplanation, but this does not mean that predicate constants are the only pred-icative expressions that must be accounted for in nominalism. In particular,any open first-order formula of a formal language L, relative to the free objectvariables occurring in that formula, can be used to define a new predicate con-stant of L.6 Such an open formula would constitute the definiens of a possibledefinition for a predicate constant not already in that language. Accordingly,an open formula must be understood as implicitly representing a predicate ex-pression of that formal language. Potentially, of course there are infinitely manysuch predicate constants that might be introduced into a formal language in thisway, and some account must be given in nominalism of their predicative role.

Question: how can nominalism, as a theory of predication, representthe predicative role of open first-order formulas.

Now an account is forthcoming by extending standard first order predicatelogic to a second order predicate logic in which predicate quantifiers are inter-preted substitutionally. That is, we can account for all of the nominalisticallyacceptable predicative expressions of an applied first-order language withoutactually introducing new predicate constants by simply turning to a secondorder predicate logic in which predicate quantifiers are interpreted substitution-ally and where predicate variables have only first-order formulas as their sub-stituents.7 There are constraints that such an interpretation imposes, of course,and in fact, as we have shown elsewhere, those constraints are precisely thoseimposed on the comprehension principle in standard “predicative” second-orderlogic.8 Here, it should be noted, the use of the word ‘predicative’ is based onBertrand Russell’s terminology in Principia Mathematica, where the restrictionin question was a part of his theory of ramified types. Apparently, because of

5Strictly speaking, nominalism deals primarily with predicate tokens, written or spoken,and not with predicates per se. The latter, however can be construed as classes as many ofpredicates of similar tokens, where classes as many are as described in chapter 11. We willignore that further complication of nominalism here.

6By an open formula we mean a formula in which some variables have a free occurrence.7Strictly speaking predicate variables will have as subtituends any formula in which no

bound predicate variable occurs, which means that free predicate variables are allowed tooccur in such substituends.

8See Cocchiarella 1980.

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the liar and other semantical paradoxes, Russell, despite his being a logical re-alist at the time, thought that only the so-called “predicative” formulas shouldbe taken as representing a property or relation.9

The restriction, simply put, is that no formula containing bound predicatevariables is to be allowed in the comprehension principle. The comprehensionprinciple (CP), in other words, is to be restricted as follows:

(∃Fn)(∀x1)...(∀xn)[F (x1, ..., xn) ↔ ϕ], (CP!)

where ϕ is a formula in which (1) no predicate variable has a bound occurrence,(2) Fn does not occur free in ϕ, and (3) x1, ..., xn are pairwise distinct objectvariables occurring free in ϕ.10

Under a substitutional interpretational the appearance of an existential positregarding the existence of a universal in the quantifier prefix (∃Fn) is just that,an appearance and nothing more. In an applied formal language, this princi-ple involves no ontological commitments under such an interpretation beyondthose one is already committed to in the use of the first-order formulas of thatlanguage. That is, by interpreting predicate quantifiers substitutionally, (CP!)will not commit us ontologically to anything we are not already committed to inour use of first-order formulas, and, as we have said, it is the logico-grammaticalrole of predicate expressions in first-order logic that is the basis of nominalism’stheory of predication.

Note that because the second-order analogue of Leibniz’s law is provableindependently of (CP), it then is provable in our nominalistic logic just as it wasin the logic for logical realism. But then, just as the universal instantiation law,(UI2), is provable in logical realism, we have a restricted version also provablein nominalism. That is, if no predicate variable has a bound occurrence in ϕ,then, by (CP!), the following is provable in nominalism,

(∀F )ψ → ψ[ϕ/F ], (UI!2)

where ϕ can be properly substituted for F in ψ. From (UI!2), we can thenderive the restricted version of existential generalization for predicates.

What these observations indicate is that a comprehension principlecan be used to make explicit what is definable in a given appliedlanguage, as well as to indicate, as in logical realism, what our exis-tential posits are regarding universals.

9Russell’s logical realism is most pronounced in his 1903 Principles of Mathematics. Hislater 1910 view in Principia Mathematica might more appropriately be described as a form ofconceptual Platonism, even though Russell never explicitly described himself this way. From1914 on, especially in his logical atomist phase, Russell is best described as a natural realist.See Cocchiarella 1991 for a description of Russell’s higher-order logic as a form of conceptualPlatonism.

10Russell used the exclamation mark as a way to indicate which formulas of his type theorywere “predicative”. We use it here in the metalanguage as a way to indicate the relevantrestriction on the comprehension principle.

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Thus, where L is a formal language, Pn is an n-place predicate constant notin L, and ψ is a first-order formula in which x1, ..., xn are all of the pairwisedistinct object variables occurring free, then

(∀x1)...(∀xn)[P (x1, ..., xn) ↔ ψ]

is said to be a possible definition in L of P . Now it is just such a possibledefinition that is posited in (CP!2). In other words, as the definiens of sucha possible definition, the first-order formula ψ is implicitly understood to be acomplex predicate of the language L. Of course, if the above were an explicitdefinition in L, then, by (EG!2), the relevant instance of (CP!) follows asprovable in L.

The kind of definitions that are excluded in nominalism but allowed in logicalrealism are the so-called “impredicative” definitions; that is, those that in realistterms represent properties and relations that seem to presuppose a totality towhich they belong. The definition of a least upper bound in real number theoryis such a definition, for example, because, by definition, a least upper bound ofa set of real numbers, is one of the upper bounds in that set.11 The exclusion ofimpredicative definitions is sometimes called the Poicare-Russell vicious-circleprinciple, because Henri and Bertrand Russell were the first to recognize andcharacterize such a principle.12

4.3 Constructive Conceptualism

The notion of an “impredicative” definition is important in conceptualism as wellas in nominalism, and it is basic to an important stage of concept-formation.Conceptualism, as we have noted, differs from nominalism in that it assumesthat there are universals, namely concepts, that are the semantic grounds forthe correct application of predicate expressions. Of course, conceptualism alsodiffers from realism in that concepts are not assumed to exist independently ofthe human capacity for thought and concept-formation.

Conceptualism is a sociobiologically based theory of the human capacity forthought and concept-formation, and, more to the point, systematic concept-formation. Concepts themselves are types of cognitive capacities, and it is theirexercise as such that underlies the speech and mental acts that constitutes ourthoughts and communications with one another. But thought and communica-tion exist only as coordinated activities that are systematically related to oneanother through the logical operations of thought; and it is with respect to theidealized closure of these operations that concept-formation is said to be sys-tematic. It is only as a result of this closure, moreover, that the unity of thoughtas a field of internal cognitive activity is possible.

11Similarly, the definition of the set of natural numbers as the intersection of all sets to which0 belongs and that are closed under the successor operator is also impredicative because it isdefined in terms of a totality (proper class) to which it belongs.

12Cf. Poincare 1906 and Russell 1906.

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The coordination and closure of concepts does not occur all at once in thedevelopment of human thought, of course, nor is the structure of the closurethe same at all stages in that development. In fact, the human child proceedsthrough stages of cognitive development that are of increasing structural com-plexity, corresponding in part to the increasing complexity of the child’s develop-ing brain. These stages, as Jean Piaget has noted, emerge as states of cognitiveequilibrium with respect to certain regulatory processes that are constitutiveof systematic concept-formation.13 Different stages proceed as transformationsfrom one state of cognitive equilibrium to another of increased structural com-plexity, where the need for such transformations arises from the child’s inter-action with his environment and the tacit realization of the inadequacy of theearlier stages to understand certain aspects of the world of his experience. Thelater stages are states of “increasing re-equilibration” of the intellect, in otherwords, so that the result is an improved representation of the world.14

Now there is an important stage of cognitive equilibrium of logical opera-tions that immediately precedes the construction of so-called “impredicative”concepts, which usually does not occur until post-adolescence. We refer to thelogic of this stage as constructive conceptualism. The later, succeeding moremature stage at which “impredicative” concept-formation is realized is calledholistic conceptualism, though we will generally refer to it later simply as con-ceptualism. It is in constructive conceptualism that “impredicative” definitionsare excluded, and this exclusion occurs in the form that the comprehensionprinciple takes in constructive conceptualism, which can be formally describedas follows:

(∀G1)...(∀Gk)(∃F )(∀x1)...(∀xn)[F (x1, ..., xn) ↔ ϕ], (CCP!)

where(1) ϕ is a pure second-order formula, i.e., one in which no nonlogical con-

stants occur,(2) F is an n-place predicate variable such that neither it nor the identity

sign occur in ϕ,(3) ϕ is “predicative” in nominalism’s purely grammatical sense, i.e., no

predicate variablehas a bound occurrence in ϕ,

(4) G1, ..., Gk are all of the distinct predicate variables occurring (free) in ϕ,and

(5) x1, ..., xn are pairwise distinct object variables.

Now we should note that by the rule for universal generalization of quanti-fiers, (UG), every instance of the conceptualist principle (CCP!) is derivablefrom the nominalist principle (CP!). But not every instance of the nominalistprinciple (CP!) is an instance of the conceptualist principle (CCP!). In anapplied formal language L, the formula ϕ in instances of (CP!) will in general

13Cp. Piaget 1977.14Ibid., p. 13 and §6 of chapter 1,

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4.3. CONSTRUCTIVE CONCEPTUALISM 89

be a first-order formula of L, and it may contain the identity sign and any pred-icate constant of L. Instances of (CCP!), on the other hand, contain neitherthe identity sign nor any predicate constants of L.

Predicate constants are excluded from instances of (CCP!) because, unlikethe situation in nominalism, a predicate constant (or first-order formula in termsof which such a constant might be defined) might not stand for a “predicative”concept, i.e., it might not stand for a value of the bound predicate variables.This is because the logic of predicate quantifiers in constructive conceptualismis like the logic of first-order quantifiers in free logic in that the logic is freeof existential presuppositions regarding predicate constants, which means thata predicate constant must stand for a “predicative” concept in order to be asubstituend of the bound predicate variables of the logic. The predicate quanti-fiers in nominalism, on the other hand, function like the objectual quantifiers ofstandard first-order logic; and that is because, as the paradigms of predicationin nominalism, predicate constants do not differ from one another in their pred-icative role, which is why, under a substitutional interpretation, all predicateconstants are substituends of the bound predicate variables.

Consider, for example, a language L containing ‘∈’ as a primitive two-placepredicate constant, and suppose we formulate a theory of membership in L withthe following as a second-order axiom:

(∀F )(∃y)(∀x)[x ∈ y ↔ F (x)]. (C)

Now, in nominalism, where predicate quantifiers are interpreted substitutionally,this axiom seems quite plausible as a thesis, stipulating in effect that everypredicate expression has an extension. But as plausible a thesis as that mightbe, it leads directly to Russell’s paradox. For, by the nominalist comprehensionprinciple, (CP!),

(∃F )(∀x)[F (x) ↔ x /∈ x], (D)

is provable under such an interpretation; and, because no predicate quantifieroccurs in x /∈ x, then, by (UI!2), x /∈ x represents a predicate expression thatcan be properly substituted for F in a universal instantiation of (C).

In constructive conceptualism, on the other hand, (D) is not an instanceof the conceptualist principle, (CCP!), and all that follows by Russell’s argu-ment from (C) is the fact that ‘∈’ cannot stand for a “predicative” (relational)concept. That is, instead of the contradiction that results when predicate quan-tifiers are interpreted substitutionally, (C), when taken as an axiom of a theoryof membership in constructive conceptualism, leads only to the result that themembership predicate does not stand for a “predicative” (relational) concept:

¬(∃R)(∀x)(∀y)[R(x, y) ↔ x ∈ y].

In other words, as a plausible thesis to the effect that every “predicative” concepthas an extension, (C) is consistent, not inconsistent, in constructive conceptu-alism.

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On the nominalist strategy, the notion of a “predicative” context is purelygrammatical in terms of logical syntax; that is, an open formula is “predica-tive” in nominalism just in case it contains no bound predicate variables. Inconstructive conceptualism, the notion of a “predicative” context is semanti-cal, which means that in addition to being “predicative” in nominalism’s purelygrammatical sense, it must also stand for a “predicative” concept. It is for thisreason that the second-order logic of constructive conceptualism must be free ofexistential presuppositions regarding predicate constants, which is why the bi-nary predicate, ‘∈’, in a theory of membership having (C) as an axiom, cannotstand for a value of the bound predicate variables.

In general, how we determine which, if any, of the primitive predicate con-stants of an applied language and theory stand for a “predicative” conceptdepends on the domain of discourse of that language and theory and how thatdomain is to be conceptually represented. In particular, those primitive predi-cates that are to be taken as standing for a “predicative” concept will be stipu-lated as doing so in terms of the “meaning postulates” of that theory, whereasthose that are not will usually occur in axioms that determine that fact.

Identity and its role in a logical theory marks another important differencebetween nominalism and constructive conceptualism. In nominalism, identity isdefinable in any applied language with finitely many predicate constants. Thisis because such a definition can be given in terms of a formula representingindiscernibility with respect to those predicate constants. Suppose, for exampleL is a language with two just two predicate constants, a one-place predicateconstant P , and a two-place predicate constant R. Then, identity can be definedin theories formulated in terms of L as follows:

a = b↔ [P (a) ↔ P (b)] ∧ [R(a, a) ↔ R(b, a)] ∧ [R(a, b) ↔ R(b, b)]∧[R(a, a) ↔ R(a, b)] ∧ [R(b, a) ↔ R(b, b)]

In other words, in any given application based on finitely many predicateconstants, which we may assume to be the standard situation, identity, in nom-inalism, is reducible to a first-order formula, which is why the identity sign isallowed to occur in instances of (CP!) under its nominalistic, substitutionalinterpretation. Such a definition will not suffice in constructive conceptualism,on the other hand, because the first-order formula in question, even were it tostand for a “predicative” concept, cannot justify the substitutivity of identicalsin nonpredicative contexts. The identity sign is not eliminable, or otherwise re-ducible, in constructive conceptualism, in other words, because, on the basis ofLeibniz’s law, identity must allow for full substitutivity even in nonpredicativecontexts. Thus, whereas,

x = y ↔ (∀F )[F (x) ↔ F (y)],

is provable in nominalism’s second-order logic, as based on its substitutionalinterpretation, the right-to-left direction of this same formula is not provable inthe logic of constructive conceptualism.

Finally, note that although “impredicative” definitions are not allowed innominalism, they are not precluded in the logic of constructive conceptualism.

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The difference is determined by the role free predicate variables have in eachof these frameworks. In nominalism, free predicate variables must be construedas dummy schema letters, which in an applied language stand for arbitraryfirst-order formulas of that theory. This means that the substitution rule,

if � ψ, then � ψ[ϕ/G(x1, ..., xn)],

is valid on the substitutional interpretation only when ϕ is “predicative” innominalism’s purely grammatical sense, i.e., only when no predicate variablehas a bound occurrence in ϕ. Indeed, the rule must be restricted in this waybecause, otherwise, by taking ψ to be the following instance of (CP!),

(∃F )(∀x1)...(∀xn)[F (x1, ..., xn) ↔ G(x1, ..., xn)],

we would be able to derive the full, unrestricted impredicative comprehensionprinciple, (CP), by substituting ϕ for G, and, thereby, transcend the substitu-tional interpretation of predicate quantifiers.

In the predicate logic of constructive conceptualism, on the other hand, theabove substitution rule is valid for all formulas, regardless whether or not theycontain any bound predicate variables.15 But, unlike the situation in nominal-ism, the validity of such a rule does not lead to the unrestricted impredicativecomprehension principle. In particular, the above instance of (CP!) is not alsoan instance of (CCP!). This means that the notion of a possible (explicit)definition of a predicate constant is broader in constructive conceptualism thanit is in nominalism, where only first-order formulas are allowed as definiens.

Nevertheless, on the basis of the rule of substitution for all formulas, it canbe shown that definitions in constructive conceptualism whose definiens con-tain a bound predicate variable will still be noncreative and will still allow forthe eliminability of defined predicate constants.16 Thus, even though construc-tive conceptualism validates only a “predicative” comprehension principle, i.e.,a comprehension principle encompassing laws of compositionality that are inaccordance with the vicious circle principle, it nevertheless allows for impred-icative definitions of predicate constants, that is, of predicate constants that donot stand for values of the bound predicate variables and that cannot thereforebe existentially generalized upon.

4.4 Ramification and Holistic Conceptualism

The difference between nominalism and constructive conceptualism, we havesaid, is similar to that between standard first-order logic and first-order logicfree of existential presuppositions regarding objectual terms. The freedom fromsuch presuppositions for predicates in constructive conceptualism indicates how

15A similar substitution rule for singular terms is also valid in free logic incidentally, i.e.,singular terms can be validly substituted for free object variables even though they cannot bevalidly substituted for bound object variables.

16See Cocchiarella 1986a, chap. 2, sec. 3, for a proof of this claim.

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concept-formation is essentially an open process, and that part of that processis a certain cognitive tension, or disequilibrium, between the predicates andformulas that stand for concepts at a given stage of concept-formation and thosethat do not. This disequilibrium in concept-formation is the real driving forceof what is known as “ramified” second-order logic, though, strictly speaking,ramified second-order logic is usually associated with nominalism and not withconceptualism.17

We can close the “gap” between predicates that stand for “predicative” con-cepts at a given stage of concept-formation and those that do not by introducingnew predicate quantifiers in addition to the original ones. These predicate quan-tifiers will still be within the confines of a restricted, constructive comprehensionprinciple. That is, a new comprehension principle would be added that allowedformulas containing predicate variables bound by the original predicate quan-tifiers, but not also formulas containing predicate variables bound by the newpredicate quantifiers. This will close the “gap” between those formulas thatstand for “predicative” concepts at the initial stage and those that do not, be-cause the latter now stand for “predicative” concepts at the new, second stage.

Of course in proceeding in this way we open up a new “gap” between theformulas that stand for “predicative” concepts at the new stage of concept-formation and those that do not. But then we can go on to close this new “gap”by introducing predicate quantifiers that are new to this stage, along with asimilar comprehension principle. This process that continues on in this way iswhat is known as “ramification”.

Formally, the process can be described in terms of a potentially infinite se-quence of predicate quantifiers ∀1, ∃1, ...,∀j , ∃j , ... (for each positive integer j),all of which can be affixed to the same predicate variables. The quantifiers (∀jF )and (∃jF ), where F is an n-place predicate variable, will then be understoodto refer to all, or some, respectively, of the n-ary “‘predicative”’ concepts thatcan be formed at the jth stage of the potentially infinite sequence of stages ofconcept-formation in question. But because open formulas representing “pred-icative” contexts of later stages will not be substituends of predicate variablesbound by quantifiers of an earlier stage, this means that the logic of the quan-tifiers ∀j and ∃j must be free of existential presuppositions regarding predicateexpressions, which is why the comprehension principle for this logic must beclosed with respect to all the predicate variables occurring free in the compre-hending formula. Thus, as applied at the jth stage, the ramified conceptualistcomprehension principle that is validated in this framework is the following:

(∀jG1)...(∀jGk)(∃jF )(∀x1)...(∀xn)[F (x1, ..., xn) ↔ ϕ], (RCCP!)

where(1) G1, ..., Gk are all of the predicate variables occurring free in ϕ;(2) F is an n-place predicate variable not occurring free in ϕ;

17This is because standard “predicative” logic has been associated with nominalism, andstandard ramified logic is an extension of standard “predicative” logic, and not of the free“predicative” logic of constructive conceptualism.

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4.4. RAMIFICATION AND HOLISTIC CONCEPTUALISM 93

(3) x1, ..., xn are distinct individual variables; and(4) ϕ is a pure ramified formula, i.e., one in which no nonlogical constants

occur and in which(a) the identity sign also does not occur and(b) in which no predicate variable is bound by a quantifier of a stage > j,

i.e., for all i ≥ j, neither ∀j nor ∃j occurs in ϕ.

The process of concept-formation that we are describing here amounts to atype of reflective abstraction that involves a projection of previously constructedconcepts onto a new plane of thought where they are reorganized under the clo-sure conditions of new laws of concept-formation characteristic of the new stagein question. This pattern of reflective abstraction is precisely what is representedby the ramified comprehension principle (RCCP!) and the logic of construc-tive conceptualism. Each successive stage of concept-formation in the ramifiedhierarchy is generated by a disequilibrium, or conceptual tension, between theformulas that stand for the “predicative” concepts of the preceding stage, asopposed to those that do not. Thus, in order to close the “gap” between for-mulas that stand for “predicative” concepts and those do not, we must proceedthrough a potentially infinite sequence of stages of concept-formation.18

Whatever the motivation for ramification in nominalism, it is clear that whatmoves us on from one stage of concept-formation to the next in constructiveconceptualism is a drive for closure, where all predicates stand for concepts.Such a closure cannot be realized in constructive conceptualism, of course, wherethe principal constraint guiding the formation of “predicative” concepts is theirbeing specifiable by conditions that are in accord with the so-called vicious circleprinciple. But the particular pattern of reflective abstraction that correspondsto this constraint is not all there is to concept-formation, and, in fact, as apattern that represents a drive for closure, it contains the seeds of its owntranscendence to a new plane or level of thought where such closure is achieved.

Concept-formation is not constrained by the vicious circle principle, in otherwords, because after reaching what Piaget calls the stage of formal operationalthought, certain new patterns of concept-formation are realizable, albeit usuallyonly in post-adolescence.19 One such pattern involves an idealized transition toa limit, where “impredicative” concept-formation becomes possible, i.e., wherethe restrictions imposed by the vicious-circle principle are transcended.

The idealized transition to a limit, in the case of our ramified logic,is conceptually similar to, but ontologically different from, an actualtransition to a limit at an infinite stage of concept-formation. Thisis a stage of concept-formation that, in effect, is not only the sum-mation of all of the finite stages of the ramified hierarchy but alsoone that is closed with respect to itself.

18See Cocchiarella 1986 for a more detailed discussion of this principle in constructive con-ceptualism.

19Cf. Piaget 1977.

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Ontologically, of course, there cannot be an infinite stage of concept-formation,but that is not to say that an idealized transition to a limit is conceptually im-possible as well. Indeed, such an idealized transition to a limit is precisely whatis assumed to be possible in holistic conceptualism, and it is possible, moreover,on the basis of the pattern of reflective abstraction represented by (RCCP!*).That is, in holistic conceptualism, the drive for closure upon which the pat-tern of reflective abstraction of ramified constructive conceptualism is based isfinally achieved, albeit only as the result of an idealized transition to a limitand not on the basis of an actual transition. In this way, conceptualism, bymeans of a mechanism of autoregulation that enables us to construct strongerand more complex logical systems out of weaker ones, is able to validate not onlythe ramified conceptualist comprehension principle but also the full, unquali-fied “impredicative” comprehension principle (CP) of “standard” second-orderlogic. There is no comparable mechanism in nominalism that can similarly leadto a validation of the impredicative comprehension principle (CP).

What is inadequate about the logic of constructive conceptualism, itis important to note, is that it cannot provide an account of the kindof impredicative concept-formation that is necessary for the develop-ment and use of the theory of real numbers, and which, as a matterof cultural history, we have in fact achieved since the nineteenthcentury.

The concept of a least upper bound, for example, or of the limit of a con-verging sequence of rational numbers, is an impredicative concept that was notacquired by the mathematical community until a little more than a century ago;and although, in our own time, it is not usually a part of a person’s conceptualrepertoire until post-adolescence, nevertheless, with proper training and con-ceptual development, it is a concept that most of us can come to acquire asa cognitive capacity. Yet, notwithstanding these facts of cultural history andconceptual development, it is also a concept that cannot be accounted for fromwithin the framework of constructive conceptualism. The constraints of the vi-cious circle principle, at least in the way they apply to concept-formation, simplydo not conform to the facts of conceptual development in an age of advancedscientific knowledge.

The validation of the full comprehension principle in holistic conceptualism,which we will hereafter refer to simply as conceptualism, does not mean thatthe logic of constructive conceptualism is no longer a useful part of conceptual-ism. What it does mean is that although all predicates stand for “predicable”concepts, not all predicates stand for “predicative” concepts, and that is a dis-tinction we can turn to in conceptualism whenever it is relevant and useful.

4.5 The Logic of Nominalized Predicates

Is there no difference then between logical realism and holistic conceptualism astheories of predication, other than the fact that the latter presupposes a logic

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4.5. THE LOGIC OF NOMINALIZED PREDICATES 95

of “predicative” concepts as a proper part? Well, in fact there is a differenceonce we consider the import of nominalized predicates and propositional formsas abstract singular terms in the wider context of modal predicate logic. Theuse of nominalized predicates as abstract singular terms is not only a part ofour commonsense framework, but it is also central to how both logical realismand conceptual intensional realism provide an ontological foundation for thenatural numbers and other parts of mathematics. This part of logical realismis sometimes called ontological logicism.

In Bertrand Russell’s form of logical realism, or ontological logicism, forexample, universals are not just what predicates stand for, but also what nom-inalized predicates, i.e., abstract nouns, denote as objectual terms.20 Here, bynominalization we mean the transformation of a predicate phrase into an ab-stract noun, which is represented in logical syntax as a objectual term, i.e., thetype of expression that can be substituted for first-order object variables. Thefollowing are some examples of predicate nominalizations:

is triangular � triangularity

is wise � wisdomis just � justice

It was Plato who first recognized the ontological significance of such a trans-formation and who built his ontology and his account of predication around it.In nominalism, of course, abstract nouns denote nothing.

In English we usually mark the transformation of a predicate into an abstractnoun by adding such suffixes as ‘-ity’, ‘-ness’, or ‘hood’, as with ‘triangularity’,‘redness’, and ‘brotherhood’. We do not need to introduce a special operatorfor this purpose in logical syntax, however. Rather, we need only delete theparentheses that are a part of a predicate variable or constant in its predicativerole. Thus, for a monadic predicate F we would have not only formulas such asF (x), where F occurs in its predicative role, but also formulas such as G(F ),R(x, F ), where F occurs nominalized as an abstract objectual term.

Note: In F (F ) and ¬F (F ), F occurs both in it predicative role andas an abstract objectual term, though in no single occurrence can itoccur both as a predicate and as a objectual term.

With nominalized predicates as abstract objectual terms, it is convenient tohave complex predicates represented directly by using Alonzo Church’s variable-binding λ-operator. Thus, where ϕ is a formula of whatever complexity and nis a natural number, we have a complex predicate of the form [λx1...xnϕ]( ),which has parentheses accompanying it in its predicative role, but which aredeleted when the complex predicate is nominalized. With λ-abstracts, the com-prehension principle can be stated in a stronger and more natural form as

(∃F )([λx1...xnϕ] = F ). (CP∗λ)

20Cf. Russell 1903, p. 43.

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96 CHAPTER 4. FORMAL THEORIES OF PREDICATION

This form is stronger than (CP) in that it implies, but is not implied by, thelatter.21 One of the rules for the new λ-operator is the rule of λ-conversion,

[λx1...xnϕ](a1, ..., an) ↔ ϕ[a1/x1, ..., an/xn] (λ-Conv∗)

The grammar of our logical syntax is now more complicated of course. Inparticular, objectual terms and formulas must now be defined simultaneously.For this purpose we will speak of a meaning expression of a given type n, wheren is a natural number. We will use 0 to represent the type of objectual terms(or just ‘terms’ for short), 1 to represent the type of formulas (propositionalforms), and n+ 1 to represent the type of n-place predicate expressions.

For each natural number n, we recursively define the meaningful expressionsof type n, in symbols, MEn, as follows:

1. every individual variable (or constant) is in ME0, and every n-place pred-icate variable (or constant) is in both MEn+1 and ME0;

2. if a, b ∈ ME0, then (a = b) ∈ME1;

3. if π ∈MEn+1 and a1, ..., an ∈ME0, then π(a1, ..., an) ∈ME1;22

4. if ϕ ∈ME1 and x1, ..., xn are pairwise distinct individual variables, then[λx1...xnϕ] ∈MEn+1;

5. if ϕ ∈ME1, then ¬ϕ ∈ME1;

6. if ϕ, χ ∈ME1, then (ϕ→ χ) ∈ME1;

7. if ϕ ∈ME1 and a is an individual or a predicate variable, then (∀a)ϕ ∈ME1;

8. if ϕ ∈ME1, then [λϕ] ∈ME0; and

9. if n > 1, then MEn ⊆ ME0.

By clause (9), every predicate expression without parentheses is a objectualterm. This includes 0-place predicates but not formulas in general unless theyare of the form [λϕ], which we take as the nominalization of ϕ, and which weread as ‘that ϕ’. For convenience, however, we shall write ‘[ϕ]’ for ‘[λϕ]’.23

It is noteworthy that this logical grammar contains what might described asthe essential parts of a theory of logical form: namely,

21The λ in (CP∗λ) indicates that a λ-abstract is part of this principle, and the ‘∗’ indicates

that nominalized predicates may occur in ϕ as singular terms.22If n = 0, we take a1, ..., an (and similarly x1, ..., xn) to be the empty sequence, resulting

in this case in a 0-place predicate expression, which, as already noted, we take to be a formula.23We could require that n be greater than 1 in clause (4)—in which case [λϕ] ∈ ME1 would

not follow—and then have clause (8) state that [ϕ] ∈ME0 when ϕ ∈ME1. But then generalprinciples—such as (CP∗

λ) and (�Ext∗) described below—that we want to apply to all n-placepredicate expressions would have to be stated separately for n = 0.

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4.5. THE LOGIC OF NOMINALIZED PREDICATES 97

• (1) the basic forms of predication, as in F (x), R(x, y), etc.;

• (2) propositional (sentential) connectives, e.g., ∧, ∨, →, and ↔;

• (3) quantifiers that reach into predicate as well as subject (or argument)positions;

• (4) nominalized predicates and propositional forms as abstract objectualterms.

These four components correspond to fundamental features of nat-ural language, and each needs to be accounted for in any theory oflogical form underlying natural language.

Now one of our goals here is to characterize a consistent second-order pred-icate logic with nominalized predicates and propositional forms as abstract ob-jectual terms. This goal is important because such a logic deals with the fourimportant features of natural language described above. Another goal is that asa framework for logical realism or (holistic) conceptualism such a logic shouldcontain all of the theorems of “standard” second-order predicate logic as a properpart.24 This means in particular that we should retain all of the theorems ofclassical propositional logic, and that all instances of the comprehension prin-ciple (CP) of “standard” second-order logic—i.e., instances in which abstractobjectual terms do not occur—should be provable. Initially, we will assumestandard first-order predicate logic with identity as well; but, as we will see, itmay be appropriate to assume “free” first-order predicate logic instead. Withstandard first-order predicate logic, we have by axiom (A8),

(∃y)(F = y)

as provable for every nominalized predicate F , and therefore also for λ-abstractsas well:

(∃y)([λx1...xnϕ] = y).

Another consequence is that the first-order principle of universal instantiationnow also applies to nominalized predicates as abstract objectual terms25:

(∀x)ϕ→ ϕ[F/x] (UI∗1)

It would be ideal, of course, if the comprehension principle (CP∗λ) can be

assumed for all formulas ϕ, including those in which nominalized predicates andpropositional forms occur as abstract objectual terms. But, if the logic is not“free of existential presuppositions” for objectual terms, such an unrestricted

24By the valid formulas of “standard” second-order logic we mean all of the second-orderformulas that are valid with respect to Henkin general models.

25We use ‘(UI∗1)’ to label this principle, with the subscript indicated that it is a first-orderobject quantifier thesis, and with the ‘∗’ to indicate that the principle applies to abstractsingular terms as well object variables and constants.

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98 CHAPTER 4. FORMAL THEORIES OF PREDICATION

second-order logic—which is similar to the system of Gottlob Frege’s Grundge-setze26—is subject to Russell’s paradox of predication, and therefore cannot beassumed as a consistent principle. Thus, e.g., where ϕ represents the Russellproperty of being identical to a property that is not predicable of itself27, whichas a λ-abstract can be formalized as

[λx(∃G)(x = G ∧ ¬G(x))],

then, by the unrestricted comprehension principle (CP∗λ),

(∃F )([λx(∃G)(x = G ∧ ¬G(x))] = F ) (1)

is provable, and therefore, by Leibniz’s Law, (LL∗), so is the weaker form,

(∃F )(∀x)[F (x) ↔ [λx(∃G)(x = G ∧ ¬G(x))](x)], (2)

which, by λ-conversion, is equivalent to

(∃F )(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]). (3)

But by (UI∗1),

(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]) → (F (F ) ↔ (∃G)[F = G ∧ ¬G(F )]),

and, by (LL∗),(∃G)[F = G ∧ ¬G(F )] ↔ ¬F (F )

are also provable, and therefore,

(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]) → [F (F ) ↔ ¬F (F )]

is provable as well. But, by sentential logic, the consequent of this last condi-tional is clearly impossible, which means that

¬(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]) (4)

is provable, and therefore, by (UG2) and a quantifier negation law,

¬(∃F )(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]), (5)

which contradicts (3) above.The above result is what is known as Russell’s paradox of predication. Rus-

sell himself later turned to his theory of ramified types to avoid the contradiction.Later, it was later pointed out that a theory of simple types sufficed, at least forthe so-called logical paradoxes such as Russell’s. The idea of a hierarchy of typesbased on the fundamental asymmetry of subject and predicate is fundamentallycorrect, we maintain. But, as we will see, the idea can be simplified even fur-ther within a strictly second-order predicate logic with nominalized predicatesas abstract objectual terms that is both consistent and equivalent to the simpletheory of types.

26Frege’s expressions for value-ranges (Wertverlaufe) were his formal counterparts of pred-icate nominalizations, i.e., formal counterparts of expressions such as ‘the concept F ’.

27See Russell 1903, p. 97.

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4.6. SUMMARY AND CONCLUDING REMARKS 99

4.6 Summary and Concluding Remarks

• A formal ontology is based on a formal theory of predication, which inturn is based on a theory of universals.

• The formal theories predication discussed in this chapter are logical real-ism (as a modern form of Platonism), nominalism, and conceptualism, wherethe latter is distinguished between a constructive conceptualism and a holisticconceptualism.

• Which comprehension principle is validated in a formal theory of predi-cation is one of its main distinguishing features. This is because a comprehen-sion principle determines what is definable in a given applied language, and italso indicates, as in logical realism, what our existential posits are regardinguniversals.

• Logical realism validates a full, unrestricted, impredicative comprehensionprinciple.

• Nominalism validates only a predicative comprehension principle, i.e., onein which only formulas free of bound predicate variables can occur as compre-hending formulas.

• Bound predicate variables can be interpreted only substitutionally in nom-inalism, which means that only first-order formulas (including those with freepredicate variables) can be properly substituted for such variables. Accordingto nominalism, it is the logico-grammatical roles that predicate expressions havein the logical forms of first-order predicate logic that explains their predicativenature.

• Constructive conceptualism validates only a predicative comprehensionprinciple but allows impredicative definitions. This is because, unlike nominal-ism, constructive conceptualism is free of “existential presuppositions” regardingfree predicate constants (and free predicate variables).

• Constructive conceptualism, unlike nominalism, also validates a ramifiedsecond-order predicate logic that is free of “existential presuppositions” regard-ing predicates. The freedom from such presuppositions indicates how concept-formation is essentially an open process. It is this open process and the concep-tual tension it generates for closure that drives concept-formation through theramified hierarchy.

• The drive for closure in constructive conceptualism contains the seeds ofits own transcendence to a new plane or level of thought where such closure isachieved.

• The conceptualist closure of the ramified hierarchy is achieved throughan idealized transition to a limit, where “impredicative” concept-formation be-comes possible, and hence where the restrictions imposed by the vicious-circleprinciple are transcended.

• The introduction of nominalized predicates as abstract singular termsmarks the critical extension of second-order predicate logics.

• The use of nominalized predicates as abstract singular terms is not only apart of our commonsense framework, but it is also central to how both logical

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100 CHAPTER 4. FORMAL THEORIES OF PREDICATION

realism and conceptual intensional realism provide an ontological foundation forthe natural numbers and other parts of mathematics.

• Nominalized predicates and propositional forms as abstract objectual termsare an essential part of a theory of logical form designed to represent naturallanguage.

• As a formal ontology logical realism or holistic conceptualism should con-tain all of the theorems of “standard” second-order predicate logic as a properpart.

• With λ-abstracts to represent complex predicates, the comprehension prin-ciple can be stated in a stronger and more natural form as (CP∗

λ) with identityin place of the biconditional.

• A second-order predicate logic with nominalized predicates, standard first-order logic, and a full, unrestricted comprehension principle leads to Russell’sparadox of predication.

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Chapter 5

Formal Theoriesof Predication Part II

5.1 Homogeneous Stratification

We saw at the end of the last chapter how Russell’s paradox of predication wasgenerated by simply extending the consistent framework of standard second-order predicate logic to include nominalized predicates as abstract terms.

The implications of this result for logical realism as a modern form of Pla-tonism were profound. How could mathematics be explained both ontologicallyand epistemologically if the formal theory of predication represented by second-order predicate logic with nominalized predicates as abstract singular termswere inconsistent?

This was the situation that confronted Russell in his 1903 Principles of Math-ematics. In fact, the form of logical realism that Russell had in mind in 1903was essentially the second-order predicate logic with nominalized predicates asabstract terms that we have described in the previous lecture.

Beginning in 1903, and for some years afterwards, Russell tried to resolvehis paradox in many different ways. It was not until 1908 that he settled on histheory of ramified types.

Now both the theory of ramified types and the later theory of simple typesavoid Russell’s paradox by setting limits on what is meaningful or significantin language. For that reason, and not without some justification, it has beenseverely criticized.

What the theory of simple logical types does is divide the predicate expres-sions (and their corresponding abstract singular terms) of the second-order logicdescribed in the previous chapter into a hierarchy of different types, and thenit imposes a grammatical constraint that nominalized predicates can occur asargument- or subject-expressions only of predicates of higher types. This purelygrammatical constraint excludes from the theory expressions of the form F (F ),or F ([λxF (x)]), as well as their negations, ¬F (F ), or ¬F ([λxF (x)]), which are

101

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just the types of expressions needed to generate Russell’s paradox.1 This wasall that Russell needed to avoid his paradox of predication; but, as a way toavoid the so-called semantical paradoxes—such as that of the liar—Russell alsodivided the predicates on each level of the hierarchy of types into a ramifiedhierarchy of orders. The simple theory of types, which is all that is needed toavoid Russell’s paradox, is based only on the first hierarchy, whereas the ramifiedtheory of types is based on both.

These grammatical constraints are undesirable, we want to emphasize, be-cause they exclude as meaningless many expressions that are not only grammat-ically correct in natural language but also intuitively meaningful, and sometimeseven true (such as ‘Being a property is a property’ or ‘Being red is not red’,etc.). Fortunately, it turns out, the logical insights behind these constraints canbe retained while mitigating the constraints themselves. In particular, we needimpose only a constraint on λ-abstracts, namely that they be restricted to thosethat are homogeneously stratified in a metalinguistic sense. Here, by a metalin-guistic characterization, we mean one that applies only in the metalanguage andnot as a distinction between types of predicates in the object language.

Retaining the same logical syntax that we described in the previous section,we say that a formula or λ-abstract ϕ is homogeneously stratified (or justh-stratified) if, and only if there is an assignment (in the metalanguage) t ofnatural numbers to the terms and predicate expressions occurring in ϕ (includ-ing ϕ itself if it is a λ-abstract) such that

• (1) for all terms a, b, if (a = b) occurs in ϕ, then t(a) = t(b);

• (2) for all n ≥ 1, all n-place predicate expressions π, and all termsa1, ..., an, if π(a1, ..., an) is a formula occurring in ϕ, then (i) t(ai) = t(aj),for 1 ≤ i, j ≤ n, and (ii) t(π) = t(a1) + 1;

• (3) for n ≥ 1, all objectual variables x1, ..., xn, and formulas χ, if [λx1...xnχ]occurs in ϕ, then (i) t(xi) = t(xj), for 1 ≤ i, j ≤ n, and (ii) t([λx1...xnχ]) =t(x1) + 1; and

• (4) for all formulas χ, if [χ] (i.e., [λχ]) occurs in ϕ and a1, ..., ak are all ofthe terms or predicates occurring in χ, then t([χ]) ≥ max[t(a1), ..., t(ak)].

The one constraint we need to impose to retain a consistent version of Rus-sell’s earlier 1903 logical realism is that, to be grammatically well-formed, all

1Instead of the λ-operator, Russell used a cap-notation, ϕ(x), to represent a propertyexpression as an abstract singular term. In Russell’s notation, type theory excluded formulasof the form ϕ(ϕ(x)) and ¬ϕ(ϕ(x)), which in our notation correspond to F ([λxF (x)]) and¬F ([λxF (x)]). In the logic described in the preceding section,

F = [λxF (x)]

is assumed to be valid.

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5.1. HOMOGENEOUS STRATIFICATION 103

λ-abstracts must be homogeneously stratified (in the metalinguistic sense de-fined). This means that formulas of the form F (F ) and ¬F (F ) are still gram-matically meaningful even though they are not h-stratified. Formulas of theform

F ([λxF (x)]), ¬F ([λxF (x)]),

[λxF (x)]([λxF (x)]), ¬[λxF (x)]([λxF (x)])

are also grammatically well-formed so long as the λ-abstracts in these formulasare h-stratified. On the other hand, the complex predicate that is involved inRussell’s paradox, namely,

[λx(∃G)(x = G ∧ ¬G(x))],

is not h-stratified, because, x and G must be assigned the same number (level)for their occurrence in x = G, whereas G must also be assigned the successor ofwhat x is assigned for their occurrence in ¬G(x).

The comprehension principle (CP∗λ) and the second-order logic of the pre-

vious section can be retained in its entirety, with the one restriction that theλ-abstracts that occur in the formulas of this logic must all be h-stratified.Because of this one restriction we will refer to the system as λHST∗.

Finally, let us note that not only is Russell’s paradox blocked in λHST∗,but so are other logical paradoxes as well. Indeed, as we have shown elsewhere,λHST∗ is consistent relative to Zermelo set theory and equiconsistent with thesimple theory of logical types.2 Also, if we were to add to λHST∗ the followingaxiom of extensionality,

(∀x1)...(∀xn)[F (x1, ...x,n ) ↔ G(x1, ...x,n )] → F = G, (Ext∗)

or, equivalently, because

Fn = [λx1...xnF (x1, ...x,n )],

is valid in λHST∗ (and in fact is taken as an axiom),

(∀x1)...(∀xn)[F (x1, ...x,n ) ↔ G(x1, ...x,n )] → (Ext∗)[λx1...xnF (x1, ...x,n )] = [λx1...xnG(x1, ...x,n )]

then the result is equiconsistent with the set theory known as NFU (New foun-dations with Urelements) as well.3

Metatheorem: λHST∗ is consistent relative to Zermelo set theory; and itis equiconsistent with the theory of simple logical types. In addition, λHST∗+(Ext∗) is equiconsistent with the set theory NFU.

2Cf. Cocchiarella 1986.3See Holmes 1999 for a development of NFU.

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5.2 Frege’s Logic Reconstructed

Gottlob Frege’s form of logical realism as described in his Grundgesetze was alsoa second-order predicate logic with nominalized predicates as abstract singularterms, and it too was subject to Russell’s paradox.4 But Frege also had ahierarchy of universals implicit in his logic, except that he assumed that allhigher levels of his hierarchy beyond the second could be reflected downwardinto the second level, which in turn was reflected in the first level of objects,which is the situation that is implicitly represented in λHST∗.5 In this respect,λHST∗ can also be used as a consistent reconstruction of Frege’s form of logicalrealism.

Unlike Russell, however, Frege did not assume that what a nominalizedpredicate denotes as an abstract singular term is the same universal that thepredicate stands for in its predicative role. In addition, Frege’s universals, whichhe called concepts (Begriffe) and relations, but which we will call propertiesand relations instead, have an unsaturated nature, and this unsaturated natureprecludes the properties and relations of Frege’s ontology from being objects.6

For this reason, Frege’s universals cannot be what nominalized predicates denoteas abstract singular terms. In other words, in Frege’s ontology what a predicatestands for in its predicative role is not what a nominalized predicate denotes asan abstract singular term.

Why then have nominalized predicates at all? In Frege’s ontology it wasnot just to explain an important feature of natural language. Rather, it was amatter of “how we are to conceive of logical objects,” and numbers in particular.7

“By what means,” Frege noted, “are we justified in recognizing numbers asobjects?” The answer, for Frege, was that we apprehend logical objects as theextensions of properties and relations (or concepts), and it is through the processof nominalization that we are able to achieve this. Here, it is the logical notionof a class as the extension of a property, or concept, that is involved, and notthe mathematical notion of a set.

Unlike a set, which has its being in its members, a class in the logicalsense has its being in the property, or concept, whose extension itis.8

Now it was Frege’s commitment to an extensional logic that led him to takeclasses as the objects denoted by nominalized predicates. A class, after all,is the extension of a predicate as well as of the property or concept that the

4See Cocchiarella 1987, chapter 2, section 6, and chapter 4, section 3, for a detailed defenseof the claim that Frege’s extensional logic of Wertverlaufe is really a logic of nominalizedpredicates.

5This reflection downward in Frege’s logic is what I have referred to elsewhere as Frege’sdouble correlation thesis. See Cocchiarella 1987, chapter two, section 9, for a discussion of thispart of Frege’s logic. Implicit in this reflection is a rejection of Cantor’s power-set theorem asapplied to Frege’s logic of extensions.

6Frege sometimes also described concepts as properties (Eigenshaften) as well.7Frege 1893, p. 143.8Cf. Frege 1979, p. 183.

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5.2. FREGE’S LOGIC RECONSTRUCTED 105

predicate stands for, and it was in terms of classes and classes of classes thatFrege proposed to construct the natural numbers. That, in fact, is the basis ofhis ontological logicism. Notationally, Frege used for this purpose the spirituslenis, or smooth-breathing symbol, as a variable-binding operator. Thus, givena formula ϕ and a variable ε, applying the spiritus lenis resulted in an expressionof the form εϕ(ε), which Frege took to be a nominalized form of the predicaterepresented by ϕ(ε). The smooth-breathing operator functioned in Frege’s logicin much the same way as the λ-operator does in the logics we have described,and for that reason we will continue to use the λ-operator here instead.

Given Frege’s commitment to an extensional logic, then it is not just λHST∗

that we should take as a consistent reconstruction of his logic, but λHST∗+(Ext∗), which, as already noted, is equiconsistent with the set theory NFUand consistent relative to Zermelo set theory.

The extensionality axiom, (Ext∗), incidentally, is one direction of Frege’swell-known Axiom V, which was critical to the way Russell’s paradox wasproved in Frege’s logic. This direction was called Basic Law Vb. The otherdirection, Basic Law Va, is actually an instance of Leibniz’s law in λHST∗.That is, by (LL∗), Frege’s Basic Law Va,

F = G→ (∀x1)...(∀xn)[F (x1, ...x,n ) ↔ G(x1, ...x,n )]

is provable in λHST∗, independently of (Ext∗), which was Frege’s Basic LawVb. Given the consistency of λHST∗+ (Ext∗) (relative to Zermelo set theory),which includes Leibniz’s law, (LL∗), as a theorem schema, it is not Frege’s BasicLaw V that was the problem for Frege so much as the way his hierarchy ofuniversals was reflected downward into the first- and second-order levels. Thiswas because Frege had heterogeneous, and not just homogeneous, relations in hislogic, including heterogenous relations between universals and objects, such asthat of predication, and these were included as part of the reflection downwardof his hierarchy.9 The hierarchy consistently represented in λHST∗, on thehand, consists only of homogeneous relations.

The representation of heterogenous relations can be retained, however, byturning to an alternative reconstruction of Frege’s logic that is closely related toλHST∗. This alternative involves replacing the standard first-order logic that ispart of λHST∗ with a logic that is free of existential presuppositions regardingobjectual terms, including nominalized predicates such as that corresponding tothe complex predicate involved in Russell’s paradox.

Now it is significant that in an appendix to his Grundgesetze Frege consideredresolving Russell’s paradox by allowing that “there are cases where an unexcep-tional concept has no extension”.10 Here, by an “unexceptional concept” Fregehad Russell’s rather exceptional concept, or property, in mind. After all, whatis exceptional about the Russell concept, or property, in Frege’s logic is thatit leads to a contradiction, unless, that is, we allow that it has no extension.But allowing that the Russell property has no extension in Frege’s logic requires

9See Cocchiarella 1987, section 9, for a more detailed discussion of this point.10Frege 1893, p. 128.

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allowing the nominalized form of the Russell predicate to denote nothing. Thatis, it requires a shift from standard first-order logic to a logic free of existen-tial presuppositions regarding objectual terms, including especially nominalizedpredicates. In other words, instead of axiom (A8) of the logic of possible objectsin chapter 2, namely,

(∃x)(a = x),

where x does not occur in a, we now have

(∀x)(∃y)(x = y),

where x, y are distinct object variables. In other words, the logic of possibleobjects is now a “free logic,” just as the logic of actual objects is a free logic.We want to use this logic as our “free” logic because although abstract objecthave being as values of the bound object variables, nevertheless, they do not“exist” in the concrete sense of existence, a restricted notion of being that wewant to retain in an extended development in chapter 6 of the second-order logicwith nominalized predicates that we are now considering.

In fact, this strategy works. By adopting a free first-order logic and yetretaining the unrestricted comprehension principle (CP∗

λ), all that follows bythe argument for Russell’s paradox is that there is no object corresponding tothe Russell property, i.e.,

¬(∃y)([λx(∃G)(x = G ∧ ¬G(x))] = y)

is provable, even though, by (CP∗λ), the Russell concept, or property, “exists”

as a concept, or property; that is, even though

(∃F )([λx(∃G)(x = G ∧ ¬G(x))] = F )

is also provable. Because the first-order logic is now a “free” logic, the originalrule of λ-conversion must be modified as follows:

[λx1...xnϕ](a1, ..., an) ↔ (∃x1)...(∃xn)(a1 = x1 ∧ ... ∧ an = xn ∧ ϕ),(∃/λ-Conv∗)

where, for all i, j ≤ n, xi does not occur in aj .Revised in this way, our original second-order logic with nominalized pred-

icates can be easily shown to be consistent. But that is because, without anyfurther assumptions, we can no longer prove that any property or relation hasan extension. That is, because the logic is free of existential presuppositions,all nominalized predicates might be denotationless, a position that a nominalistmight well adopt.

But in Frege’s ontological logicism some properties and relations must haveextensions, and, indeed, it would be appropriate to assume that all of the prop-erties and relations that can be represented in λHST∗ have extensions in thisalternative logic. That in fact is exactly what we allow in our alternative recon-struction of Frege’s logic.

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The added assumption can be stipulated in the form of an axiom schema.But to do so we need to first define the key notion of when an expression of ourlogical grammar can be said to be bound to objects .

Definition: If ξ is a meaningful expression of our logical grammar, i.e.,ξ ∈ MEn, for some natural number n, then ξ is bound to objects if, andonly if, for all predicate variables F , and all formulas ϕ ∈ ME1, if (∀F )ϕ is aformula occurring in ξ, then for some object variable x and some formula ψ, ϕis the formula [(∃x)(F = x) → ψ].

To be bound to objects, in other words, every predicate quantifier occurringin an expression ξ must refer only to those properties and relations (or concepts)that have objects corresponding to them, which in Frege’s logic are classes asthe extensions of the properties or relations in question. The axiom schema weneed for this is given as follows:

(∃y)(a1 = y) ∧ ... ∧ (∃y)(ak = y) → (∃y)([λx1...xnϕ] = y), (∃/HSCP∗λ)

where,

• (1) [λx1...xnϕ] is h-stratified,

• (2) ϕ is bound to objects,

• (3) y is an object variable not occurring in ϕ, and

• (4) a1, ..., ak are all of the object or predicate variables or nonlogical con-stants occurring free in [λx1...xnϕ].11

Because of it close similarity to our first reconstructed system, λHST∗, wewill refer to this alternative logic as HST∗

λ.12 As we have shown elsewhere,HST∗

λ is equiconsistent with λHST∗, and therefore with the theory of simpletypes as well. It is of course also consistent relative to Zermelo set theory.

Finally, let us note that although HST∗λ + (Ext∗) can be taken as a recon-

struction of Frege’s logic and ontology, it cannot also be taken as a reconstruc-tion of Russell’s early (1903) ontology, even without the extensionality axiom,(Ext∗). This is because Russell rejected Frege’s notion of unsaturatedness andassumed that nominalized predicates denoted as singular terms the same con-cepts and relations they stand for as predicates. In other words, unlike Frege,Russell cannot allow that some predicates stand for properties and relations (orconcepts), but that as objectual terms their nominalizations denote nothing.Of course, we do have the logical system λHST∗, which can be taken as areconstruction of Russell’s early ontological framework.

11We understand the conditional posited in this axiom schema to reduce to just the conse-quent if k = 0, i.e., if [λx1...xnϕ] contains no free variables or nonlogical constants.

12For a detailed account of all of the axioms of these systems see Appendix 1 of chapter 6,and for an account of their various properties see Cocchiarella 1986, chapter V.

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5.3 Conceptual Intensional Realism

In conceptualism, predicable concepts are cognitive capacities that underlie ourrule-following abilities in the use of the predicate expressions of natural lan-guage, and, as such, concepts determine the truth conditions of that use. More-over, as capacities that can be exercised by different persons at the same time,as well as by the same person at different times, concepts cannot be objects,e.g., ideas or mental images as particular mental occurrences. In other words,as intersubjectively realizable cognitive capacities, concepts are objective andnot merely subjective entities.

Moreover, as essential components of predication in language and thought,concepts as cognitive capacities have an unsaturated nature, and it is this un-saturated nature that is the basis of predication in language and thought. Inparticular, it is the exercise of a predicable concept in a speech or mental actthat informs that act with a predicable nature, a nature by means of which wecharacterize and relate objects in various ways.

The unsaturatedness of a concept as a cognitive capacity is not the same asthe unsaturatedness of a universal in Frege’s ontology. For Frege, a property(Begriff, Eigenshaft) or relation is really a function from objects to truth values,and it is part of the nature of every function, according to Frege, even thosefrom numbers to numbers, to be unsaturated. Predication in other words, isreduced to functionality in Frege’s ontology. In conceptualism, on the otherhand, it is predication that is more fundamental than functionality.

We understand what it means to say that a function assigns truth valuesto objects, after all, only by knowing what it means to predicate concepts, orproperties and relations, of objects. The unsaturated nature of a concept isnot that of a function, but of a cognitive capacity that could be exercised bydifferent people at the same time as well as by the same person at differenttimes, and it might in fact not even be exercised ever at all. When such acapacity is exercised , however, what results is not a truth value, whatever sortof entity that might be, but a mental event, and if expressed overtly in languagethen a speech-act event as well.

Now we should note that conceptual thought consists not just of predicableconcepts, but of referential and other types of concepts as well. Referentialconcepts, for example, are cognitive capacities that underlie our ability to re-fer (or really to purport to refer) to objects, and, as such, they too have anunsaturated cognitive structure. More importantly, referential concepts have astructure that is complementary to that of predicable concepts, so that each,when exercised or applied jointly in a judgment, mutually saturates the other,resulting thereby in an act (event) that is informed with a referential and apredicable nature. As we will explain in a later chapter, it is the complemen-tarity of predicable and referential concepts as unsaturated cognitive structuresthat is the basis of the unity of our speech and acts, and which explains why atranscendental subjectivity need not be assumed as the basis of this unity. Inthis lecture, however, we will restrict ourselves to the predicable concepts thatare the counterparts of the universals of logical realism.

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5.3. CONCEPTUAL INTENSIONAL REALISM 109

If predicable concepts are unsaturated cognitive structures, then what pointis there in having a logic of nominalized predicates as abstract objectual terms?One point is the same as it was for Frege, namely to account for the ontology ofthe natural numbers as logical objects. Another is to explain the significance ofnominalized predicates in natural language, including especially complex formsof predication containing infinitives, gerunds, and other abstract nouns. Thereis a difference with Frege’s ontology, however, in that whereas Frege was com-mitted to an extensional logic, conceptualism, once it admits abstract objectsinto its ontology, is committed to an intensional logic. Instead of denoting theextensions of concepts, in other words, nominalized predicates in conceptual-ism denote the intensional contents of concepts, which is why we refer to thisextension of conceptualism as conceptual intensional realism.

Now by the intensional content of a predicable concept we understand anabstract intensional object corresponding to the truth conditions determinedby the different possible applications of that concept, i.e., the conditions underwhich objects can be said to fall under the concept in any possible context ofuse, including fictional contexts. Of course, there are some predicable concepts,such as that represented by the Russell predicate,

[λx(∃G)(x = G ∧ ¬G(x))],

that determine truth conditions corresponding to which, logically, there canbe no corresponding abstract object, on pain of contradiction. This does notmean that such a predicable concept does not determine truth conditions andtherefore does not have intensional content; rather, it means only that such acontent cannot be “object”-ified, i.e., there cannot be an abstract object cor-responding to the content of that concept, the way there are for the contentsof other concepts. The real lesson of Russell’s paradox is that some rather ex-ceptionable, “impredicatively” constructed concepts determine truth conditionsthat logically cannot be “object”-ified, whereas most predicable concepts areunexceptionable in this way.

In the alternative ontology of conceptual Platonism, incidentally, the ab-stract object corresponding to the intensional content of a predicable concept isa Platonic Form, which traditionally has also been called a property or relation—a terminology that we can allow as well in conceptual realism so long as we donot confuse these properties and relations with the natural properties and rela-tions of conceptual natural realism. There is an important ontological differencebetween conceptual Platonism and conceptual intensional realism, however, de-spite the similarity of both to logical realism.

Unlike logical realism, conceptual Platonism is an indirect and not a directPlatonism. That is, in conceptual Platonism, but not in logical realism, abstractobjects are cognized only indirectly through the concepts whose correlates theyare. This means that our representation of abstract objects is seen as a reflexiveabstraction corresponding to the process of nominalization. In other words, eventhough abstract objects according to conceptual Platonism exist in a realm thattranscends space, time and causality, and in that sense preexist the evolution of

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110 CHAPTER 5. FORMAL THEORIES OF PREDICATION PART II

consciousness and the cognitive capacities that we exercise in thought and ouruse of language, nevertheless, from an epistemological point of view, no suchentity can be cognized otherwise than as the correlate of a concept, i.e., as anabstract intensional object corresponding to the truth conditions determined bythat concept.

In conceptual intensional realism, on the other hand, all abstract objects,despite having a certain autonomy, are products of language and culture, and inthat respect do not preexist the evolution of consciousness and the cognitive ca-pacities that we exercise in language.13 They are, in other words, evolutionaryproducts of language and culture, and therefore depend ontologically on lan-guage and culture for their “existence,” or being. Of course, abstract objects,especially numbers, are also an essential part of the means whereby furthercultural development becomes possible. Nevertheless, as cultural products, the“existence” of abstract objects is primarily the result, and development of, thekind of reflexive abstraction that is represented by the process of nominaliza-tion. It was through the institutionalization of this process that abstract objectsachieved a certain autonomy and, in time, became reified as objects. Abstractobjects do not exist in a Platonic realm outside of space, time, and causality,on our interpretation, but are in fact the result, in effect, of an ontological pro-jection inherent in the development and institutionalization in language of theprocess of nominalization.

The fundamental insight into the nature of abstract objects according to con-ceptual intensional realism is that we are able to grasp and have knowledge ofsuch objects as the “object”-ified truth conditions of the concepts whose contentsthey are, i.e., as the object correlates of those concepts. This “object”-ificationof truth conditions is realized, moreover, through a kind of reflexive abstractionin which we attempt to represent what is not an object—in particular an un-saturated cognitive structure underlying our use of a predicate expression—asif it were an object. In language this reflexive abstraction is institutionalized inthe rule-based linguistic process of nominalization.

Finally, we note that we can take the system HST∗λ as the core part of the

formal ontology of conceptual realism (or of conceptual Platonism), as well asof Frege’s form of logical realism. Of course Frege’s logical realism differs fromconceptual realism in having the extensionality axiom, (Ext∗), as part its coreas well. There are other difference as well, which we take up in the next chapter.

5.4 Hyperintensionality

There might be a problem with the systems λHST∗ and HST∗λ, it should be

noted, in that they do not seem to adequately respect intentional contexts suchas belief, desire, etc.—at least not if such contexts must be “hyperintensional,”

13Cf. Popper and Eccles 1977, chap. P2, for a similar view. We should note, however,that although our view of abstract objects supports the Popper-Eccles interactionist theoryof mind, it does not also depend on that theory.

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5.4. HYPERINTENSIONALITY 111

or more “fine-grained” in structure than is allowed, e.g., by logical equivalence.The claim is that logically equivalent formulas do not in general preserve truthwhen interchanged in intentional contexts, and that only a hyperintensionallogic will suffice for that purpose. A hyperintensional logic is one whose identitycriteria for intensional objects is as “fine-grained” as possible, which means, forexample, that conjunctive or disjunctive complex predicates are identical onlywhen the first and second conjunct, or disjunct, of each is identical with thefirst and second conjunct, or disjunct, respectively, of the other; and the samecondition applies to conditionals and biconditionals, and quantifier phrases aswell.

The argument that neither λHST∗ nor HST∗λ can be developed as a hy-

perintensional logic is a result of a so-called “hyperintensional paradox”, whichdepends on adding the following hyperintensional assumptions as axioms tothese systems:14

(Hyper1):[λx1...xn(Qa)ϕ] = [λx1...xn(Qa)ψ] → (∀a)([λx1...xnϕ] = [λx1...xnψ])

where Q is either ∀ or ∃ and a is a predicate or objectual variable.(Hyper2):

[λx1...xn(ϕ� ψ)] = [λx1...xn(ϕ′ � ψ′)] → [λx1...xnϕ] = [λx1...xnϕ′]∧

[λx1...xnψ] = [λx1...xnψ′]

where � is any of the sentential connectives ∧,∨,→, or ↔.(Hyper3):

(∀F )(∀G)((∀a)([λx1...xnF (a, x1, ..., xn)] = [λx1...xnG(a, x1, ..., xn)]) →[λx1...xnF (x1, ..., xn)] = [λx1...xnG(x1, ..., xn)]),

where a is a predicate or objectual variable.(Hyper3) is an implausible assumption, as we explain below, and (Hy-

per1) is in conflict with our conceptual realist account of the copula, which wewill take up in Part II.15 We do not, indeed cannot, accept these assumptions,but for now we can state the thesis in question as the following theorem.Theorem: λHST∗+ (Hyper1) + (Hyper2) + (Hyper3) and

HST∗λ+(Hyper1) + (Hyper2) + (Hyper3) are inconsistent.

Proof: Let

H = [λy(∃F )(y = [λz(∀G)(F (G) → G(z)) ∧ ¬F (y))]

14See Bozon 2004 for the original formulation of this paradox, which is patterned after onegiven for simple type theory.

15In section 8.5 we introduce Is as a special predicate for the copula and in HST∗λ take the

following as a meaning postulate

[λxIs(x, [∃yA])] = [λx(∃yA)(x = y)].

The Is predicate is not h-stratifiable. Our use of it, however, is only for the initial level ofanalysis of predication in speech and mental acts; that is, we do not need its objectification ornominalization. In any case, applying (Hyper3) to this meaning postulate would yield theresult that (∃yA)(x = y), which says that x is an A, is identical with x = [∃yA], where [∃yA]is the intensional content of the phrase ‘an A′, which is false when x is a concrete object.

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112 CHAPTER 5. FORMAL THEORIES OF PREDICATION PART II

andH∧ = [λz(∀G)(H(G) → G(z))].

Both H and H∧ are h-stratified,16 and therefore well-formed in λHST∗ and“object”-fiable in HST∗

λ. Suppose first that H(H∧). Then, by definition of Hand λ-conversion,

(∃F )(H∧ = [λz(∀G)(F (G) → G(z)] ∧ ¬F (H∧)).

That is,H∧ = [λz(∀G)[F (G) → G(z)] ∧ ¬F (H∧)),

for some value of F , and hence by definition of H∧ and Leibniz’s law,

[λz(∀G)(H(G) → G(z))] = [λz(∀G)(F (G) → G(z))].

Therefore, by (Hyper1),

(∀G)([λz(H(G) → G(z))] = [λz(F (G) → G(z))]),

and hence, by (Hyper2),

(∀G)([λzH(G)] = [λzF (G)] ∧ [λzG(z)] = [λzG(z)]),

and in particular,(∀G)([λzH(G)] = [λzF (G)]).

But then, by (Hyper3),[λzH(z)] = [λzF (z)], and therefore, by axiom (Id∗λ)

of both λHST∗ and HST∗λ,17 it follows that (H = F ) and hence, by Leibniz’s

law, ¬H(H∧), contrary to our initial assumption, which is impossible.But now given ¬H(H∧), it follows by definition of H and λ-conversion (or

(∃/λ-Conv∗ in HST∗λ),

(∀F )(H∧ = [λz(∀G)(F (G) → G(z))] → F (H∧)),

and therefore by universal instantiation,

H∧ = [λz(∀G)(H(G) → G(z))] → H(H∧),

from which, by definition ofH∧, it follows thatH(H∧), which is impossible given¬H(H∧). In other words, the above hyperintensional axioms together with thedefinitions of H and H∧ lead to the following impossible result: H(H∧) ↔¬H(H∧).

The above use of (Hyper3) in the inference from (∀G)([λzH(G)] = [λzF (G)])to [λzH(z)] = [λzF (z)] indicates how implausible (Hyper3) is as a logical prin-ciple. After all, even if H(z) and F (z) are identical for all values of z that are

16Both H and H∧, however, are impredicatively specified and would not be allowed inconstructive conceptualism’s ramified logic, which can be developed into a hyperintensionallogic.

17See appendix 1 of chapter 6 for a description of the axioms of λHST∗ and HST∗λ ,

including (Id∗λ).

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5.4. HYPERINTENSIONALITY 113

properties, it does not follow that H(z) and F (z) are identical for all objects asvalues for z, i.e., that [λzH(z)] = [λzF (z)]. If we reject this use of (Hyper3),then one important step in the above proof is no longer acceptable.

The supposed conclusion of this “paradox”, as we have said, is that, likethe simple theory of types, neither λHST∗ nor HST∗

λ can serve as the basisof a hyperintensional logic if (Hyper1)–(Hyper3) are assumed as principlesof hyperintensionality. The more restrictive predicative logic of constructiveconceptualism described in section five of the previous chapter can suffice forthis purpose, however, because the above argument depends essentially on im-predicative expressions that are not applicable in predicative logic, ramified orotherwise.

Despite the implausibility of (Hyper3) for λHST∗ and HST∗λ, and of

(Hyper1) for HST∗λ, the question we want to turn to here is not whether or

not (Hyper1)–(Hyper3) are appropriate principles of hyperintensionality, butwhether or not either logical realism or conceptual (intensional) realism mustbe committed to hyperintensionality. This is a relevant question because, intu-itively, the most natural interpretation for a hyperintensional logic is a strictlysyntactical one in which identity means identity of constituent expressions, ex-cept perhaps for rewrite of bound variables. That interpretation, of course, isinappropriate for either logical or conceptual realism. A closely related inter-pretation is one that assumes an ontology in which properties and relations areeither absolutely simple or intrinsically complex, and in which the latter arebuilt up from the former by means of ontologically real counterparts of the log-ical constants that occur in their syntactic representations in a formal ontology.We will call this a simples⊕complex ontology to distinguish it from one, suchas logical atomism, in which there are simple properties and relations but nocomplex ones.

Now a simples⊕complexes ontology does not seem appropriate for Frege’svariant of logical realism. That is because the properties—or what Frege called“concepts”—and relations of Frege’s realism are functions from objects to truthvalues, and as such they do not in themselves contain ontological counterparts ofthe logical constants as constituents. The property, or function, correspondingto the predicate [λx(F (x)∨G(x))], e.g., does not contain a disjunction operationas a constituent of the function, even though the values of the function are de-termined in terms of a disjunction. The function itself is “simply” a correlationof truth values with objects. A property, or function, may be represented bya complex predicate expression, in other words, without itself having an onto-logical complexity corresponding to that of the predicate expression. A similarobservation applies, of course, to properties and relations represented by con-junctive and other complex predicate expressions. Functions in general consistonly of correlations from arguments to values and, unless a function is an ar-gument or a value of another function, it is not a constituent of that functioneven if its values are arguments of that function.

Now Bertrand Russell in his early form of logical realism as described inhis 1903 Principles of Mathematics certainly seems to have assumed an ontol-ogy of simples and complexes, where by a complex he means an object whose

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“being presupposes the being of certain other terms,” namely objects that areconstituents of the complex.18 But his complexes are of only two kinds, namely,nonempty classes as ones, which are complexes in that the being of such a classpresupposes the being of its members. Of course, hyperintensionality cannotbe said to apply to classes as ones, which are strictly extensional entities. Theother kind of complex entities recognized by Russell are propositions. “For ex-ample, ‘A differs from B ’, or ‘A’s difference from B ’, is a complex,” accordingto Russell, “of which the parts are A and B and difference.”19 Nothing is saidin the Principles about properties or relations also being complex, and Russell’sstatement that there are only two kinds of complexes, namely classes as onesand propositions, seems to exclude there being complex properties and relations.

What about propositional functions? Are they not complex properties andrelations? Well, certainly not in Russell’s 1903 ontology. That is because, in1903, propositional functions are not themselves objects, or entities of any type,and therefore they could not be properties or relations. Thus, according to thisearly Russell, “the [propositional function] ϕ in ϕx is not a separate and distin-guishable entity: it lives in the propositions of the form ϕx and cannot surviveanalysis.”20 On the other hand, in Russell’s later logical realism, as described inPrincipia Mathematica (1910–13), propositional functions do seem to be part ofhis ontology, whereas propositions are no longer themselves “single entities”.21

In fact, I myself have identified the propositional functions of Principia withproperties and relations, though I now think an alternative interpretation ismore appropriate.22 The connection of propositional functions with proper-ties and relations was explicitly made by Russell in 1907 when he stated two“principles” that he said were “indispensable if we are to avoid contradictionsand ... preserve ordinary mathematics,” principles that seem to be implicit inPrincipia:23

Any propositional function of x is equivalent to one assigning aproperty to x.

Any propositional function of x and y is equivalent to one as-signing a relation between x and y.

Note that Russell does not say here that a propositional function of onevariable is identical with a property, but only that it is equivalent to one, andsimilarly that a relation is only equivalent, not identical, with a propositionalfunction with several variables. This does not mean that there are now two typesof universals, namely propositional functions on the one hand and propertiesand relations on the other, and that the two are somehow equivalent. Thatwould be both redundant as a formal ontology and ambiguous about the role

18Russell 1903, §133, p. 137.19Ibid., §135, p. 139.20Ibid., §85, p.88.21See Cocchiarella 1987 chapter one for a discussion of this change in Russell’s views.22See Cocchiarella 1987, chapter 1, §§9–10 and chapter 5, §2, for the claim that the propo-

sitional functions of Principia are properties and relations.23Russell 1907, p. 281.

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5.4. HYPERINTENSIONALITY 115

of universals in predication. On our alternative interpretation, however, whatRussell is really doing here is taking propositional functions as expressions thatstand for, or represent, properties and relations, and, although he could havebeen clearer about the matter, that is what he really means by equivalence inthis context. Something like that is what Russell himself claims in his laterbook, My Philosophical Development, where he wrote that “Whitehead and Ithought of a propositional function as an expression containing an undeterminedvariable and becoming an ordinary sentence as soon as a value is assigned to thevariable: ‘x is human’, for example, becomes an ordinary sentence as soon as wesubstitute a proper name for ‘x’. In this view ... the propositional function is amethod of making a bundle of such sentences.”24 If this view is correct, then thecomplexity of propositional functions is only a complexity of expression, i.e., ofsyntax, and not one of ontology.

The fact that properties and relations are described by complex predicateexpressions, in other words, does not mean that the properties and relationsare themselves complex and somehow contain simple properties and relationsand logical operations as constituents. Just as the union or intersection oftwo sets A and B does not contain the union or intersection operation as aconstituent, so too a property that is represented by a complex conjunctiveor disjunctive predicate expression does not itself contain the conjunction ordisjunction operation as a constituent. The fact that A ∪ B contains both Aand B, i.e., that A,B ⊆ A∪B, or that A∩B is contained in both A and B, i.e.,that A∩B ⊆ A,B, does not mean that A and B are themselves constituents ofA ∪B the way, e.g., the members of A and B are constituents (members) of Aand B (and of A ∪ B as well), or that A and B are constituents (members) ofA∩B. Nor of course are the union operation, ∪, and the intersection operation,∩, constituents of A∪B and A∩B respectively. Similarly, neither the propertiesF and G, nor the disjunction and conjunction operations, are real constituentsof the properties [λx(F (x)∨G(x))] and [λx(F (x)∧G(x))], even though there isa logical relation based upon these operations between F and G and [λx(F (x)∨G(x))] and [λx(F (x) ∧ G(x))]—a relation that is the basis of an abstract facthaving these properties as constituents.

The only complexes in Russell’s 1910–13 logical realist ontology are facts—and events, though Russell does not clearly distinguish events from concrete,physical facts—including abstract facts having universals as constituents.25 Thus,according to Russell, “the statement ‘two and two are four’ deals exclusivelywith universals,” and the complex that makes it true is an abstract fact.26 Ab-stract facts are needed in Russell’s logical realism as the “truth makers” of thestatements of logic and mathematics that contain no descriptive constants. Thisontology is quite different from an ontology of sets (with or without urelements),i.e., one in which pure mathematics is reduced to set theory as opposed to logic.In set theory there are no set-theoretical facts about pure sets, i.e., sets whosetransitive closures contain no urelements other than the empty set. This is be-

24Russell 1959, p.124.25See Russell 1912, p. 137.26Ibid., p.105.

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cause a set has its being in its members, and a set’s being is all that is neededto account for the truth or falsehood of statements about membership in thatset. In other words, the being of a set consists in its having just the membersthat it has, and no fact over and above the being of the set itself is needed toaccount for membership in that set. A property or relation (in intension), onthe other hand, does not have its being in its “instances,” and therefore its beingcannot of itself account for the truth of statements about objects having thatproperty or relation. Propositions, it should be noted, are not objective truthsor falsehoods in this ontology; rather, according to Russell, there are no propo-sitions other than sentences, i.e., propositions, like propositional functions, arenow only expressions. Russell’s 1910–13 logical realist ontology, in other words,is not a simples⊕complexes ontology as characterized above, and in that regardthe hyperintensional assumptions (Hyper1)–(Hyper3) are inapplicable to thisversion of logical realism. What a complex predicate represents in this ontologyis not a complex property or relation, but only another property or relationthat stands in a certain logical relation between the properties and relationsthat are represented by the component parts of that complex predicate. Thelogical relation, together with these properties and relations, are constituents ofan abstract fact, rather than of a complex property or relation. And abstractfacts, like concrete, physical facts, are extensional entities and not hyperinten-sional entities. The above hyperintensional “paradox”, accordingly, does notapply to either Russell’s or Frege’s versions of logical realism.

Of course, one might argue that there is still the problem of explaininghyperintensional contexts, i.e., of how a logical realist formal ontology suchFrege’s or Russell’s can account for the logic of such intentional contexts asbelief, desire, etc. Be that as it may, in any case one cannot reject these formalontologies on the basis of the above hyperintensionality paradox.

One might, on the other hand, reject our interpretation of propositional func-tions as expressions, or one might just insist on maintaining a simples⊕complexesontology regardless of what Frege’s or Russell’s own views were. In that case,however, an explanation must be given of how a property or relation can con-tain a logical operation as well other properties or relations as constituents. Itis not enough to simply assume that this is so without giving an ontologicalaccount of how it is possible, i.e., of how there can be such complexes.27 Butthen, assuming such an ontology will bring one back to the problem of how thehyperintensional paradox is to be avoided.

Finally, with regard to conceptual realism, it is noteworthy that althoughconcepts are formed, or constructed, on the basis of other concepts, conceptsthemselves, as cognitive capacities, do not contain other concepts or logicaloperations as constituents. In other words, with respect to concepts as cognitivecapacities, conceptual realism is not a simples⊕complexes ontology. Of course,with respect to the intensional contents of concepts, including propositions asthe contents of our speech and mental acts, the situation might well be different.

27An algebraic or set-theoretical semantics for hyperintensionality, we should note, does notof itself amount to an ontological account.

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5.4. HYPERINTENSIONALITY 117

In other words, we might well allow intensional objects to be either simpleor complex, though some account will then be needed of the sense in whichthey might contain objects, including the intensional counterparts of logicaloperations, as constituents.

But even assuming that a simples⊕complexes ontology applies to the inten-sional objects of conceptual realism, nevertheless, the hyperintensional assump-tion (Hyper3) does not apply to conceptual realism as represented by λHST∗

or HST∗λ for the reason given above. Moreover, none of the above hyperin-

tensional principles (Hyper1)–(Hyper3) apply to conceptual realism at thelevel of the logical forms that represent the cognitive structure of our speechand mental acts; nor can most of the steps in the above argument be taken asrepresenting the cognitive structure of a speech or mental act. This is importantbecause it is only on this level of analysis that hyperintensionality has a roleto play. In other words, it is only on this initial level of analysis, as opposedto the second level where deductive transformations are represented, must com-plex predicates be given a fine-grained representation. A speech or mental actin which, e.g., being round and red, i.e., [λx(Round(x)∧Red(x))], is predicatedof an object is not the same as a speech or mental act in which being not eithernot-round or not-red, i.e., [λx¬(¬Round(x) ∨ ¬Red(x))] is predicated of thatobject, even though the predicate expressions representing these concepts arelogically equivalent. Thus, although hyperintensionality does apply on the levelof analysis on which the cognitive structure of our speech and mental acts arerepresented, it does not apply on the level of deductive transformations, suchas those involved in the above paradox.

Hyperintensionality, or fine-grained structure, applies only to cognitive struc-ture, which, in conceptual realism is represented only at the initial level of anal-ysis. Deductive transformations, which may involve logical forms that in nosense can be taken as representing the cognitive structure of our speech andmental acts—such as the predicate forms [λz(H(G)] and [λz(F (G)] that occurin the above proof of the paradox—are represented on a second and differentlevel of analysis. Logical forms on this level allow for a variety of transforma-tions that show the deductive consequences of our speech and mental acts, butdo not themselves represent such acts. Deductive transformations also allowfor the study of the consequences of scientific hypothesis, or of mathematicaltheories, which in general are not intended or designed to represent features ofcognition. On this level, there is no basis for allowing (Hyper1)–(Hyper3) asdeductive principles, and in fact given the above argument there is good reasonto reject these assumptions altogether.

Finally, we should note that there are other, equally important reasons whywe must distinguish an initial level of analysis regarding the cognitive structureof our speech and mental acts from a second level at which deductive trans-formations are allowed to occur. These other considerations have to do withconceptual realism’s theory of reference and the deactivation of the referentialexpressions that occur as the direct objects of transitive verbs. We will returnto this issue in section nine of chapter seven.

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5.5 Summary and Concluding Remarks

• As a way of avoiding Russell’s paradox, the theory of simple logical typesdivides predicate expressions and their corresponding abstract singular termsinto a hierarchy of different types, and then imposes a grammatical constraintthat nominalized predicates can occur as argument- or subject-expressions onlyof predicates of higher types.

• The grammatical constraints of type theory exclude as meaningless manyexpressions that are not only grammatically correct in natural language but alsointuitively meaningful, and sometimes even true.

• The logical insights of type theory, and in particular the asymmetry be-tween predicate and subject expressions, can be retained while mitigating thegrammatical constraints of the theory. Within standard second-order predi-cate logic with nominalized predicates as abstract singular terms we need onlyimpose a constraint on complex predicates (λ-abstracts), namely that theybe restricted to those that are homogeneously stratified in a metalinguisticsense.

• By retaining the full comprehension principle (CP∗λ) of second-order pred-

icate logic with nominalized predicates, but excluding λ-abstracts that are nothomogeneously stratified, we obtain the system λHST∗, which is equivalent tosimple type theory and consistent relative to Zermelo set theory.

• λHST∗ can be taken as a consistent reconstruction of Frege’s and Russell’searly 1903 form of logical realism.

• For Frege’s extensional ontology, an extensionality axiom, (Ext∗), can beadded to λHST∗. This extensionality axiom is Frege’s Basic Law Vb. Theother direction, Basic Law Va, is an instance of Leibniz’s law in λHST∗. ThusFrege’s basic law V is consistent in λHST∗.

• By replacing standard first-order (possibilist) logic by free logic, λ-abstracts,whether homogeneously stratified or not, can be allowed in the comprehensionprinciple (CP∗

λ). Russell’s paradox then only shows that the λ-abstract for theRussell property, when transformed into an abstract singular term, must failto denote. A new axiom schema, (∃/HSCP∗

λ), is added in order to includethe objects denoted by nominalized λ-abstracts that are homogeneously strati-fied. The resulting system HST∗

λ is equivalent to λHST∗and can be taken asa better reconstruction of Frege’s ontology, but not also of Russell’s.

• The difference between the systems λHST∗ and HST∗λ shows in a clear

and precise way one of the important differences between Russell’s and Frege’sformal ontologies.

• The concepts of conceptual realism are rule-following cognitive capacities inthe use of predicate expressions and as such have an unsaturated nature similarto but also different from the unsaturated concepts and relations of Frege’sontology. It is the unsaturated nature of a predicable concept that informs aspeech or mental act with a predicable nature.

• Unlike Frege’s ontology where nominalized predicates denote the extensionsof concepts, in conceptual realism nominalized predicates denote the intensionalcontent of concepts—if in fact that content can be “object”-ified. The “object”-

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5.5. SUMMARY AND CONCLUDING REMARKS 119

ification of the intensional content of a concept is an abstract intensional objectthat represents the truth conditions determined by that concept.

• The intensional content of most concepts can be “object”-ified, i.e., pro-jected onto the level of objects, as abstract intensional objects. But the in-tensional content of the concept that the Russell predicate stands for, as wellas certain others, cannot be “object”-fied, i.e., the nominalized predicates thatstand for those concepts must be denotationless.

• The system HST∗λ can be taken as the core part of the formal ontology

for conceptual realism as well as of Frege’s form of logical realism. Frege’s log-ical realism differs from conceptual realism in having the extensionality axiom,(Ext∗), as part its core as well.

• The principle of rigidity (PR), which is discussed in the next chapter, isvalid in logical realism but not in conceptual realism. This is another importantdifference between the two formal ontologies.

• Logically equivalent formulas do not in general preserve truth when inter-changed in intentional contexts, and only a hyperintensional logic will sufficefor that purpose.

• One objection to the systems λHST∗ and HST∗λ (and the theory of sim-

ple types as well) is that they do not adequately respect the fine-grained, or“hyperintensional,” structure of such intentional contexts as belief, desire, etc.This is because formulas provably equivalent in these systems do not in generalpreserve truth when interchanged in such contexts. The claim is that only ahyperintensional logic will suffice for that purpose.

• The argument that neither λHST∗ nor HST∗λ (nor simple type theory)

cannot be considered a hyperintensional logic is a result of a so-called “hyper-intensional paradox”, which depends on certain hyperintensional assumptionsthat result in a contradiction in λHST∗ and HST∗

λ (and simple type theory).• One of these assumptions, (Hyper3), is implausible for both λHST∗ and

HST∗λ, and another, (Hyper1), is implausible for HST∗

λ.• A hyperintensional logic is based on either a strictly syntactical interpreta-

tion of properties and relations or an ontology in which properties and relationsare either absolutely simple or intrinsically complex, and in which the latter arebuilt up from the former by means of ontologically real counterparts of the log-ical constants that occur in their syntactic representation. Both interpretationsare not appropriate for either Frege’s or Russell’s logical realism, and it mightapply at best to the intensional objects of conceptual realism. It does not applyto predicable concepts however.

• Although predicable concepts are formed, or constructed, on the basis ofother concepts in conceptual realism, concepts themselves, as cognitive capaci-ties, do not contain other concepts or logical operations as constituents.

• None of the hyperintensional assumptions (Hyper1)–(Hyper3) apply toconceptual realism at the level of the logical forms that represent the cognitivestructure of our speech and mental acts.

• It is only on the level of the logical forms of our speech and mental acts,as opposed to the level where deductive transformations are represented, thatcomplex predicates must be given a fine-grained representation.

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120 CHAPTER 5. FORMAL THEORIES OF PREDICATION PART II

• Hyperintensionality, or fine-grained structure, applies only to cognitivestructure, which, in conceptual realism is represented only at the initial level ofanalysis.

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Chapter 6

Intensional Possible Worlds

We have seen in chapter two how actualism and possibilism can be distinguishedfrom one another in tense logic and with respect to the logic of the temporalmodalities of Aristotle and Diodorus. A deeper analysis can be given, however,in terms of the second-order theories of predication of both logical realism andconceptual realism. We could, for this purpose, restrict ourselves to second-order tense logic, which would be appropriate for conceptual realism because,as already noted, thought and communication, as forms of conceptual activity,are inextricably temporal phenomena. Tense logic, in other words, is implicitlyassumed as a fundamental part of the formal ontology of conceptual realism. Acausal or natural modality is also a fundamental part of conceptual (natural)realism, and with this modality comes the distinction between what is actualand what is possible in nature. That is, with a causal or natural modalitywehave a broader and perhaps an even sharper distinction between actualism andpossibilism. We will take up this type of modality in chapter twelve.

Logical realism does not, or at least need not, reject tense logic and the tem-poral modalities; nor, unlike logical atomism, must it reject a causal or naturalmodality. But, except for Frege’s extensional ontology, one might think thatintensional versions of logical realism are implicitly, if not explicitly, committedto the logical modalities, i.e., logical necessity and possibility. But, as we haveargued in chapter three, only logical atomism can provide an unproblematicaccount of the logical modalities, despite the fact that logical atomism is anextreme form of natural realism rather than of logical realism. Unlike logicalrealism, the comprehension principle (CP) is not valid in logical atomism whereonly simple properties and relations are the nexuses of the atomic states of af-fairs that make up its ontology. Every possible world in logical atomism is madeup of the same simple material objects and properties and relations, as thosethat make up the actual world, the only difference being between those statesof affairs that obtain in a given world and those that do not. Different possibleworlds in logical atomism, as we have noted, amount to different permutationsof being-the-case and not-being-the-case of the same atomic states of affairs,and hence the same simple material objects and properties and relations, that

121

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122 CHAPTER 6. INTENSIONAL POSSIBLE WORLDS

make up the actual world. As a result, in such an ontology we have a precisenotion of what it means to refer to “all possible worlds,” and dually to “somepossible world”, which are the key notions characterizing how necessity and pos-sibility are to be understood. What results from such an account is a strictlyformal notion of necessity and possibility, i.e., a notion that has no material orontological content, as it must be in logical atomism, and which is why therecan be no causal modality in the framework.

This logical atomist notion of a possible world is not at all suitable for log-ical realism with its commitment to a realm of abstract objects and a networkof complex logical relations between them. The kind of modality that an in-tensional form of logical realism is committed to is an ontological, rather thana strictly formal, notion of logical necessity and possibility, a notion that canbe referred to as metaphysical rather than formal. A metaphysical modality,apparently, is stronger than a physical or natural modality, and yet, becauseit has ontological content, it is not a purely formal logical modality such as isfound in logical atomism.

This notion of a metaphysical modality is difficult to explicate, however,because, as we explained in §§3.6-3.7, we do not have clear and precise crite-ria by which to determine the semantical conditions appropriate for a logic ofmetaphysical necessity and possibility. What we need is some way by which tounderstand the key notion of a metaphysically possible world—if in fact such anotion is really different from what is possible in nature.

Question: What notion of a metaphysically possible world is ap-propriate for logical realism?1

A similar difficulty applies to conceptual (intensional) realism insofar as oneis tempted to introduce a notion of conceptual necessity or possibility. Heretoo we have a complex network of concepts and a realm of abstract intensionalobjects that cannot be accounted for within logical atomism, and for whichstrictly formal notions of necessity and possibility therefore cannot be given.Temporal and causal, or natural, modalities do not seem to suffice to accountfor the network of relations between concepts and the intensional objects thatare their objectual counterparts, nor of the logico-mathematical relations be-tween abstract objects in general. A conceptual modality based on the strictlypsychological abilities of humans seems inadequate to account for these complexnetworks, however, and yet how otherwise to formally account for the kinds ofmodel structures that could represent conceptually possible worlds is not allclear.

We will not attempt to deal with these difficulties for logical and conceptualrealism here. We will assume instead that some explication can in principle begiven. We will also assume that all metaphysically possible worlds are equally

1What is involved in answering this question is determining the appropriate conditionsfor sets of models as set-theoretic counterparts of possible worlds in the semantic clauses fornecessity and possibility. Allowing arbitrary sets of models as possible-world counterparts,where one can add or take away a model from such a set, does nothing by way explainingwhat is meant by metaphysical necessity.

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accessible from other metaphysically possible worlds, so that the modal logic formetaphysical necessity contains at least the laws of S5, and similarly that allconceptually possible worlds are accessible from each other so that an S5 modallogic applies to conceptual realism as well. We will assume, in other words,that the notions of metaphysical necessity and possibility correspond, at leastroughly, to the equally difficult notions of conceptual necessity and possibility.Despite their similarity in this regard, there is an important difference betweenthe conceptual and the metaphysical modalities, and hence a difference betweenlogical realism and conceptual realism that we will explain later.

For convenience, we will use � and ♦ as modal operators for both metaphys-ical and conceptual necessity and possibility. We will also speak of possibilismand actualism as a distinction applicable to both logical and conceptual real-ism. The second-order modal predicate logics with nominalized predicates asabstract singular terms that result from this addition to λHST∗ and HST∗

λ

are called �λHST∗ and HST∗λ�.2

6.1 Actualism Versus Possibilism Redux

We assume that the logic of possible objects described in chapter two has beenmodified in accordance with §§5.2–5.3, so that axiom (A8), namely,

(∃x)(a = x),

where x does not occur in a, has been changed to

(∀x)(∃y)(x = y),

where x, y are distinct object variables. In other words, the logic of possibleobjects is now a “free logic,” just as the logic of actual objects is a free logic.The difference between them is that whereas what is true of every possibleobject is therefore also true of every actual object, the converse does not alsohold; that is,

(∀x)ϕ → (∀ex)ϕ

is a basic law, but the converse is not.Strictly speaking, our so-called logic of “possible objects” now deals with

more than merely possible objects, i.e., objects that actually exist in some pos-sible world or other. Here, by existence—or for emphasis, actual existence—wemean existence as a concrete object, as opposed to the being of an abstract in-tensional object. Abstract intensional objects do not ever exist as actual objectsin this sense even though they have being and are now taken as values of thebound object variables along with all possible objects. We can formulate thisdifference between actual existence and being as follows:

(∀Fn)¬E!(F ). (¬E!(Abst))2For a detailed axiomatization and discussion of �λHST∗ and HST∗

λ� see Cocchiarella1986.

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124 CHAPTER 6. INTENSIONAL POSSIBLE WORLDS

Now it is noteworthy that in actualism there is no distinction between be-ing and existence; that is, actualism is committed to there being only actuallyexisting objects. Actualism can allow for nominalized predicates, but only asvacuous, i.e., nondenoting, objectual terms. In that case, actualism will vali-date something like ¬E!(Abst), but expressed in terms of an actualist predicatequantifier as described below. In possibilism, or rather in what we are nowcalling possibilism, there is a categorial distinction between being and possibleexistence, which is expressed in part by the validity of ¬E!(Abst). In fact, thisparticular categorial distinction is one of the many—perhaps indeterminatelymany—conditions needed to characterize a metaphysical, or conceptual, possi-ble world. For convenience, however, we will continue to speak of the first-orderlogic of being, which includes possible existence, as simply the logic of possibil-ism.

The actualist quantifiers ∀e and ∃e when applied to predicate variables referto those concepts (or properties) and relations that only existing, actual objectscan fall under, or have, in any given possible world.3 The following, accordingly,are valid theses of the second-order logic of possible objects:

(∀eFn)ϕ↔ (∀Fn)(�(∀x1)...∀xn)[F (x1, ..., xn) → E!(x1) ∧ ... ∧ E!(xn)] → ϕ),

and

(∃eFn)ϕ↔ (∃Fn)(�(∀x1)...∀xn)[F (x1, ..., xn) → E!(x1) ∧ ... ∧ E!(xn)] ∧ ϕ).

We call the concepts, or properties, and relations that only actual existing ob-jects can fall under at any time in any possible world existence-entailing concepts(properties) and relations.

Now many concepts (or properties) and relations are such that only ac-tual existing objects can have, or fall under, them. In fact these are the morecommon concepts (or properties) and relations that we ordinarily apply in ourcommonsense framework. Thus, for example, an object cannot be red, or green,or blue, etc., at any time in a given possible world unless that object exists inthat world at that time. Similarly an object cannot be a pig or a horse at atime in a world unless it exists at that time in that world; nor, we should add,can there be a winged horse or a pig that flies unless it exists. Of course, inmythology there is a winged horse, namely Pegasus, and there could as well bea pig in fiction that flies. But that is not at all the same as actually being awinged horse or an actual pig that flies. Indeed, as far as fiction goes, there caneven be a story in which there is an impossible object, such as a round square.4

But a fictional or mythological horse is not a real, actually existing horse, and3We should keep in mind that in this section we are characterizing actualism and “possi-

bilism” (which now includes abstract objects as well) for both logical realism and conceptualrealism, even though logical realism takes predicates to stand for properties and relations,whereas conceptual realism takes predicates to stand for concepts (whether monadic or rela-tional). In logical realism, nominalized predicates denote the same properties and relationsthey stand for in their role as predicates, whereas in conceptual realism nominalized predicatesdenote the intensional contents of the concepts they stand for in their role as predicates.

4See, e.g., the story Romeo and Juliet in Flatland in Cocchiarella 1996, § 7.11.

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one of the tasks of a formal ontology is to account for the distinction betweenmerely fictional and actually existing objects. Later, in a subsequent chapterwe will explain how the ontology of fictional, or mythological, characters andentities such as the winged horse Pegasus, can be accounted for as intensionalobjects. For now it is important to distinguish actual existence from being andmerely possible existence.

The two main theses of actualism are:(1) quantificational reference to objects can be only to objects thatactually existence, and(2) quantificational reference to properties, concepts, or relationscan be only to those that “entail” existence in the above sense, i.e.,the only properties, concepts or relations there are according to ac-tualism are those that only actually existing objects can have or fallunder.

What this means is that in actualism the quantifiers ∀e and ∃e, must betaken as primitive symbols when applied to object or predicate variables. Thefollowing, moreover, is a basic theorem of actualism.

(∀eFn)[F (x1, ..., xn) → E!(x1) ∧ ... ∧E!(xn)].

In regard to the concept of existence, note that the statement that everyobject exists, i.e., (∀ex)E!(x), is a valid thesis of actualism. In possibilism,however, the same statement is false, and in fact, given the being of abstractintensional objects, none of which ever exist as actual objects, it is logicallyfalse that every object exists; that is, ¬(∀x)E!(x), is a valid thesis of possibilismas we understand it here.5 What is true in both possibilism and actualism,on the other hand, is the thesis that to exist is to possess, or fall under, anexistence-entailing concept or property; that is,

E!(x) ↔ (∃eF )F (x)

is valid in both actualism and possibilism.In possibilism, in fact, by taking quantification over existence-entailing con-

cepts or properties as primitive, or basic, we can define existence as follows:

E! =df [λx(∃eF )F (x)].

What this definition indicates in conceptualism is that the concept of existenceis an “impredicative” concept. In other words, because existence is itself anexistence-entailing concept—i.e., if a thing exists, then it exists—then, the con-cept of existence is formed or constructed in terms of a totality to which itbelongs. This, in fact, is why according to conceptual realism the concept of ex-istence is so different from ordinary existence-entailing concepts, such as beingred, or green, a horse, a tree, etc.

5The universal and null concepts [λx(x = x)] and [λx(x �= x)] are h-stratified and thereforeprovably have intensional objects as their object-correlates in both λHST∗ and HST∗

λ.

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126 CHAPTER 6. INTENSIONAL POSSIBLE WORLDS

Identity, incidentally, coincides with indiscernibility in both possibilism andactualism. In possibilism this thesis that identity coincides with indiscernibilityis formulated as follows:

(∀x)(∀y)(x = y ↔ (∀F )[F (x) ↔ F (y)]).

In actualism, the thesis is stated in a more restricted way, namely, as

(∀ex)(∀ey)(x = y ↔ (∀eF )[F (x) ↔ F (y)]).

There is another difference on this matter if we formulate the thesis without theinitial quantifiers. That is, whereas in possibilism, the following

x = y ↔ (∀F )[F (x) ↔ F (y)]

is valid, in actualism the related formula

x = y ↔ (∀eF )[F (x) ↔ F (y)]

is actually false in the right-to-left direction when neither x nor y exist. Inother words, if x and y do not exist, then they vacuously fall under all the sameexistence-entailing concepts, namely none; and yet it does not follow that thatx = y. Neither Pegasus nor Bellerophon actually exist, and yet it would be falseto conclude that therefore Pegasus is Bellerophon. In other words,

¬E!(x) ∧ ¬E!(y) → (∀eF )[F (x) ↔ F (y)]

is valid in both possibilism and actualism even when x �= y.The principle of universal instantiation for actualist quantifiers can be for-

mulated as follows in possibilism:

(∃eFn)([λx1...xnϕ] = F ) → ((∀eG)ψ → ψ[ϕ/G(x1, ..., xn)]). (∃/UIe2)

In actualism, where λ-abstracts cannot be nominalized6, the formulation, asfollows, is somewhat more complex regarding its antecedent condition:

(∃eFn)�(∀x1)�(∀x2)...�(∀xn)�[F (x1, ..., xn) ↔ ϕ] → ((∀eG)ψ → ψ[ϕ/G(x1, ..., xn)]).

The comprehension principle for the actualist quantifiers is not as simple asthe comprehension principle (CP∗

λ) for possibilism. It amounts to a kind ofAussonderungsaxiom for existence-entailing properties, concepts and relations:

(∀eGk)(∃eFn)([λx1...xn(G(x1, ..., xk) ∧ ϕ)] = F ), (CPeλ)

where k ≤ n, and Gk and Fn are distinct predicate variables that do not occur inϕ. In the monadic case, for example, what this principle states is that although

6Strictly speaking, actualism can allow for nominalized predicate expressions, but only byassuming as a basic principle that they denote nothing.

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we cannot expect every open formula ϕx to represent an existence-entailing con-cept or property, nevertheless conjoining ϕx with an existence-entailing conceptor property G(x) does represent a concept or property that entails existence.The formulation in actualism, where the nominalized λ-abstract is not allowed(as a denoting singular term), is again more complex, but we will forego thosedetails here.7 A simpler comprehension principle that is also valid in actualismis:

(∃eFn)�(∀ex1)...(∀exn)[F (x1, ..., xn) ↔ ϕ].

Before concluding this section we should note that although many of ourcommonsense concepts are concepts that only existing objects can have, never-theless there are some concepts, especially relational ones, that can hold betweenobjects that do not exist in the same period of time in our world or even in thesame world. All animals, for example, have ancestors whose lifespans do notoverlap with their own, and yet they remain their ancestors. An acorn that wechoose to crush under our feet will never grow into an oak tree in our world,and yet, as a matter of natural possibility (as based, e.g., on the many-worldsinterpretation of quantum mechanics), there is a world very much like ours inwhich we choose not to crush the acorn but leave it to grow into an oak tree. Itis only a possible, and not an actual, oak tree in our world, but there is still arelation between it and the acorn that we crushed, just as there is a ancestralrelation between animals whose lifespans do not overlap.

6.2 Intensional Possible Worlds

The actual world, according to conceptual realism, consists of physical objects,states of affairs, and events of all sorts that are structured in terms of the lawsof nature and the natural kinds of things there are in the world. There areintensional objects as well, to be sure, but these have being only as naturalproducts of the evolution of consciousness, language and culture; and withoutlanguage and thought they would not be at all. Abstract intensional objects, inother words, have a mode of being dependent on language and culture, and arenot part of what makes up the physical world. Possible worlds are like the actualworld, moreover, i.e., they consist of physical objects and events structured interms of the laws of nature as described earlier, e.g., in §§3.6.1–3.6.2.

In speaking of possible worlds here—and even of the actual world—we shouldbe cautious to note that possible worlds are not “objects,” as when we speak ofthe various possible kinds of objects and events in the world. In other words,in conceptual realism possible worlds are not values of the bound object vari-ables, and therefore they are not quantified over as objects. It is true thatwe seem to quantify over possible worlds in our semantical set-theoretic meta-language. But, to be precise, what we really quantify over are set-theoretical

7See Cocchiarella 1986, chapter III, §9, for a detailed axiomatization of actualism. Fora completeness theorem for the second-order logic of actual and possible objects withoutabstract objects see Cocchiarella 1969.

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model structures that we call possible worlds. These set-theoretical model struc-tures are not possible worlds in any literal metaphysical sense; rather, they areabstract mathematical objects that we use to represent the different possiblesituations described in the object-language of our formal ontology by meansof modal operators. Unlike the model-theoretic “possible-world” parameters ofthe set-theoretical metalanguage, the modal operators of a formal ontology canbe iterated and can occur within the scope of other occurrences of the same,or of a dual, operator. In addition, unlike the “external relations” expressed inthe metalanguage between model-theoretic “possible worlds” and objects in thedomains of those worlds, modal operators in the language of the formal ontologyexpress “internal relations” between objects and the properties or concepts theyfall under.8

Possible worlds in logical realism are not really different from those in con-ceptual realism, i.e., they are made up of the same physical objects and events,and, as in conceptual realism, they do not themselves “exist” in different pos-sible worlds. They are not themselves a kind of concrete object in addition tothe possible situations represented by the modal operators.9 It is significant,however, that there are abstract objects in logical realism that are the inten-sional counterparts of possible worlds. This is because, unlike the situation inconceptual realism, intensional objects have a mode of being in logical realism(Platonism) that is independent of the natural world, and even of whether ornot there is a natural world, and in that sense they are independent of all possi-ble worlds. That is, even though abstract intensional objects in general do not“exist” as concrete objects—and hence unlike the latter do not “exist” in somepossible worlds and fail to “exist” in others—nevertheless, in logical realismthey have a mode of being (namely, being simpliciter) that is independent ofall possible worlds. These abstract intensional objects have being in a Platonicrealm that is so richly structured logically that it even includes objects that areintensional counterparts of possible worlds.

There are no intensional counterparts of possible worlds in conceptual re-alism, on the other hand, because in this framework all abstract intensionalobjects are ontologically grounded in terms of the human capacity for thoughtand concept-formation. The intensional counterparts of possible worlds in logi-cal realism, other words, cannot be accounted for in conceptual realism becausethey far exceed our cognitive abilities in the construction, or projection, of suchcounterparts. It is in the ontological status of these kinds of entities that webegin to see an important difference between logical realism and conceptual(intensional) realism.

Now there are at least two kinds of intensional objects in the ontology oflogical realism that are counterparts of metaphysically possible worlds. For

8In logical atomism, which provides the only ontology suitable for the logical modalities,all “internal” relation are strictly formal relations.

9In Plantinga 1974, which we take to be a kind of Platonic realism, the only possibleworlds considered are what we describe below as the intensional counterpart to a possibleworld. David Lewis, on the other hand, definitely takes possible worlds to be concrete objects.Lewis, however, is not a Platonist.

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convenience, we will call both kinds intensional possible worlds. Our first suchkind is a “world” proposition; that is, a proposition P that is both possible,i.e., ♦P , and maximal in the sense that for each proposition Q, either P entailsQ or P entails ¬Q, where by “entailment” we mean necessary material impli-cation, i.e., either �(P → Q) or �(P → ¬Q).10 Where P is a possible-worldcounterpart in this sense, we read �(P → Q) as ‘Q is true in P ’. This notionof an intensional possible world can be defined by the following λ-abstract:

Poss-Wld1 =df [λx(∃P )(x = P ∧ ♦P ∧ (∀Q)[�(P → Q) ∨ �(P → ¬Q)])].

Note that this λ-abstract is a homogeneously stratified, and therefore Poss-Wld1 stands for a property in logical realism. It also stands for a concept inconceptual (intensional) realism, where it is not the concept that is impossibleto construct but the being of the propositions (intensional objects) that fallunder the concept. This is because, as already noted, such intensional objectsgo beyond what we can construct by means of our cognitive abilities. In logicalrealism, on the other hand, there is not only such a property as a value of thebound predicate variables of both �λHST∗ and HST∗

λ�, but, in addition, itsnominalization denotes in both of these systems a value of the bound objectualvariables as well.

Of course, the fact that Poss-Wld1 is a well-formed predicate that standsfor a property, or concept, does not mean that it must be true of anything,i.e., that there must be propositions that have this property, or fall under thisconcept. In fact, as already noted, any proposition that falls under this conceptas an intensional object has content that so far exceeds what is cognitivelypossible for humans to have as an object of thought that the “existence,” orrather being, of such a proposition can in no sense be validated in conceptualrealism. But then, in logical realism, propositions are Platonic entities existingindependently of the world and all forms of human cognition, and thereforethe possible objectual being of a proposition P such that Poss-Wld1(P ) is notconstrained in logical realism by what is cognitively possible for humans to haveas an object of thought.

Nevertheless, we cannot prove that there are intensional possible worlds inthis sense in either �λHST∗ and HST∗

λ�, unless some axiom is added to thateffect. One such axiom would be the following, which posits the being of a propo-sition corresponding to each possible world (or model of the metalanguage).

�(∃P )[Poss-Wld1(P ) ∧ P ]. (∃Wld1)

Of course, one immediate consequence of (∃Wld1) is that some intensionalpossible world now obtains, i.e., (∃P )[Poss-Wld1(P ) ∧ P ], which we can referto as “the intensional (counterpart of the) actual world.”

A criterion of adequacy for this notion of a possible world is that it shouldyield the type of results we find in the set-theoretic semantics for modal logic.One such result is that a proposition is true, i.e., now obtains, if, and only if,

10See Prior & Fine, 1977 for a discussion of this approach to intensional possible worlds.

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it is true in the actual world. This result is in fact provable on the basis of(∃Wld1), i.e.,

Q↔ (∃P )[Poss-Wld1(P ) ∧ P ∧ �(P → Q)],

is provable in both �λHST∗and HST∗λ� if (∃Wld1) is added as an axiom.

From this another appropriate result follows; namely, that a proposition Qis possible, i.e., ♦Q, if it true in some possible world; that is,

♦Q↔ (∃P )[Poss-Wld1(P ) ∧ �(P → Q)],

is provable in both �λHST∗ + (∃Wld1) and HST∗λ� + (∃Wld1).

Finally, another appropriate consequence is that if Q and Q′ are true in allthe same possible worlds (models), then they are necessarily equivalent; and,conversely, if they are necessarily equivalent, then they are true in all the samepossible worlds; that is,

(∀P )(Poss-Wld1(P ) → [(�[P → Q] ↔ �[P → Q′]) ↔ �(Q↔ Q′)])

is provable in �λHST∗ + (∃Wld1) and HST∗λ� + (∃Wld1). It does not fol-

low, however, that propositions are identical if they are true in all of the sameintensional possible worlds.

There is yet another notion of an intensional possible world that can haveinstances in logical realism but not in conceptual realism. This is the notionof an intensional possible world as a property in the sense of “the way thingsmight have been.” David Lewis, for example, claimed that possible worlds are“ways things might have been,” but, curiously, according to Lewis the “waysthat things might have been” are concrete objects, not properties.11 RobertStalnaker, who is an actualist, pointed out, however, that “the way things areis a property or state of the world, [and] not the world itself,” as Lewis wouldhave it.12 In other words, according to Stalnaker, “the ways things might havebeen” are properties. For an actualist such as Stalnaker this means that thereare possible worlds qua properties, but, except for the actual world, which isconcrete, they are all uninstantiated properties. This is because, accordingto Stalnaker, only concrete worlds could be instances of such properties, andas an actualist the only such concrete instance is the actual world. In otherwords, although there are possible worlds qua properties, according to Stalnaker,nevertheless there can be no possible worlds qua instances of those propertiesother than the actual world, because such instances would then be concrete andyet not actual, which is what actualism rejects. In logical realism, however, thesituation is quite different.

In the ontology of logical realism, the “ways things mighthave been” are properties, but they are not properties ofmetaphysically possible worlds as concrete objects. Rather,

11See Lewis 1973, p. 84. The relevant text is reprinted in Loux, 1979, p. 182.12Stalnaker 1976, p. 228.

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they are properties of propositions as intensional objects;in particular, they are properties of all and only the propo-sitions that are true in the possible world in question.

Now a property of all and only the propositions that are true in a possibleworld (model) A could not be an intensional counterpart of the world A if theextension of that property were different in different possible worlds (models).Possible worlds are different, in other words, if the propositions true in thoseworlds are different. What is needed is a property of propositions that does notchange its extension from possible world to possible world. We will call such aproperty a “rigid” property, which we “define” as follows:13

Rigidn(Fn) : (∀x1)...(∀xn)[�F (x1, ...xn) ∨ �¬F (x1, ...xn)].

Note that because a rigid property will have the same extension in every possibleworld (model), the extension of that property can then be identified ontologi-cally with the property itself.

The type of intensional possible world that is now under consideration isthat of a rigid property that describes “the ways things might have been” ina given given metaphysically possible world. It does this by rigidly holding ofall and only the propositions that are true in that world. This notion can bespecified by a homogeneously stratified formula, which means that the propertyof being an intensional possible world in this sense can be defined by means ofa λ-abstract as follows:

Poss-Wld2 =df [λx(∃G)(x = G∧Rigid1(G)∧♦(∀y)[G(y) ↔ (∃P )(y = P ∧P )])].

Now, as with our first notion of an intensional possible world, the claimthat there are possible worlds in this second sense is also not provable in either�λHST∗ or HST∗

λ�, unless we add an assumption to that effect. One suchassumption is the following, which says that there is such a possible-world prop-erty G, i.e., Poss-Wld2(G), that holds in any possible world of all and only thepropositions that are true in that world:

�(∃G)(Poss-Wld2(G) ∧ (∀y)[G(y) ↔ True(y)]). (∃Wld2)

Here by True(y) we mean that y is a proposition that is the case, i.e.,

True =df [λy(∃P )(y = P ∧ P )],

which, because the λ-abstract in question is homogeneously stratified, specifiesa property, or concept, in both �λHST∗ and HST∗

λ�.Now it turns out that we do not have to assume either of the new axioms,

(∃Wld1) or (∃Wld2), to prove that there are intensional possible worlds inthe formal ontology of logical realism in either of these two senses. Both, in

13Rigidity can be λ-defined as a predicate in �λHST∗, but not in HST∗λ� , where it must

be construed only as an abbreviation in the principle of rigidity described below. That is whywe present rigidity in the present form.

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fact, can be derived in the modal systems �λHST∗ and HST∗λ� from what we

will call the principle of rigidity, (PR), which stipulates that every property,or concept, F , is co-extensive in any possible world with a rigid property, i.e., arigid property that can in effect be taken as the extension of F (in that world).14

We formulate the principle of rigidity as follows:

�(∀Fn)(∃Gn)(Rigidn(G) ∧ (∀x1)...(∀xn)[F (x1, ...xn) ↔ G(x1, ...xn)]). (PR)

The thesis that there is a rigid property corresponding to any given propertyis intuitively valid in logical realism where properties have a mode of being thatis independent of our ability to conceive or form them as concepts. In conceptualrealism, on the other hand, the thesis amounts to a “reducibility axiom” claimingthat for any given concept or relation F we can construct a corresponding rigidconcept or relation that in effect represents the extension of the concept F .Such a “reducibility axiom” is much too strong a thesis about our abilities forconcept-formation.

That (∃Wld2) is derivable from (PR) follows from the fact that True rep-resents a property in these systems; that is,

(∃G)(Rigid2(G) ∧ (∀y)[G(y) ↔ True(y)])

is provable on the basis of (CP∗λ) in both �λHST∗+(PR) and HST∗

λ�+(PR),and therefore, by the rule of necessitation and obvious theses of S5 modal logic,it follows that (∃Wld2) is derivable from (PR). A similar argument, which wewill not go into here, shows that (∃Wld1) is also derivable from in �λHST∗+(PR) and HST∗

λ�+ (PR).The fact that with (PR) we can prove in both of the theories of predication

�λHST∗ and HST∗λ� that there are intensional possible worlds in either the

sense of Poss-Wld1 or Poss-Wld2 is significant in more than one respect. Onthe one hand it indicates the kind of ontological commitment that logical real-ism has as a modern form of Platonism. On the other hand, it also indicatesa major kind of difference between logical realism and conceptual realism, be-cause, unlike logical realism, the principle of rigidity is not valid in conceptualrealism. What it claims about concept-formation is not cognitively realizablefor humans. Nor can there be intensional possible worlds in the sense eitherof Poss-Wld1 or Poss-Wld2 in conceptual realism, because such intensionalobjects are not cognitively realizable in human thought and concept-formation.Here, with the principle of rigidity and the notion of a proposition as the inten-sional counterpart of a possible world, we have clear distinction between logicalrealism as a modern form of Platonism and conceptual realism as a modernform of conceptualism, i.e., as a form of conceptual intensional realism. Also,what these differences indicate is that the notions of metaphysical necessity

14The idea of representing the extension of a property by means of a rigid property wasfirst suggested by Richard Montague.A type-theoretical version of the thesis was used as aprinciple of “extensional comprehension” by Dan Gallin in his development of Montague’sintensional logic. See Montague 1974, p.132, and Gallin 1975, p. 77.

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and possibility in logical realism are not the same as the notions of concep-tual necessity and possibility in conceptual realism, at least not if conceptualpossibility is grounded in what is cognitively realizable in human thought andconcept-formation.

Logical realism: The principle of rigidity, (PR), is valid, and there-fore so are (∃Wld1) and (∃Wld2). That is, there are intensional pos-sible worlds in logical realism in the sense of Poss-Wld1 as well asof Poss-Wld2.Conceptual realism: The principle of rigidity, (PR), is not valid,and there can be no intensional possible worlds in the sense of ei-ther Poss-Wld1 or Poss-Wld2, because such intensional objects ex-ceed what is cognitively realizable in human thought and concept-formation.Therefore, metaphysical necessity and possibility are thenot the same as conceptual necessity and possibility.

6.3 Summary and Concluding Remarks

• How the notion of a metaphysically possible world can be characterizedis problematic, and it is difficult to determine what notion of a metaphysicallypossible world, if any, is appropriate for logical realism. The notion of a con-ceptually possible world is similarly problematic, and it is similarly difficult todetermine what notion of a conceptually possible world, if any, is appropriatefor conceptual realism.

• We assume, for convenience of comparison, that the notions of metaphysi-cal necessity and possibility correspond, at least roughly, to the equally difficultnotions of conceptual necessity and possibility, and that both can be representedby the modal logic S5.

• Actualism and possibilism can be given a fuller ontological explanation interms of a distinction between concepts, or properties and relations, that entail(concrete) existence and those that do not.

• To exist is to possess, or fall under, an existence-entailing concept in con-ceptual realism, or an existence-entailing property in logical realism. Existence,accordingly, is an impredicative concept or property.

• Abstract objects have being but do not exist, where by existence we meanconcrete existence. In logical realism abstract objects “exist” independently ofwhether there is a world or not. In conceptual realism, abstract objects donot “exist” independently of the world, and in particular they do not “exist”independently of consciousness, language and culture.

• In both logical and conceptual realism, possible worlds, including the actualworld, are made up exclusively of physical objects and events. Possible worldsare not objects in either ontology and do not themselves “exist” in differentworlds.

• In logical realism, but not in conceptual realism, there are abstract in-tensional objects that are the counterparts of possible worlds. These abstract

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objects, like all abstract objects, have being only in a Platonic realm separatefrom all possible worlds. This is a fundamental difference between logical andconceptual realism.

• There are two kinds of abstract possible-world counterparts in logical re-alism, both of which are called intensional possible worlds. The first kind is aworld-proposition, i.e., a proposition P that is both possible and maximal inthe sense that for each proposition Q, either P entails Q or P entails ¬Q. Thesecond kind of intensional possible world is a “rigid” property that describes“the ways things might have been” in a given metaphysically possible world byrigidly holding of all and only the propositions that are true in that world.

• The “existence” (or really being) of both kinds of intensional possibleworlds is derivable in the modal systems �λHST∗ and HST∗

λ� from the prin-ciple of rigidity, (PR), which stipulates that every property, or concept, F , isco-extensive in any possible world with a rigid property, i.e., a rigid propertythat can in effect be taken as the extension of F (in that world).

• The principle of rigidity, (PR), is valid in logical realism. It is not validin conceptual realism because the intensional objects it posits exceed what iscognitively realizable in human thought and concept-formation. This is a fun-damental difference between the formal ontologies of logical and conceptualrealism.

6.4 Appendix 1

The well-formulas of the logic λHST∗ are as defined in chapter five, with theproviso that only h-stratified λ-abstracts are well-formed in λHST∗. The ax-ioms of λHST∗ consist of those of standard first-order logic as described inchapter two together with the comprehension principle

(∃F )([λx1...xnϕ] = F ), (CP∗λ)

but, again, with the understanding that the λ-abstract [λx1...xnϕ] must beh-stratified. There are also an axiom regarding the distribution of a universalpredicate quantifier over a conditional and an axiom regarding vacuous predicatequantifiers:

(∀F )[ϕ→ ψ] → [(∀F )ϕ→ (∀F )ψ],

(∀F )[ϕ→ ψ] → [ϕ→ (∀F )ψ], where F is not free in ϕ.The rule of λ-conversion is also an axiom scheme of λHST∗,

[λx1...xnϕ](a1, ..., an) ↔ ϕ(a1/x1, ..., an/xn) (λ-Conv∗)

where each ai is free for xi, for 1 ≤ i ≤ n, as well as an axiom for rewrite ofbound variables,

[λx1...xnϕ] = [λy1...ynϕ(y1/x1, ..., yn/xn)],

and a final axiom[λx1...xnF (x1, ..., xn)] = F, (Id∗

λ)

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where F is an n-place predicate variable. The only inference rules of λHST∗

are modus ponens and universal generalization for both predicate and individualvariables.

The formulas of HST∗λ are the same as those for λHST∗, except that λ-

abstracts that are not h-stratified are allowed as well. The first-order axioms arethose of standard first-order logic as described in chapter two, except that axiom(A8), i.e., (∃x)(a = x), where x is not a, is replaced by the free-logic axiom(∀x)(∃y)(x = y) together with the identity axiom (a = a) for all object terms a.In other words, standard first-order logic is replaced by a logic free of existentialpresuppositions regarding object terms. The comprehension principle (CP∗

λ)for HST∗

λ is as above, except, again, non-h-stratified λ-abstracts are no longerexcluded. The remaining axioms and inference rules are the same as those forλHST∗, except for the rule of λ-conversion, which is replaced by the followingfree-logic version:

[λx1...xnϕ](a1, ..., an) ↔ (∃x1)...(∃xn)(a1 = x1 ∧ ... ∧ an = xn ∧ ϕ),(∃/λ-Conv∗)

where, for all i, j ≤ n, xi does not occur in aj .We can extend either λHST∗ or HST∗

λ so as to contain a logic of existenceby adding the actualist quantifier ∀e (with ∃e defined as usual) and then redefinethe notion of a well-formed formula so as to include those containing the actualistquantifier as well. We also add the modal operator � (and, by definition, ♦)with axioms for S5 modal logic. The axioms of first-order actualism as describedin chapter two are then added as well as a distribution over conditionals axiomand a vacuous quantifier axiom for ∀e when applied to predicate variables:

(∀eF )[ϕ→ ψ] → [(∀eF )ϕ→ (∀eF )ψ],

(∀eF )[ϕ→ ψ] → [ϕ→ (∀eF )ψ], where F is not free in ϕ.

There are two special axioms for the existence-entailing concepts or proper-ties, namely,

(∃eF )([λx1...x2(∃eG)G(x1, ..., xn)] = F ),

where F ,G are distinct n-place predicate variables, and

(∀eGk)(∃eFn)([λx1...x2(∃eG)(G(x1, ..., xk) ∧ ϕ)] = F ),

where n ≤ k, F is a n-place predicate variable and G is a k-place predicatevariable, neither of which occur free in ϕ. The first axiom stipulates, e.g., inthe monadic case that falling under an existence-entailing concept, or having anexistence-entailing property, is itself an existence-entailing concept or property.The second axiom is a schema that amounts in effect to an Aussonderungs axiomfor existence-entailing concepts or properties and relations.

The hierarchy of predicative concepts described in the previous chapter canbe retained in conceptual realism in case we want to represent the impredica-tivity of certain concepts, such as existence defined as follows:

E! = [λx(∃eF )F (x)].

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136 CHAPTER 6. INTENSIONAL POSSIBLE WORLDS

That is, for each positive integer n, we will have,

¬(∃jGn)(E! = G),

as a valid thesis for each predicative stage j. Similarly, we can note the impred-icativity of the Russell concept, or property, as follows:

¬(∃jGn)([λx(∃F )(x = F ∧ ¬F (x))] = G).

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Part II

Conceptual Realism

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Chapter 7

The Nexus of Predication

A universal, we have said, is what can be predicated of things.1 But what exactlydo we mean in saying that a universal can be predicated of things? In particular,how, or in what way, do universals function in the nexus of predication?

In nominalism, there are no universals, and the only nexus of predicationis the linguistic nexus between subject and predicate expressions (or tokens ofsuch). What this means in nominalism is that only predicates can be true orfalse of things.

But what are the semantic grounds for predicates to be true or false ofthings? Are there really no concepts as cognitive capacities involved in suchgrounds? What then accounts for the unity of a sentence in nominalism asopposed to a mere sequence of words? Can nominalism really explain the unityof the linguistic nexus?

In logical realism, which is a modern form of Platonism, universals existindependently of language, thought, and the natural world, and even of whetheror not there is a natural world. Bertrand Russell and Gottlob Frege, as wehave noted, described two of the better known versions of logical realism.2 InRussell’s early form of logical realism, for example, universals are constituentsof propositions, where the latter are independently real intensional objects. Thenexus of predication in such a proposition, according to Russell, is a relationrelating the constituents and giving the proposition “a unity” that makes itdifferent from the sum of its constituents.3 Thus, according to Russell, “aproposition ... is essentially a unity, and when analysis has destroyed this unity,

1Cf. Aristotle, De Interpretatione, 17a39. Some of the material in this chapter occurredin my paper in Metalogicon, VI, 2003. Some critical points have been revised.

2We have in mind here mainly the 1903 Russell of The Principles of Mathematics. Russell’slater turn in 1914 to logical atomism is a turn to a form of natural realism.

3See Russell 1903, §55, p. 52. Russell is unclear in 1903 about what relation is the unityof the proposition expressed by ‘Socrates is human’ and others of this type. A solution isproposed in Cocchiarella 1987, chapter 2, §5, where this proposition is rephrased as ‘Socratesis a human being’, and where the verb ‘is’ stands for the relation of identity and ‘a humanbeing’ stands for what Russell in 1903 called a denoting concept. The result on this analysisexpresses the proposition that Socrates is identical with a human being.

139

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140 CHAPTER 7. THE NEXUS OF PREDICATION

no enumeration of constituents will restore the proposition. The verb [i.e., therelation the verb stands for], when used as a verb, embodies the unity of theproposition ....”4

Now a relation, in Russell’s modern form of Platonism, can also occur in aproposition as a term, i.e., as one of the constituents related. That is why wecan have formulas of the form R(x,R) as well as R(x, y). But then how can arelation occur as a term in some propositions and in others, and perhaps evenin the same proposition, as the unifying relating relation? That is, how cana relation have a predicative nature holding the constituents of a propositiontogether and also be one of the objects held together by the relating relation ofthat proposition? This was something Russell was unable to explain.5

Frege introduced a fundamental new idea regarding the unity of a propositionand the nexus of predication.6 This was his notion of an unsaturated function,which applies to the nexus of predication in language as well as to propositionsas abstract entities. On the unsaturated nature of a predicate as the nexus ofpredication of a sentence, Frege claimed that “this unsaturatedness ... is neces-sary, since otherwise the parts [of the sentence] do not hold together”.7 On theunsaturated nature of the nexus of predication of a proposition, Frege similarlyclaimed that “not all parts of a proposition can be complete; at least one mustbe ‘unsaturated’, or predicative; otherwise, they would not hold together.”8

It is the unsaturated nature of a predicate and the properties and relations itstands for that accounts for both predication in language and the unity of aproposition, according to Frege.9

Now in Frege’s ontology properties and relations of objects are functions thatassign the truth values “the true” or “the false” to objects. These truth val-ues are abstract objects, but, apparently, they are not the properties truth andfalsehood that propositions have in Russell’s form of Platonism. In any case, allfunctions, including functions from numbers to numbers, have an unsaturatednature according to Frege. Objects, on the other hand, and only objects, havea saturated nature, and therefore functions, being unsaturated, cannot be ob-jects. This distinction between functions and objects is fundamental in Frege’sontology, and, as we will see, it has a counterpart in conceptualism.

Predication in Frege’s ontology, as we have noted, is explained in terms offunctionality, which is contrary to the usual understanding of functionality interms of predication, i.e., in terms of many-one relations.10 But conceptually itis predication that is more fundamental than functionality.

We understand what it means to say that a function assigns truth values

4Russell 1903, p. 50.5Ibid.6Frege used the word ‘Gedanke’ for what we are here calling a proposition. A Gedanke

in Frege’s ontology is not a thought in the sense of conceptualism but an independently realintensional object expressed by a sentence.

7Frege 1979, p. 177.8Frege 1952, p. 54.9Frege usually referred to properties (Eigenshaften) as concepts; but we will avoid that

terminology here so as not confuse Frege’s realism with conceptualism.10Compare Russell 1903, p. 83.

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7.1. PREDICATION IN NATURAL REALISM 141

to objects, for example, only by knowing what it means to predicate concepts,or properties and relations, of objects. Nevertheless, aside from this reversalof priority between predication and functionality, Frege’s real contribution tothe analysis of the nexus of predication is his view of the unsaturated nature ofuniversals as the ground of their predicative nature.

Something like this view is basic to the way the nexus of predication isexplained in conceptualism.

7.1 Predication in Natural Realism

Natural realism is different from logical realism, we have noted, in that fornatural realism universals do not exist independently of the natural world andits causal matrix. Universals exist only in things in nature, or at least in thingsthat could exist in nature, and whether or not a predicate stands for such auniversal is strictly a factual, and not a logical, matter.

Logical atomism is a form of natural realism that provides a clear and use-ful account of predication in reality. In particular, in the Tractatus Logico-Philosophicus, Wittgenstein replaced Frege’s unsaturated logically real proper-ties and relations (as functions from objects to truth values) with unsaturated“material”, i.e., natural, properties and relations as the modes of configura-tion of atomic states of affairs. Reality, on this account, is just the totality ofatomic facts—i.e., the states of affairs that obtain in the world; and the nexusof predication of a fact is the material property or relation that is the mode ofconfiguration of that fact (atomic state of affairs). This is similar to Russell’stheory of a relating relation as what unifies a proposition, except that insteadof a proposition as an abstract intensional entity we now have facts, or statesof affairs, and instead of a logically real relation we have a natural propertyor relation as the nexus, or mode of configuration, of such a state of affairs.Also, because natural properties and relations have an unsaturated nature asthe nexuses of predication, they cannot themselves be objects in states of affairs,unlike the situation in Russell’s early Platonist ontology.

One of the major flaws of logical atomism, however, is its ontology of simplematerial objects (bare particulars?). The idea that the complex natural world isreducible to ontologically simple objects and atomic states of affairs is a difficult,if not impossible, thesis to defend. It is even more difficult to defend the addedclaim, which is also made in logical atomism, that all meaning and analysismust be based on ontologically simple objects and the atomic states of affairsin which they are configured.

But having natural properties and relations as modes of configuration ofstates affairs—i.e., as the nexuses of predication in reality—is an importantand useful view. In fact, we can retain this view of natural properties andrelations even though we reject the idea of simple objects. Something very muchlike this is exactly what we have in conceptual natural realism, where insteadof the simple material objects of logical atomism we have complex physicalobjects as the constituents of states of affairs. Conceptual natural realism, as

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we noted in our introductory lecture, is a modern counterpart to Aristotle’snatural realism, just as conceptual intensional realism is a mitigated, moderncounterpart to conceptual Platonism; and both are taken as part of what wemean by conceptual realism. Also, if we add to the logic of conceptual naturalrealism the modal operator �c

for a causal or natural necessity and also add alogic of natural kinds, then we get a modern form of Aristotelean essentialism.11

But this is a topic we will turn to and develop in more detail in a later lecture.

7.2 Conceptualism

What underlies our capacity for language and predication in language, accordingto conceptualism, is our capacity for thought and concept formation, a capacitythat is grounded in our evolutionary history and the social and cultural envi-ronment in which we live. Predication in thought is more fundamental thanpredication in language, in other words, and this is so because what holds theparts of a sentence together in a speech act are the cognitive capacities thatunderlie predication in thought.

There are two major types of cognitive capacities that char-acterize the nexus of predication in conceptualism. Theseare:

(1) a referential capacity, and(2) a predicable capacity.

These capacities underlie our rule-following abilities in the use of referentialand predicable expressions. Predicable concepts, for example, are the cognitivecapacities that underlie our abilities in the correct use of predicate expressions.When exercised in a speech or mental act, a predicable concept is what informsthat act with a predicable nature—a nature by which we characterize or relateobjects in a certain way. A predicate expression whose use is determined in thisway is then said to stand for the concept that underlies its use.

Referential concepts, on the other hand, are cognitive capacities that underlieour use of referential expressions. Referential concepts are what underlie theintentionality and directedness of our speech and mental acts. When exerciseda referential concept informs a speech or mental act with a referential nature.A referential expression whose use is determined in this way is said to stand forthe concept that underlies that use.

Referential and predicable concepts are a kind of knowledge, more specifi-cally a knowing how to do things with referential and predicable expressions.They are not a form of propositional knowledge, i.e., a knowledge that certainpropositions about the rules of language are true, even though they underlie therule-following behavior those rules might describe. Referential and predicableconcepts are objective cognitive universals.

11See chapter 12 for a more detailed account of conceptual natural realism as a modernform of Aristotelian essentialism.

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The objectivity of referential and predicable concepts does not consist intheir being independently real universals, however; that is, they do not have thekind of objectivity universals are assumed to have in logical realism.

The objectivity of referential and predicable concepts con-sists in their being intersubjectively realizable cognitive ca-pacities that enable us to think and communicate with oneanother.

As intersubjectively realizable cognitive capacities, moreover, concepts are notmental objects—e.g., they are not mental images or ideas as in the traditionalconceptualism of British empiricism—though when exercised they result in ob-jects, namely speech and mental acts, which are certain types of events. Inparticular, as cognitive capacities that (1) may never be exercised, or (2) thatmay be exercised at the same time by different people, or (3) by the same personat different times, concepts are not objects at all but have an unsaturated na-ture analogous to, but not the same as, the unsaturated nature concepts are saidto have in Frege’s ontology. Unlike the concepts of Frege’s ontology, however,which are functions from objects to truth values, the concepts of conceptualismare cognitive capacities that when exercised result in a speech or mental act(which may be either true or false).

Another important feature of predicable and referential concepts is that eachhas a cognitive structure that is complementary to the other—a complementar-ity that is similar to, but also different from, that between the functions thatpredicates stand for and those that quantifier phrases stand for in Frege’s on-tology.

In conceptualism, it is the complementarity between pred-icable and referential concepts that underlies the mentalchemistry of language and thought. In particular, as com-plementary, unsaturated cognitive capacities, predicable andreferential concepts mutually saturate each other when theyare jointly exercised in a speech or mental act.

In conceptualism, in other words, the nexus of predication is the joint exerciseof a referential and a predicable concept, which interact and mutually saturateeach other in a kind of mental chemistry.

A judgment or basic speech act of assertion, for example, is the result ofjointly exercising a referential and a predicable concept that underlie the use,respectively, of a noun phrase (NP) as grammatical subject and a verb phrase(VP) as predicate:

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144 CHAPTER 7. THE NEXUS OF PREDICATION

S↙ ↘

NP ... VP⇑

(nexus of predication)

In conceptualist terms this act can be represented as follows:

Assertion(Judgment)

↙ ↘referential act ..... predicable act

⇑(nexus of predication)(mutual saturation)

Here, of course, by a referential act we mean the result of exercising a referentialconcept, and by a predicable act the result of exercising a predicable concept.

7.3 Referential Concepts

Now by a referential expression, i.e., the kind of expression that stands for areferential concept, we do not mean just proper names and definite descrip-tions, such as ‘Socrates’ and ‘The man who assassinated Kennedy’, but any ofthe types of expressions that functions in natural language as grammatical sub-jects, which includes quantifier phrases such as ‘All citizens’, ‘Most democrats’,‘Few voters’, ‘Every raven’, ‘Some raven’, etc.12 In fact, in conceptual realism,only a quantifier phrase has the kind of unsaturated structure that is comple-mentary to a predicate expression the way the structure of a referential conceptis complementary to that of a predicable concept. For this reason we will rep-resent all of the different kinds of referential expressions in conceptual realismas quantifier phrases.

Referential concepts are what quantifier phrases stand forin conceptual realism, just as predicable concepts are whatpredicate expressions stand for.

Consider, for example, a judgment that every raven is black. In conceptualrealism, this judgement is analyzed as the result of jointly exercising, and mu-tually saturating, (a) the predicable concept that the predicate phrase ‘is black’stands for with (b) the referential concept that the referential phrase ‘Everyraven’ stands for.

12We will not deal with the logic of determiners such as ‘most’, ‘few’, ‘several’, etc., in thisbook. Instead we restrict ourselves to the universal and existential quantifier phrases.

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7.3. REFERENTIAL CONCEPTS 145

[Every raven]NP [is black]V P

↙ ↘(∀xRaven) ... Black(x)

↘ ↙(∀xRaven)Black(x).

A negative judgment expressed by ‘Some raven is not black’ is analyzed similarlyas:

[Some raven]NP [is not black]V P

↙ ↘(∃xRaven) ... [λx¬Black(x)]( )

↘ ↙(∀xRaven)[λx¬Black(x)](x).

The negation in this judgment is internal to the predicate, which is analyzed asthe complex predicate expression [λx¬Black(x)]( ).

Now what this view of referential expressions requires is that the logicalgrammar of conceptual realism must be expanded to include a category of com-mon nouns, or what we instead call common names.13 Common names, as theabove examples indicate, will occur as parts of referential-quantifier phrases.

Actually it is not just common names that can occur as parts of quantifierphrases, but proper names as well. In other words, instead of a category ofcommon names, what we now add to the logical grammar of conceptual realismis a category of names, which includes proper names as well as common names.The nexus of predication in conceptual realism, as we have said, is the mutualsaturation of a referential act with a predicable act, which means that objectualreference, e.g., the use of a proper name as a grammatical subject, is not essen-tially different from general reference, such as the use of the quantifier phrases‘Every raven’ and ‘Some raven’ in the above examples. Thus, instead of propernames and common names being different types of expressions, in conceptualrealism we have just one logical category of names , with common names andproper names as two distinct subcategories.

Names↙ ↘

proper names common names

What the difference is between proper names and common names is a matterwe will take up in the next section in our discussion of objectual reference.14

13We will restrict ourselves to common names that are common count nouns. The logic ofmass nouns will not be covered in this book.

14See see chapter 10, or Cocchiarella 2002, for a detailed formal description and separatedevelopment of the logic of names.

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146 CHAPTER 7. THE NEXUS OF PREDICATION

Now in addition to complex predicates, which are accounted for by λ-abstracts,we also need to account for complex referential expressions. What we mean by acomplex referential expression is a quantifier phrase containing a complex com-mon name, i.e., a common name restricted by a defining relative clause. Tosyntactically generate a complex common name, we use a forward slash, ‘/’, asa binary operator on (a) expressions from the category of common names and(b) formulas as defining relative clauses. For example, by means of this operatorwe can symbolize the restriction of the common name ‘citizen’ to ‘citizen (whois) over eighteen’, or more briefly, ‘citizen (who is) over-18’, as follows:

Citizen (who is) over 18↓ ↓

Citizen whox is over 18↘ ↙

Citizen/Over-18(x)

An assertion of the sentence ‘Every citizen (who is) over eighteen is eligible tovote’ can then be symbolized as:

[Every citizen (who is) over eighteen]NP [is eligible to vote]V P

↙ ↘(∀xCitizen/Over-18(x)) Eligible-to-vote(x)

↘ ↙(∀xCitizen/Over-18(x))Eligible-to-vote(x)

Now there is a difference in conceptual realism, we should note, betweenan initial level at which the logical analysis of a speech or mental act of agiven context is represented, and a subsequent, lower level where inferences andlogical deductions can be applied to those analyses. This means that we needrules to connect the logical forms that represent speech and mental acts withthe logical forms that represent the truth conditions and logical consequences ofthose acts in a more logically perspicuous way. For example, where the standardquantifier phrases of our previous chapters are now understood at least implicitlyas containing the ultimate, superordinate common name ‘object’, i.e., where thequantifier phrases

(∀x) and (∃x)

are now read as

(∀xObject) and (∃xObject).

then we can connect our new way of representing speech and mental acts onthe initial level of logical analysis with the more standard way on the lower,deductive level, by means of such rules as the following:

(∀xA)F (x) ↔ (∀x)[(∃yA)(x = y) → F (x)] (MP1)

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(∃xA)F (x) ↔ (∃x)[(∃yA)(x = y) ∧ F (x)] (MP2)

For example, by means of these rules we can see why the argument:(∀xA)F (x)(∃yA)(b = y)

∴ F (b)is valid in this logic.

Complex referential expressions can also be decomposed by such rules so thatthe relative clause is exported out. The following rules suffice for this purpose:

(∀xA/G(x))F (x) ↔ (∀xA)[G(x) → F (x)], (MP3)

(∃xA/G(x))F (x) ↔ (∃xA)[G(x) ∧ F (x)]. (MP4)

Thus, with these rules we can see why the argument:

(∀xA/G(x))F (x)(∃yA)(b = y) ∧G(b)

∴ F (b)

is also valid in this logic.

7.4 Singular Reference and Proper Names

The previous examples involve forms of general reference, in particular to everyraven and to some raven, respectively. This is different from most moderntheories of reference, which deal exclusively with singular reference, such asin the use of a proper name to refer to someone. The sentence ‘Socrates iswise’, for example, is usually symbolized as Wise(Socrates), or more simply asF (a), where F represents the predicate ‘is wise’ and a is an objectual constantrepresenting the proper name ‘Socrates’. Some philosophers have even arguedagainst the whole idea of general reference, claiming that logically there can beonly singular reference.15 We will turn to such arguments in the next chapter.

Now, as we have noted, a proper name can be used either with or without anexistential presupposition that the name denotes. As it turns out, it is concep-tually more perspicuous and logically appropriate that we use the quantifiers ∃and ∀ to indicate which type of use is being activated in a given speech or mentalact. Thus, for example, we can use (∃xSocrates) to represent a referential actin which the proper name ‘Socrates’ is used with existential presupposition, i.e.,with the presupposition that the name denotes.

[Socrates]NP [is wise]V P

↙ ↘(∃xSocrates) Wise(x)

↘ ↙(∃xSocrates)Wise(x)

15See, e.g., Geach 1980. A refutation of Geach’s arguments against general reference isgiven in chapter 9 and in Cocchiarella 1998.

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148 CHAPTER 7. THE NEXUS OF PREDICATION

In this initial-level analysis, i.e., the level of analysis at which we are representingthe cognitive structure of a speech or mental act, the existential quantifier phrase(∃xSocrates) indicates that the referential act that is a part of this structureis being made with the presupposition that the name ‘Socrates’ actually namesan object. In a secondary level of analysis, where deductive transformationsoccur, both proper and common names, as we will explain in a later chapter,can be transformed into objectual terms and allowed to occur in the argumentpositions of predicates in place of objectual variables, i.e., aside from occurringas parts of quantifier phrases, proper and common names will later be allowedto occur as obectual terms as well. In this lower-level logical framework, theabove expression is equivalent to the form it has in first-order “free” logic; i.e.,the following is valid in the lower-level logical framework:

(∃xSocrates)Wise(x) ↔ (∃x)[x = Socrates ∧Wise(x)].

Note that although the right-hand side has the same truth conditions as theleft, it does not represent the cognitive structure of the speech or mental act inquestion. What the right-hand side says is:

Some object is identical with Socrates and it is wise.

Now just as the existential quantifier, ∃, indicates that a proper name isbeing used with existential presupposition, so too the universal quantifier, ∀,indicates that the name is being used without existential presuppositions. Areferential use of the proper name ‘Pegasus’, for example, might well be with-out an existential presupposition that the name denotes, in which case it isappropriate to represent that use as (∀xPegasus). Thus, the sentence ‘Pegasusflies’, where the name ‘Pegasus’ is not being used with existential presuppositioncan be symbolized as

(∀xPegasus)Flies(x),

which in our lower-level logical framework is equivalent to

(∀x)[x = Pegasus→ Flies(x)].

Again, although the latter has the same truth conditions as ‘Pegasus flies’, itdoes not represent the cognitive structure of that speech act. Rather, what itsays is,

Every object is such that if it is (identical with) Pegasus, then it flies.

7.5 Definite Descriptions

Like proper names, definite descriptions can also be used to refer with, or with-out, existential presuppositions. For example, there can be a context in whicha father who asserts,

The child of mine who gets the best report card will receive a prize,

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7.5. DEFINITE DESCRIPTIONS 149

might not in fact presuppose that just one of his children will get a reportcard better than the others. The father realizes, in other words, that two ormore of his children might do equally well, in which case his use of the definitedescription is not intended to refer to exactly one child. In other words, thefather’s referential act is without existential presuppositions. Logically,what thefather asserts in that context has the same truth conditions as,

If there is just one child of mine who gets a report cardbetter than the others,then s/he will receive a prize.

But, as we have already noted, having the same truth conditions in conceptualrealism is not the same as representing the same cognitive structure of a speechor mental act.

The distinction between using a definite description with and without ex-istential presuppositions requires the introduction of two new quantifiers, ∃1

and ∀1. For example, where A is a common name and F and G are monadicpredicates, an assertion of the form, ‘The A that is F is G’ can be analyzed asfollows:

[The A that is F ]NP [is G]V P ,↙ ↘

(∃1xA/F (x)) G(x)↘ ↙(∃1xA/F (x))G(x)

On the other hand, an assertion of the same sentence but in which the use ofthe definite description is without existential presupposition will be symbolizedas

[The A that is F ]NP [is G]V P ,↙ ↘

(∀1xA/F (x)) G(x)↘ ↙(∀1xA/F (x))G(x)

In neither case, we want to emphasize, is the definite description being inter-preted as a “singular” term. In this regard, our analyses are similar to BertrandRussell’s in his famous 1905 paper, “On Denoting”. In that paper, and there-after, Russell did not represent definite descriptions as singular terms but ana-lyzed them in context in terms of quantifiers and formulas. Of course, Russelldid not distinguish between using a definite description with, as opposed towithout, existential presuppositions, but, instead, he interpreted them all asbeing used with existential presuppositions. Russell’s theory is easily emended,however, so as to include that distinction as well.

There is a difference between our analysis and Russell’s in that our analy-sis represents the cognitive structure of the speech or mental act in question,

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whereas Russell’s represents only the truth conditions of that act. The twoanalyses are logically equivalent, but only one represents the cognitive structureof the speech or mental act. Thus, given a slight reformulation of Russell’scontextual analysis, we can formulate the equivalence as rules connecting thelogical forms of the initial level of analysis, i.e., where the cognitive structureof our speech and mental acts are analyzed, with the logical forms of the lowerlevel where truth conditions and deductive relations are represented. Thus, inthe case where the definite description is used with existential presupposition,we have the following rule that connects our analysis with Russell’s:

(∃1xA/F (x))G(x) ↔ (∃xA)[(∀yA)(F (y) ↔ y = x) ∧G(x)].

The right-hand side of this biconditional says that there is one and only one Athat is an F , and that A is G. In the case where the definite description is usedwithout existential presupposition we have the related but somewhat differentrule:

(∀1xA/F (x))G(x) ↔ (∀xA)[(∀yA)(F (y) ↔ y = x) → G(x)],

which says that if there is one and only one A that is F , then that A is G.It is instructive to note why it is that although Russell’s contextual analysis

provides a perspicuous representation of the truth conditions of the speech ormental act in question, it does not at all represent the cognitive structure ofthat act. First, note that regardless of whether or not the referential act iswith or without existential presuppositions, it is in either case the same predi-cable concept that is being exercised, a fact that is explicitly represented by thelogical forms given in our analyses above for conceptual realism. On Russell’scontextual analysis, however, the predicable expressions, as represented by thebracketed formulas on the right-hand-side of each of the above biconditionals,are different.

Secondly, note that the referential import of the speech or mental act in ques-tion is properly represented in either case by a complex referential expression—namely, (∃1xA/F (x)) or (∀1xA/F (x)) —whereas the predicable aspect is repre-sented by a simple predicate expression—namely, G(x). The referential expres-sions used in Russell’s analyses, on the other hand, are the simple quantifiersphrases (∃xA) in the one case, and (∀xA) in the other, and, as just noted, thepredicate aspects are represented by complex formulas.

Russell’s contextual analysis is not wrong in how it represents the truthconditions of a speech or mental act in which a definite description is used as areferential expression; but, unlike the analyses that are given in conceptual real-ism, it does not provide an appropriate representation of the cognitive structureof that act.

In conceptual realism, the distinction between logical formsthat represent the cognitive structure of a speech or mentalact and those that give a logically perspicuous representa-tion of the truth conditions of that act is fundamental andinvolves different levels of analysis.

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The one type of logical form occurs on an initial level of analysis and is aboutthe cognitive structure of our speech and mental acts, whereas the other occurson a lower level where it is the truth conditions and logical consequences ofthose act that are perspicuously represented by logical forms.

7.6 Nominalization as Deactivation

Not all speech or mental acts are assertions in which a referential and a predi-cable concept are exercised. A denial, for example, that some raven is white isnot an assertion in which a referential act is exercised. Similarly, in expressinga conditional, such as that if Pegasus exists, then there is a winged horse, one isnot asserting either the antecedent or the consequent of the conditional. Unlikea basic assertion in which the nexus of predication is the mutual saturation ofa referential and a predicable concept, no referential concept is being exercisedin a conditional assertion.16

Unlike the negative judgment that some raven is not black, a denial thatsome raven is white, as might be expressed by the sentence ‘No raven is white’,is not an act in which reference is made to any raven, no less to every raven,even though the denial is equivalent to asserting of each raven that it is notwhite. Grammatically, the denial can be analyzed as follows,

[That some raven is white]NP [is not the case]V P

where the sentence ‘Some raven is white’ has been nominalized and transformedinto a grammatical subject. In this transformation the quantifier and predicatephrases of the sentence ‘Some raven is white’ are “deactivated,” indicating thatthe referential and predicable concepts these phrases stand for are not beingexercised. The denial is not about a raven but about the propositional contentof the sentence—namely, that it is false, i.e., not the case.

We could make this deactivation explicit by symbolizing the denial as,

Not-the-Case([(∃xRaven)White(x)]),

where the brackets around the formula (∃xRaven)White(x) indicate that thesentence has been transformed into an abstract singular term—i.e., an expres-sion that can occupy the position of an object variable where it (purports to)denotes the propositional content of the sentence. It is more convenient, how-ever, to retain the usual symbolization, namely,

¬(∃xRaven)White(x),

so long as it is clear that, unlike the equivalent sentence,

(∀xRaven)¬White(x)

16See Russell 1903, §38, for a similar view, and on how ‘If p, then q’ differs from ‘p; thereforeq’, where in the latter case both p and q are asserted, whereas neither is asserted in the former.

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152 CHAPTER 7. THE NEXUS OF PREDICATION

which is read (in non-idiomatic English) as ‘Every raven is such that it is notwhite’, no reference is being made to ravens in the speech or mental act inquestion. In conceptualism, as already noted, we distinguish the initial level ofanalysis at which a logical form represents the cognitive structure of a speechor mental act from the lower level of deductive relations at which a logicallyequivalent logical form gives a perspicuous representation of the truth conditionsof that act.

Now deactivation applies to a predicate not only when it occurs within anominalized sentence, but also when its infinitive or gerundive form occurs ina speech act as part of a complex predicate. In other words, deactivation alsoapplies directly to nominalized predicates occurring as parts of other predicates.Consider, for example, the predicate phrase ‘is famous’, which can be symbolizedas the λ-abstract [λxFamous(x)] as well as simply by Famous( ). The λ-abstract is preferable as a way of representing the infinitive ‘to be famous’,which is one form of nominalization:

to be famous↓

to be an x such that x is famous↓

[λxFamous(x)]

Now the sentence ‘Sofia wants to be famous’ does not contain the active form ofthe predicate ‘is famous’ but only a nominalized infinitive form as a componentof the complex predicate ‘wants to be famous’. When asserting this sentencewe are not asserting that Sofia is famous, in other words, where the predicableconcept that ‘is famous’ stands for is activated, i.e., exercised; rather, what thecomplex predicate ‘wants to be famous’ indicates is that the predicable conceptthat ‘is famous’ stands for has been deactivated. The whole sentence can besymbolized as

[Sofia]NP [wants [to be famous]]V P

↓ ↓ ↓(∃ySofia)[λyWants(y, [λxFamous(x)])](y)

Nominalized predicates do not denote the concepts the predicates stand forin their role as predicates, as we have already noted in a previous chapter, be-cause the latter, as cognitive capacities, have an unsaturated nature and cannotbe objects. As an abstract singular term, i.e., as an objectual term that purportsto denote a single abstract object, what a nominalized predicate denotes is theintensional content of the predicable concept the predicate otherwise stands for.In conceptual realism, what we mean by the intensional content of a predicableconcept is the result of a projection onto the level of objects of the truth con-ditions determined by the concept’s application in different possible contexts ofuse.

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Now it is important to note here that the complex predicate

[λyWants(y, [λxFamous(x)])]

does not represent a real relation between Sofia and the intensional object thatthe infinitive ‘to be famous’ denotes. What the complex predicate stands for isa predicable concept, which as a cognitive capacity has no more internal com-plexity than any other predicable concept.17 What is complex is the predicateexpression and the truth conditions determined by the concept it stands for—i.e., the conditions under which the predicate can be true of someone in anygiven possible context of use.

It is a criterion of adequacy of any theory of predication that it must ac-count for predication even in those cases where a complex predicate containsa nominalized predicate as a proper part, as well as the more simple kinds ofpredication where predicates do not have an internal complexity. What thiscriterion indicates is one of the reasons why conceptualism alone is inadequateas a formal ontology and needs to be extended to include an intensional realismof abstract objects as the intensional contents of both denials and assertions aswell as of our predicable concepts.

7.7 The Content of Referential Concepts

The fundamental insight into the nature of abstract objects , according to con-ceptual realism, is that we are able to grasp and have knowledge of such objectsas the objectified truth conditions of the concepts whose contents they are. This“object”-ification of truth conditions is realized through a reflexive abstractionin which we attempt to represent what is not an object—e.g., an unsaturatedcognitive structure underlying our use of a predicate expression—as if it werean object. In language this reflexive abstraction is institutionalized in the rule-based linguistic process of nominalization.

As already noted, we do not assume an independent realm of Platonic formsin conceptual realism in order to account for abstract objects and the logic ofnominalized predicates. Conceptual realism is not the same as either logicalrealism or conceptual Platonism. Some of the reasons why this is so are:

1. The abstract objects of conceptual realism are not entities that are predi-cated of things the way they are in logical realism and conceptualPlatonism—i.e., they are not unsaturated entities and therefore they donot have a predicative nature in conceptual realism.

2. The abstract objects of logical realism and conceptual Platonism existindependently of the evolution of culture and consciousness, whereas inconceptual realism all abstract objects, including numbers, are productsof the evolution of language and culture. Nevertheless, although they are

17Learning how to correctly use such an expression may of course be a more difficult cognitiveprocess than learning simpler predicate expressions.

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products of cultural evolution, they also have both a certain amount of au-tonomy and an essential role in the continuing evolution and developmentof knowledge and culture.18

3. In logical realism, abstract objects are objects of direct awareness, whereasin conceptual realism all knowledge must be grounded in psychologicalstates and processes. In other words, we cannot have knowledge of abstractobjects if our grasp of them as objects must be through some form ofdirect awareness. According to conceptual realism we are able to graspand have knowledge of abstract objects only as the intensional contentsof the concepts that underlie reference and predication in language andthought. That is, we are able to grasp abstract objects as the “object”-ified truth conditions of our concepts as cognitive capacities.

The reflexive abstraction that transforms the intensional content of an un-saturated predicable concept into an abstract object is a process that is notnormally achieved until post-adolescence. An even more difficult kind of re-flexive abstraction also occurs at this time. It is a double reflexive abstractionthat transforms the intensional content of a referential concept into a predicableconcept, and then that predicable concept into an abstract object.

The full process from referential concept to abstract object is doubly complexbecause it involves a reflexive abstraction on the result of a reflexive abstraction.Where A is a name (proper or common, and complex or simple), and Q is aquantifier (determiner), we define the predicate that is the result of the firstreflexive abstraction as follows:

[QxA] =df [λx(∃F )(x = F ∧ (QxA)F (x))].

In this definition the quantifier phrase (QxA) is transformed into a complexpredicate (λ-abstract), which can then be nominalized in turn as an abstractsingular term that purports to denote the intensional content of being a conceptF such that (QxA)F (x).

Consider, for example, an assertion of the sentence ‘Sofia seeks a unicorn’,which can be analyzed as follows19:

[Sofia]NP [seeks [a unicorn]]V P

↓ ↓ ↓(∃xSofia)[λxSeek(x, [∃yUnicorn])](x)

No reference to a unicorn is being made in this assertion. Instead, the referen-tial concept that the phrase ‘a unicorn’ stands for has been deactivated in thespeech act. This deactivation is represented on the initial level of analysis bytransforming the quantifier phrase into an abstract singular term denoting itsintensional content. Note that the relational predicate ‘seek’ in this example is

18See Popper 1967, p. 106, and chapter P2 of Popper & Eccles 1983 for a similar view ofabstract objects.

19This example is from Montague 1974.

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not extensional in its second argument position. What that means is that onthe lower level of representing truth conditions and logical consequences, thesentence does not imply that there is a unicorn that Sofia seeks. But the dif-ferent assertion that Sofia finds a unicorn, which is symbolized in an entirelysimilar way:

SofiaNP [finds [a unicorn]]V P

↓ ↓ ↓(∃xSofia)[λxFind(x, [∃yUnicorn])](x)

does imply that there exists a unicorn, and moreover that it has been found bySofia. That is, the following

(∃yUnicorn)(∃xSofia)Finds(x, y).

is a logical consequence of the above sentence. Thus, even though the twodifferent sentences,

(∃xSofia)[λxSeek(x, [∃yUnicorn])](x)

(∃xSofia)[λxFind(x, [∃yUnicorn])](x)

have the same logical form, and therefore represent essentially the same cognitivestructure of a speech or mental act, even though only one of them implies thatthere is a unicorn.

The reason why the one sentence implies that there is a unicorn and theother does not is that the relational predicate ‘find’, but not the predicate ‘seek’,is extensional in its second argument position. The extensionality of ‘find’ isrepresented by the following meaning postulate20:

[λxFinds(x, [∃yA])] = [λx(∃yA)Finds(x, y)].

By identity logic and λ-conversion, the following is a consequence of this meaningpostulate,

(∃xSofia)[λxFinds(x, [∃yA])](x) ↔ (∃xSofia)(∃yA)Finds(x, y)

Of course, there is no meaning postulate like this for the intensional predicate‘seek’.

20Actually, this is but an instance of the more general meaning postulate in question, namelythat for any determiner (quantifier) Q,

[λxF inds(x, [QyA])] = [λx(QyA)F inds(x, y)].

Thus if Sophia finds all, some, most, several, etc., unicorns, then, respectively, all, some,several, etc., unicorns are such that Sophia finds them.

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7.8 The Two Levels of Analysis

Our analysis of the deactivation of quantifier phrases occurring as direct ob-jects of transitive verbs such as ‘seek’ and ‘find’ is similar to an analysis givenby Richard Montague in his paper “The Proper Treatment of Quantificationin Ordinary English,” except that Montague’s framework is a type-theoreticalform of logical realism.21 Unfortunately, there is a problem with Montague’sanalysis that seems to apply to our approach as well. The problem arises whena quantifier phrase occurring as a direct-object of a complex predicate appliesto two argument positions implicit in that predicate.

Consider, for example, an assertion of the sentence ‘Gino bought and ate anapple’, which has the quantifier phrase ‘an apple’ occurring as the direct objectof the complex predicate ‘bought and ate’. Now the complex predicate ‘boughtand ate’ implicitly has two argument positions for the direct-object expression‘an apple’, one associated with ‘bought’, and the other associated with ‘ate’. Theproblem is how can we distinguish in logical syntax a nonconjunctive assertionof the form

[x]NP [(bought and ate) an apple]V P

from the different conjunctive assertion of

[x]NP [bought an apple]V P and [x]NP [ate an apple]V P

where, as the direct object, the quantifier phrase ‘an apple’ has been deacti-vated in each assertion. This is a problem because although the nonconjunctivesentence ‘x (bought and ate) an apple’, implies on the deductive level the con-junctive sentence ‘x bought an apple and x ate an apple’, nevertheless the twosentences are not logically equivalent, which means that the nonconjunctivesentence should not be analyzed as the conjunctive sentence.

Now if we assume that the analysis of ‘x (bought and ate) an apple’, whichhas a deactivated occurrence of the quantifier phrase ‘an apple’ as the direct-object argument of the predicate,

to be a y such that x bought-and-ate y

is the same as that for the complex predicate

to be a y such that x bought y and x ate y.

↓[λy(Bought(x, y) ∧Ate(x, y))],

then, the sentence ‘Gino bought and ate an apple’ would be analyzed as

21See Montague 1970b.

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7.8. THE TWO LEVELS OF ANALYSIS 157

[Gino]NP [(bought and ate) an apple]V P

↙ ↓ ↘(∃xGino) [λx[λy(Bought(x, y) ∧Ate(x, y))]([∃yApple])]

↓ ↓(∃xGino)[λx[λy(Bought(x, y) ∧Ate(x, y))]([∃yApple])](x)

But then, by λ-conversion, this analysis not only implies that Gino bought anapple and Gino ate an apple, it is also implied by the latter, i.e., on this analysisthe two are equivalent, which is contrary to the result we want.

Intuitively, what we want is to first “reactivate” the quantifier phrase ‘anapple’, i.e., to transform

[λy(Bought(x, y) ∧Ate(x, y))]([∃yApple])

to

(∃yApple)[λy(Bought(x, y) ∧Ate(x, y))](y),

before applying λ-conversion. This reactivation is justified by the fact that‘bought’ and ‘ate’ are both extensional in their direct-object positions, i.e.,because for an arbitrary common name A,

[λxBought(x, [∃yA])] = [λx(∃yA)Bought(x, y)]

and[λxAte(x, [∃yA])] = [λx(∃yA)Ate(x, y)]

are (instances of) meaning postulates for the predicates ‘bought’ and ‘ate’.22

Then, by a rule to the effect that a complex conjunctive predicates is extensionalif it is made up of two extensional predicates, It follows that the conjunctivepredicate ‘bought and ate’ is also extensional; i.e., then

[λx[λy(Bought(x, y)∧Ate(x, y))]([∃yA])] = [λx(∃yA)(Bought(x, y)∧Ate(x, y))]

is valid as well.One way to resolve the problem, which we will adopt here, is to not only

distinguish the initial level of analysis where logical forms are designed to rep-resent the cognitive structure of a speech or mental act from the deductive levelwhere deductive transformations such as λ-conversion can be applied, but alsoto mandate that before a formula can be transformed at the deductive level todetermine its logical consequences we must first apply Leibniz’s law as based onthe meaning postulates for the predicate expression represented at this initial

22As noted in the case for ‘find’, the meaning postulates apply to determiners Q in general:

[λxBought(x, [QyA])] = [λx(QyA)Bought(x, y)],

[λxAte(x, [QyA])] = [λx(QyA)Ate(x, y)].

In referring to meaning postulates hereafter, we will, for convenience of discussion, generallymention only the instances.

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level. Such an application is logically obligatory in order to transform the logicalform in question into a form that is appropriate for deductive transformationson the lower level of analysis.

As a way of understanding the rationale for this distinction note that if therewere a simple verb in English, say, e.g., ‘bouate’, that is synonymous with thecomplex verb ‘bought and ate’ the way that ‘period of time of fourteen days’ issynonymous with ‘fortnight’, then instead of the verb phrase ‘bought and atean apple’, we might also have the simpler phrase ‘bouate an apple’23, so thatinstead of an assertion of ‘Gino bought and ate an apple’ we might have had anassertion of ‘Gino bouate an apple’, the cognitive structure of which would berepresented as follows:

(∃xGino)[λxBouate(x, [∃yApple])](x).

We would then of course have a meaning postulate regarding the extensionalityof the predicate [λxBouate(x, [∃yApple])], namely,

[λxBouate(x, [∃yApple])] = [λx(∃yApple)Bouate(x, y)]

in which case, by this meaning postulate, we would then have

(∃xGino)[λx(∃yApple)Bouate(x, y)](x).

This last, by a meaning postulate connecting ‘bouate’ with ‘bought and ate’, isequivalent to

(∃xGino)[λx(∃yApple)[λy(Bought(x, y) ∧Ate(x, y))](y)](x),

which is the result we want of course for an assertion of ‘Gino bought and atean apple’. Our position for the distinction between the levels of analysis is thatwhat applies in a natural and plausible way for a simple predicate like ‘bouate’should also apply for a complex predicate like ‘bought and ate’ as well.

The point of this strategy is that every predicable con-cept expressed by a predicate or verb phrase in a speech ormental act should be considered, qua predicable concept,on a par with any other predicable concept, regardless ofthe complexity of the predicate or verb phrase used to ex-press the concept in that act. The complexity, we havesaid, is not in the concept as a cognitive capacity, but inthe predicate or verb phrase used to express that concept.

There is another way of resolving the problem, incidentally; but in the endit just brings us back to this distinction we have made between the two levelsof analysis. On this alternative, we would assume that the sentence ‘Ginobought and ate an apple’ is synonymous with ‘Gino bought an apple and ate

23Of course the more likely new verb would be in the present tense corresponding to ‘buysand eats’, with ‘bouate’ as its past tense form.

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it’, which makes explicit the two direct-object positions, one occupied by thequantifier phrase ‘an apple’ and the other by the co-referential pronoun ‘it’.This alternative reading is also synonymous with ‘Gino bought an apple andate that apple’, which makes explicit the two direct-object positions as well.

Now we have given elsewhere a conceptualist analysis of co-referential pro-nouns in terms a variable-binding ‘that’-operator, T , as in ‘that apple’, which wesymbolize as (T yApple).24 Thus, by means of the T -operator, we can symbolize‘Gino bought an apple and ate that apple’ as follows:

[Gino]NP [bought an apple and ate that apple]V P

↙ ↓(∃xGino) [λx(Bought(x, [∃yApple]) ∧Ate(x, [TyApple])]

↓ ↓(∃xGino)[λx(Bought(x, [∃yApple]) ∧Ate(x, [TyApple])](x),

where both the quantifier phrase ‘an apple’ and its co-referential phrase ‘that ap-ple’ occur deactivated in direct-object positions. Then, given that both ‘Bought’and ‘Ate’ are extensional in their second-argument positions, the above sentenceis equivalent to

(∃xGino)[λx((∃yApple)Bought(x, z) ∧ (TyApple)Ate(x, z))](x).

Now, by the following meaning postulate for the T -operator,25

[(∃yA)ϕy ∧ (TyA)ψy] = [(∃yA)(ϕy ∧ ψy)], (MPT1)

what follows by Leibniz’s law from the above sentence is

(∃xGino)[λx(∃yApple)(Bought(x, z) ∧Ate(x, z))](x),

which is the result we wanted.26 This last implies, but is not equivalent to,

(∃xGino)(∃yApple)Bought(x, z) ∧ (∃xGino)(∃yApple)Ate(x, z)).

The problem with this alternative is that unless the same distinction andrestriction is made about the different levels of analysis, the meaning postulate(MPT1) could be used to generate a contradiction. Suppose, e.g., we have a

24See Cocchiarella 1998, §7.25This rule says that the sentence ‘Some A is ϕ and that A is ψ’ is equivalent to ‘Some A is

such that it is ϕ and ψ’. An example of the rule is the equivalence between ‘Some man brokethe bank at Monte Carlo and that man died a pauper’ and ‘Some man is such that he brokethe bank at Monte Carlo and he died a pauper’.

To avoid problems that could otherwise arise, this rule, as we note below, must be appliedbefore other logical operations, such as simplification to separate conjuncts.

26Not all complex predicates like ‘bought and ate’ are extensional, incidentally. For example,in ‘Gino seeks and wants to kill a dragon’, which we might take to be synonymous with ‘Ginoseeks a dragon and wants to kill that dragon’, neither ‘seeks’ nor ‘wants to kill’ are extensional.In that case neither of the deactivated quantifier phrases ‘a dragon’ and ‘that dragon’ can bereactivated in the context in question.

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context in which a sentence of the form (∃yA)F (y) ∧ (TyA)G(y) is asserted,but in which it is also true that no A that is H is G, i.e., ¬(∃yA)[H(y)∧G(y)],even though some A is in fact H , i.e., (∃yA)H(y). But by simplification of(TyA)G(y) from the assertion in question and conjoining it with (∃yA)H(y),we would then have, by (MPT1), (∃yA)[H(y) ∧ G(y)], which is impossible inthe context in question. The way to resolve this problem is again to distinguishbetween the level of analysis at which the purpose is to represent the cognitivestructure of a speech or mental act and the level of deductive transformationswhere rules such as those for simplifying and putting together a conjunction canbe applied. Leibniz’s law and only Leibniz’s law as based on meaning postulatescan be applied on the initial level of analysis, and in fact this use of Leibniz’slaw is required before we can turn to the deductive consequences of our speechand mental acts.

The restriction to Leibniz’s law based on meaning postulates for complexpredicates, such as those for reactivating quantifier phrases that occur as argu-ments of extensional predicates, or those (like (MPT1) above) that eliminate aco-referential ‘that’-phrase, does not mean that we can replace any predicate byany logically equivalent predicate, e.g., a conjunctive predicate [λx(ϕx ∧ ψx)]by the predicate [λx¬(ϕx ∨ ψx)] obtained by replacing a conjunction by thenegation of a disjunction, or even by the predicate [λx(ψx ∧ ϕx)] obtained bysimply commuting the conjunctive components of the predicate. The purposeof the initial level of analysis is not only to represent the cognitive structure of aspeech or mental act but also to represent the “intentional content” of that act,where by “intentional content” we mean a strong sense of intensionality that re-spects cognitive structure, and in that regard a sense that is stronger than logicalequivalence. It is at this level, and only at this level, that issues about the iden-tity conditions for “intentional content” arise. It is our view that “intentionalcontent” is preserved under the transformations of predicates by meaning pos-tulates for reactivating nominalized quantifier phrases of extensional predicates,and for transforming predicates involving co-referential ‘that’-phrases into pred-icates without a co-referential ‘that’-phrase. There are other transformationsthat preserve “intentional content” as well, moreover, such as, e.g., transforminga predicate representing a passive verb into a predicate representing the activeform of that verb. But “intentional content” is not in general preserved underthe relation of logical equivalence, which is why mere logical equivalence doesnot apply to the initial level of analysis regarding the cognitive structure of ourspeech and mental acts, but only to the second level of deductive consequences.

7.9 Ontology of the Natural Numbers

Stringent identity conditions for intensional objects, including the propositionsthat are the contents of our speech and mental acts, apply at the initial level ofanalysis, we have said, i.e., the level at which we are concerned with representingthe cognitive structure of our speech and mental acts. Less stringent identityconditions are allowed for abstract objects on the level of strictly deductive

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transformations, however, where we are not concerned with the fine-grainedfeatures of cognitive structure. Logical equivalence, for example, and in somecontexts perhaps even extensional equivalence, suffices for most of the abstractobjects dealt with in mathematics, objects that initially at least are cognized asthe contents of concepts. On this level abstract objects are not only products ofcultural evolution, but are themselves the means by which the further evolutionof culture is possible.

The question of how we are to account for the ontological status of thenatural numbers and our epistemological grasp of them is an important onein philosophy and cognitive science, even if not in pure mathematics itself.Significantly, it is an issue that can be explained by the kind of analyses wehave given of the abstract objects that are the correlates of referential concepts.We note in particular in this regard that there are three ontological aspectsin which the natural numbers have in general been described by philosophersof logic and mathematics, and each of them, it turns out, corresponds to oneor another of the three types of expressions involved in the double reflexiveabstraction, or double-correlation thesis, of conceptual realism.

The first type of expression is that of numerical quantifier phrases. The en-tities associated with these expressions are usually called “quantities”, e.g., fivechairs, two cats, ten people, one president, etc. Their basic use is as referentialexpressions, and what they stand for are referential concepts. The objectualquantities that are usually associated with these expressions, according to con-ceptual realism, are the abstract objects that are the correlates of the referentialconcepts they stand for in their basic use. That is to say that quantities, onour proposal, are the contents of certain referential concepts, namely, thoseexpressed by numerical quantifier phrases.

Now some numerical quantifier phrases do not refer to any particular kindof object, but to objects in general; and for that reason we shall call them purenumerical quantifier phrases. The ones that we are interested in here can becontextually defined on the deductive level as follows:

(0x)ϕx = df¬(∃x)ϕx,(1x)ϕx = df (∃x)(ϕx ∧ (0y)[y �= x ∧ ϕ(y/x)]),(2x)ϕx = df (∃x)(ϕx ∧ (1y)[y �= x ∧ ϕ(y/x)]),

etc.

Under the ontological aspect of quantities, the natural numbers, according toconceptual realism, are none other than the pure quantities that are the concept-correlates of the referential concepts expressed by pure numerical quantifierphrases.

The second type of expression is that of the cardinal number predicates,e.g., ‘has twelve instances’, or even ‘has twelve members’. The entities associ-ated with these predicates are usually called cardinal number properties. Thus,to take one of Frege’s examples, the class of Apostles has twelve members, orequivalently, the property of being an Apostle has twelve instances. In Bertrand

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Russell’s ‘no-classes’ version of logicism, the natural numbers were in effect iden-tified with the finite cardinal number properties. Of course, in conceptualismpredicates stand for concepts and not for properties; but then in conceptual re-alism the properties in question are just the intensional objects denoted by thenominalized forms of those predicates as abstract singular terms. The predicatesthemselves can be defined in terms of the corresponding quantifier phrases asfollows. (We retain the parentheses here so as to emphasize that we are definingpredicates in these cases and not abstract singular terms.)27

0( ) = df [λF (0x)F (x)]( ),1( ) = df [λF (1x)F (x)]( ),2( ) = df [λF (2x)F (x)]( ),

etc.

These definitions, as we have already noted, indicate the general way inwhich a predicable (first-level) concept is to be correlated with a referential(second-level) concept in accordance with the Frege’s double-correlation thesis,which corresponds in conceptual realism to a type of double abstraction. Also,in accordance with conceptual realism’s version of Frege’s thesis, note that thecardinal number properties that are the intensions or concept-correlates of thepredicable concepts that the above predicates stand for are none other thanthe pure quantities already identified as the natural numbers. In other words,the natural numbers under either of these ontological aspects come to the samething, as far as conceptual realism is concerned.

Finally, the third and last type of expression for the natural numbers are thenumerals as abstract objectual terms. These are the expressions most favored inmathematics since the entities associated with them are the natural numbers intheir most simple and direct ontological aspect. But their simplicity of expres-sion is misleading and leads to the epistemological problem of how the naturalnumbers are conceptually accessible to us. This problem, along with the prob-lem of how intensional objects are in general conceptually accessible to us, isresolved in conceptual realism through its version of Frege’s double-correlationthesis. Abstract objectual terms, including numerals, according to conceptualrealism, are ultimately explained on the basis of a nominalizing transformationof predicates. Thus, the numerals in particular can be defined as follows:

0 = df [λF (0x)F (x)],1 = df [λF (1x)F (x)],2 = df [λF (2x)F (x)],

etc.

Note that on this analysis the natural numbers are epistemically accessible tous in just the same way that concept-correlates in general are—namely, through

27Reminder: A λ-abstract of the form [λFϕ] is an abbreviation for a λ-abstract of the form[λx(∃F )(x = F ∧ ϕ)]).

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the concepts whose correlates they are. That is, we are conceptually able tothink about the natural numbers as objects because we are conceptually ableto lay hold upon them as the intensions of the (predicable concepts determinedby the) referential concepts that pure numerical quantifier phrases stand for.And this conceptual ability, we have said, is just what in conceptual realism isexplained by Frege’s double-correlation thesis.

Finally, that the natural numbers denoted by numerals as abstract singularterms turn out to be just the pure quantities that are the intensions of thereferential concepts expressed by pure numerical quantifiers is a natural, and nota fortuitous, result. For it explains why children first learn about quantities, e.g.,two apples, four cows, ten trees, etc., and only later, after they have formed theconcept of an object simpliciter, learn about the numbers themselves. That is,children learn to think about the natural numbers as abstract objects by learningfirst to objectify the content of the referential concepts expressed by numericalquantifier phrases, and then, by means of the concept of an object simpliciter,they learn to think about the natural numbers as the objectified contents ofthe referential concepts expressed by pure numerical quantifier phrases. Andthat this is possible is just what is explained in conceptual realism by Frege’sdouble-correlation thesis.

7.10 Ontology of Fictional Objects

The natural numbers are not the only entities that can be accounted for in termsof the deactivation of referential concepts. In fictional discourse, for example,our apparent reference to the objects of a story is not a form of direct referencebut of indirect reference, i.e., indirect, or deactivated, reference with respect toan implicit operator such as ‘In the story ...’. The propositions that make upthe content of a story are not the same as states of affairs, which are part of thecausal order of the natural world. As intensional objects, propositions enableus to construct a “bracketed world” of intensional content within which we areable to freely speculate and construct various hypotheses and theories aboutthe natural world. In fact, whether true or false, all theories about the naturalworld consist of a system of propositions, which we are able to contemplateindependently of whether or not there are any states of affairs in the naturalworld corresponding to them. In this way, as intensional objects, propositionsserve to advance the development of science and technology, and thereby thefurther evolution of culture.

Propositions make up the content of our fables and myths, and, in fact,they are the content of stories of all kinds, both true and false. In this waypropositions and the abstract objects that are the contents of deactivated ref-erential expressions also serve the literary and aesthetic purposes of culture. Inreading a fictional story, for example, we are given to understand that none ofthe references made in the story are to be taken literally, i.e., that all of thereferential expressions occurring in the sentences of the story are understoodto be deactivated, by which we mean that we are dealing with the intensional

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content of those referential expressions and not with any real objects that thoseexpressions might otherwise be used to refer to in direct discourse. The same istrue of stories that are put forward as descriptions of reality—except in thosecases, assuming we believe the stories in question, we indirectly re-activate thereferential function of the expressions used in those stories by indicating, evenif only implicitly, that the stories are to be taken as true. Here, in fact, we seethe significance of the law,

that φ is true ↔ φ,

or, in symbols,True([ϕ]) ↔ ϕ,

whereby an assertoric occurrence of a propositional form φ is connected with anominalized occurrence of φ. All stories are to be interpreted in this regard asa form of indirect discourse—such as the contexts that occur within the scopeof an ‘In-the-story’ operator, which often is only implicit when we read, or arebeing told, a story. For it is only by first understanding the content of a storythat we can then raise the question of its veracity, i.e., the question of whetheror not there are states of affairs in the space-time causal manifold correspondingto the propositions that make up that story.

All fictional characters, on this account, are intensional objects—namely, theintensional objects that are the correlates of the referential expressions used inthe fiction in question. These intensional objects are accounted for in conceptualrealism, as we have said, through the deactivation of referential concepts, whichlogically is represented by a double correlation, first of referential concepts withpredicable concepts, and then of the latter with their concept-correlates.

In a specific story, say, A, both the propositions and the intensional objectsinvolved in the referential expressions of that story may be relativized as follows,

[QxS]A =df [λy(∃F )(y = F ∧ In(A, [(QxS)F (x)])],

where ‘[(QxS)F (x)]’ is a nominalization of the formula ‘(QxS)F (x)’, and‘In(A, [...])’ represents the formula-operator ‘In (the story) A, ...’. Thus, thereferential expression ‘Sherlock Holmes’ will be taken to have one intensionalobject as its content in Conan Doyle’s novel The Hound of the Baskervilles anda different intensional object in Conan Doyle’s The Valley of Fear.

Now because the singular term ‘Sherlock Holmes’ is used with existentialpresupposition in the fictional worlds of both novels, it is represented as hav-ing the logical form ‘(∃xSherlock-Holmes)’ in the sentences that make up thewritten text of those novels; and therefore the intensional objects that arethe constituents of the propositions making up the content of those novelsare represented by, e.g., ‘[∃xSherlock-Holmes]Baskervilles’ and ‘[∃xSherlock-Holmes]V alley ’, respectively. Though these intensional objects are not identical,they are counterparts to one another in much the sense of David Lewis’s coun-terpart theory. Indeed, it is here among the intensional objects of our variousstories—and not among the concrete objects that exist in, and across, different

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causally possible worlds—that David Lewis’s counterpart theory has its properapplication.

It is the relativization of intensional objects in this way that explains theso-called “incompleteness” of fictional objects. There are many predicate expres-sions of English, for example, that can be meaningfully applied to humans butthat are neither affirmed nor denied of the character Sherlock Holmes in any ofConan Doyle’s novels. Neither the formula ‘In(A, [(∃xSherlock-Holmes)F (x)])’nor ‘In(A, [(∃xSherlock-Holmes)¬F (x)])’ will then be true of the concept (asa value of ‘F ’) that such a predicate might stand for, in order words, regardlesswhich of Conan Doyle’s novels we consider as a value of ‘A’; and therefore, nei-ther ‘[∃xSherlock-Holmes]A(F )’ nor ‘[∃xSherlock-Holmes]A([λx¬F (x)])’ willbe true as well—which is to say that, in the story A, the character SherlockHolmes falls under neither the concept F nor its complement, and is, therefore,“incomplete” in that regard.

Alexius Meinong’s impossible objects, when construed as fictional charactersor objects (or as intensional objects of someone’s belief-space), are also “incom-plete” in this way.28 Thus, whereas ‘The round square is round and square’is false as a form of direct discourse, nevertheless, it could be true in a givenfictional context. Suppose, for example, we construct a story called, Romeo andJuliet in Flatland , which takes place in a two-dimensional world (Flatland) ata time when two families, the Montagues and the Capulets, are having a feud.In Flatland, the Capulets, one of whom is Juliet, are all circles, and the Mon-tagues, one of whom is Romeo, are all squares. (Juliet has curves and Romeohas angles.) Unknown to the two families, Romeo and Juliet have an affair anddecide to live together in secret. In time, Juliet becomes pregnant and, giventhe difference in genetic makeup between Romeo and herself, she gives birth toa round square. Although Romeo and Juliet both love their baby, the roundsquare, the two families, the Montagues and the Capulets, become enraged whenthey discover what has happened. They kill Romeo and Juliet, and their baby,the round square. But, not wanting it to be known that a round square—which,given the cruel social mores of Flatland society, would have been considered amonster—was born into either family, the Montagues and Capulets keep thebirth, and death, of the round square a secret. They then pass it around thatRomeo and Juliet were ill-starred lovers who committed suicide in despair of theopen hostility between their respective families. The story ends with Romeo andJuliet being eulogized and buried together—but without their baby, the roundsquare, whose body was cremated and reduced to ashes.

As this story makes clear, we can meaningfully talk about “impossible”objects as if they were actual objects—although such talk can be true onlywhen relativized to a context of indirect discourse, such as a story, and perhapsthe belief-space of someone with inconsistent beliefs. Thus, for example, as partof the story, Romeo and Juliet in Flatland, it is true to say that the round square

28See Meinong 1904. Also, see Parsons 1980 for a logical reconstruction of Meinong’s on-tology and Cocchiarella 1982 for a discussion of Parson’s reconstruction and an alternativeaccount to Meinong’s ontology.

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is round and square, which, formally, can be represented as follows:

In(R&J-in-Flatland, [(∃1xSquare/Round(x))[λxRound(x) ∧ Square(x)](x)]).

Thus, even though both

[∃1xSquare/Round(x)]([λxRound(x)]),

and[∃1xSquare/Round(x)]([λxSquare(x)]),

are false regarding the intensional content of ‘The round square’ simpliciter,nevertheless, both

[∃1xSquare/Round(x)]R&J-in-Flatland([λxRound(x)]),

and[∃1xSquare/Round(x)]R&J-in-Flatland([λxSquare(x)]),

are true of the intensional content of ‘The round square’ relativized to thestory, Romeo and Juliet in Flatland. Nevertheless, as an object of a fictional,intensional world—as opposed to the objects of the actual world of nature—suchan “impossible” object will be “incomplete” with respect to the different kindsof things that are in fact said of it in its fictional world. It is in this way thatconceptual realism is able to explain the “incomplete” and “impossible” objectsof Meinong’s theory of objects.29

7.11 Summary and Concluding Remarks

• In conceptual realism the nexus of predication is what accounts for theunity of a speech or mental act that is the result of jointly exercising a referentialand predicable concept.

• A unified account of both general and singular reference can be given interms of this nexus. Such a unified account is possible because the category ofnames includes both proper and common names.

• A unified account can also be given in terms of this nexus for predicateexpressions that contain abstract noun phrases, such as infinitives and gerunds.

• The same unified account also applies to complex predicates containingquantifier phrases as direct-object expressions of transitive verbs, such as thephrase ‘a unicorn’ in ‘Sofia seeks a unicorn’. Conceptually, the content of sucha quantifier phrase and the referential concept it stands for is “object”-ifiedthrough a double reflexive abstraction that first generates a predicable conceptand then the content of that concept by deactivation and nominalization. Alldirect objects of speech and thought are intensionalized in this way so that aparallel analysis is given for ‘Sofia finds a unicorn’ as for ‘Sofia seeks a unicorn’.

29See Cocchiarella 1987, chapter 3, for a more detailed account of how Meinong’s theorycan be reconstructed in the kind of framework we have in mind here.

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And yet, relations, such as Finds, that are extensional in their second argumentpositions can still be distinguished from those that are not, such as Seeks, bymeaning postulates.

• The same double reflexive abstraction explains the three different typesof expressions that represent the natural number concepts, namely first, asnumerical quantifier phrases, such as three men, two cows, five cups, etc., then,second, as the cardinal number predicates ‘has n instances’, or ‘has n members’,and the third as the numerals ‘1’, ‘2’, ‘3’, etc., i.e., as objectual terms thatpurport to name the natural numbers as abstract objects.

• The deactivation of referential expressions is also involved in fictional dis-course and in stories in general. The objects of fiction are none other than theintensional objects that deactivated referential expressions denote as abstractobjectual terms. This account of the ontology of fictional objects explains their“incompleteness” as well as their status as intensional content.

Finally, we note that there is much more involved in a conceptualist analysisof language and thought beyond our account of the nexus of predication. Onesuch issue, which we will take up in chapter eleven, is how both proper andcommon names can be transformed into objectual terms occurring as denotativearguments of predicates, which is different from their referential role as parts ofquantifier phrases. Such objectual terms denote “classes as many,” as opposedto sets as “classes as ones”. In addition to providing another account of “theone and the many”, classes as many also provide truth conditions for pluralreference and predication. Classes as many also lead to a natural representationof the natural numbers as properties of classes as many.

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Chapter 8

Medieval Logic andConceptual Realism

Reference in conceptual realism, as we have explained in the previous chapter,is not restricted to so-called singular terms, e.g., proper names and definite de-scriptions, but involves quantifier phrases containing common or proper names,where the former can be complex as well as simple.1 This uniform accountof both singular and general reference is similar in a number of respects tothe medieval suppositio theory of the 14th century. In fact, we maintain thatwith minor modifications the medieval theory can be reconstructed within theframework of conceptual realism. Indeed, conceptual realism provides a generalphilosophical framework within which we can reconstruct and explicate not justthe suppositio theory as a theory of reference, but a number of other issues thatwere central to medieval logic as well, such as the identity theory of the cop-ula (for categorical propositions) and the difference between real and nominaldefinitions. One of the benefits of such a reconstruction is that we can defendmedieval logic against the kinds of arguments that Peter Geach has given againsta uniform account of general and singular reference in his book Reference andGenerality.2

8.1 Terminist Logic and Mental Language

The medieval logic we will be concerned with here is what came to be called “ter-minist logic” because it was primarily concerned with the semantics or “prop-erties of terms” (proprietates terminorum), i.e., of adjectives and proper and

1This chapter is based on my 2001 paper, “A Logical Reconstruction of MedievalTerminist Logic in Conceptual Realism,” in Logical Analysis and History of Philosophy,vol. 4.

2See T.K. Scott [1966b] for criticism of Geach’s account of Ockham’s suppositio theory.A refutation of Geach’s general arguments against the kind of uniform account of referencegiven in conceptual realism will be presented in the next chapter.

169

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common count nouns, the latter of which we will continue to speak of as com-mon names. These “terms” are also called categorematic expressions, as op-posed to the syncagorematic logical expressions, such as the quantifier words‘all’ and ‘some’, and the connectives ‘or’, ‘and’, ‘if then’, etc. This approachto logic began in the 13th century with such logicians as Peter of Spain, RogerBacon, Lambert of Auxerre, and William of Sherwood. Around 1270, however,terminist logic “went into a kind of hibernation” and was replaced by an alter-native movement known as “speculative grammar”.3 The hibernation ended inthe early 14th century when “terminist-style semantic theory woke up again”.4

The major terminist logicians of this later period are William of Ockham, JohnBuridan, Walter Burley, and Gregory of Rimini. Ockham and Buridan are gen-erally described as nominalists, but on our account really were conceptualists.Burley is said to be a realist, but his realism was really a form of conceptualnatural realism (which we will describe in a later chapter).

The conceptualism of the terminist logicians is clearly seen in their assump-tion that underlying spoken and written language there exists a mental lan-guage made up of both categorematic and syncategorematic concepts and men-tal propositions, or the mental acts that in conceptual realism we prefer to calljudgments—or thoughts when they are only entertained and not asserted orjudged.5 This “language of thought”, which is generally referred to today asMental , was assumed to be common to all humans.6 Unlike spoken and writtenlanguages, which were said to be “conventional” languages, Mental was said tobe a “natural” language, by which was meant a language somehow establishedby nature. Apparently, what makes Mental natural is that its categorematicconcepts (mental terms) “get their signification by nature and not by conven-tion.”7

Signification is the basic semantical relation of Mental, but it applies onlyto categorematic concepts, syncategorematic concepts being said not to signifyat all. The signification of a categorematic concept is not an intensional ob-ject, but rather the things that fall under the concept, by which was meant,in a narrow sense, the things that now fall under the concept, but which, ina wider sense, included the things that could fall under the concept, i.e., thethings that can fall under the concept, and therefore the things that did, do, orwill fall under the concept as well. This distinction between narrow and widesignification was possible because our thoughts (mental acts) occur in time and,by means of tense and modal modifications, can be oriented toward the pastor the future, as well as the present, and even toward what is merely possible.

3Spade [1996], p.43. Spade gives a useful account of the history of this period. We will berelying on this text throughout this essay.

4Ibid.5In conceptual realism a proposition is the intensional content of a speech or mental act,

and not, as in terminist logic, the assertion or judgment made. We will allow some laxity inthe use of ‘proposition’ in our account of terminist logic, however.

6Geach in [1957], p.102, is one of the first to use ‘Mental’ this way. See Trentman [1970],Normore [1990], and Spade [1996], chapter 4, for an account of Mental. Normore [1985], p.189,explicitly refers to it as “a language of thought.”

7Spade [1996], p.93.

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Mental is a tensed and modal language, in other words, containing among itssyncategorematic concepts certain operators that correspond to the tenses andmodal modifications of verbs, or what the medieval logicians called ampliation.Moreover, because mental terms can signify in a wider sense, Mental is ontolog-ically committed to a form of possibilism, though the possible objects signified(in the wide sense) by a concept seemed to be only those that are possible innature.8

That mental terms get their signification by nature is based on the idea thatthere is a “natural likeness” between concepts and the things they signify, anatural likeness that is caused, apparently, by the things signified.9 This sug-gests that concepts are something like images, which is much too restrictivea view in that it excludes those concepts of things that cannot be “imaged,”i.e., things other than the visible objects of the macrophysical world and forwhich we cannot conceive a “likeness.” Scientific concepts of things, processesand events in the microphysical world can be mathematically modelled, and inthat sense “imagined,” but, because they are smaller than the wavelength oflight, they cannot be “imaged” or pictured literally, and therefore there can beno “natural likeness” between our concepts and such things. Nor, of course,can there be a “natural likeness” (or a causal relation) between concepts andmathematical objects, be they numbers, sets, fields, or whatever.10 The idea ofa “natural likeness” also suggests that the concepts corresponding to commonnames must all be sortal concepts, i.e., concepts that have identity criteria asso-ciated with them. But the concepts corresponding to the common names ‘thing’,‘object’, ‘individual’, and even ‘physical object’, and ‘abstract object’, do nothave identity criteria associated with them (the way, e.g., the common names‘man’, ‘dog’, ‘carrot’, ‘chair’, etc., do), and yet such (nonsortal) concepts wereexplicitly allowed by the terminist logicians. Even the common name ‘furniture’,unlike different sorts of furniture (such as tables and chairs) and the commonname ‘event’, unlike sorts of events (such as a running, kicking, or kissing) donot have identity criteria associated with them. In what sense is there a “nat-ural likeness” between things in general and the common-name concept thing,or between physical objects and the complex common-name concept physicalobject? We can conceive of a “likeness” to, or form an image of, a chair or table,i.e., of a particular sort of furniture, but we cannot form an image of furniturein general, nor can we form an image of events in general, though we can forman image of a kissing (between, say, Abelard and Heloise, or Bill and Monica).

8See Normore [1985], p.191. It is not clear Normore would agree that all possibilia inOckham’s and Buridan’s ontology are possible in nature, i.e. that the modality in questionis a natural possibility (as opposed, e.g., to a logical or metaphysical possibility). A naturalpossibility and necessity seems to be what Burley had in mind, however.

9According to Spade, “Ockham says the things a concept signifies are all such that: (a)The concept is like every one of them. (b) It is not like any one of them any more than it islike any other one; it is equally like all of them. (c) It is like any one of them more than it islike anything else—anything that is not signified by the concept.” ([1996], pp.153f)

10It seems that mathematical objects were not accounted for at all in terminist logic. Thereconstruction we give below in terms of conceptual realism provides a way of giving anaccount as indicated in the previous chapter.

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The notion that all categorematic concepts have a “natural similarity” withthe things they signify is much too restrictive and is one of the semantic featuresof Mental that we do not assume applies to conceptual realism. For this reason,we will not refer hereafter to Mental as a “natural” language, but will insteadfollow contemporary practice and speak of historically real languages such asLatin, English, French, etc., as natural languages. We prefer the contemporaryusage partly because it is contemporary, but also because we want to distinguishartificial languages (such as HST∗

λ), which are also “conventional,” from thehistorically developed natural languages spoken by a linguistic community.

In regard to the relation between Mental and spoken (and written) languages,Ockham’s view differs somewhat from Buridan’s. According to Ockham, for ex-ample, a linguistic term used in an assertion signifies the same things that aresignified by the concept corresponding to the term. Buridan, on the other hand,says that the spoken linguistic term signifies the concept corresponding to theterm, and only indirectly, through the concept, signifies the things that it signi-fies.11 In conceptualism, we take a position similar to Ockham’s in that, e.g., areferential expression of English that occurs as the noun phrase of an assertionin English refers to the same things that are referred to by the referential con-cept that the expression stands for—and, in fact, the linguistic act is just themental act expressed overtly in English.12 On the other hand, we do want tosay in conceptualism that a referential expression of a spoken language standsfor (stat pro) a referential concept, and similarly that a predicable expressionstands for a predicable concept, that might be exercised in a given speech act,which might seem in some respects similar to what Buridan says. But then,Ockham does have a notion of subordination, which he says holds between alinguistic term and the corresponding concept, which seems to be essentiallywhat is meant in saying that the term stands for the concept. It is Ockham’sposition, in other words, that is closer to conceptualism as we understand ithere.

The relation between a conventional language such as Latin or English anda language of thought such as Mental is sometimes said to be analogical, espe-cially by writers sympathetic to nominalism. Peter Geach, for example, held the“general thesis” that “language about thoughts is an analogical development oflanguage about language,”13 and, in particular, that “the concept judging is... an analogical extension of the concept saying.”14 Similarly, Wilfrid Sellarsclaimed that our view of thoughts as “inner episodes” is a theoretical construc-tion modelled upon our view of meaningful linguistic behavior, and in particularthat “concepts pertaining to the intentionality of thoughts,” such as that of ref-erence, are “derivative from concepts pertaining to meaningful speech.”15 The

11Cf. Normore [1985], p.190, and Spade [1996], chapter 3.12This is not to say that there are no “inner episodes” of thinking, i.e. of referring and

predicating, that are nonlinguistic.13Geach [1957], p.98.14Ibid., p.7515Sellars [1981], p. 326. Also, see “Empiricism and the Philosophy of Mind” in Sellars

[1963].

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thesis does seem to be one that Ockham followed in that some of the featuresof Latin are assumed to carry over into Mental. Indeed, according to Geach,“the grammar of Mental turns out to be remarkably like Latin grammar. Thereare nouns and verbs in Mental; nouns have cases and numbers, and verbs havevoice, mood, tense, number, and person.”16

Despite his general thesis, Geach warns us not to carry the analogy toofar. Ockham, in particular, according to Geach, “merely transfers features ofLatin grammar to Mental, and then regards this as explaining why such fea-tures occur in Latin.”17 Except for mood and tense, Geach maintained that“the grammatical properties ascribed by Ockham to Mental words may all beeasily dismissed.”18 The exception is noteworthy because tense “does enter intothe content of our thoughts,” and, according to Geach, “there are modal dif-ferences between thoughts—though the moods of a natural language like Latinare a very inadequate indication of this, being cluttered with a lot of logicallyinsignificant idiomatic uses.”19 This of course is a position very much like theone we described in chapter two.

Geach’s claim that Ockham carried the analogy of thought to language toofar has been attacked on several fronts.20 Nevertheless, his critics agree that theproper comparison is not of Mental with Latin but of Mental with the kind of“ideal languages” that logicians and philosophers have constructed in the twenti-eth century, i.e., with a logistic system from the point of view of logic as languageas opposed to the view of logic as calculus.21 Thus, according to J. Trentman,“Ockham’s real criterion ... for admitting grammatical distinctions into Mentalamounts to asking whether the distinctions in question would be necessary in anideal language—ideal for a complete, true description of the world.”22 Similarly,according to Paul Spade, “mental language is to be a kind of ideal language,which has only those features it needs to enable it to discern the true from thefalse, to describe the world adequately and accurately.”23 Of course, an ideallanguage should account not only for an adequate and accurate description ofthe world, but also for valid reasoning about the world. In other words, Mentalshould be constructed as a logically ideal language, relative to which analysesof natural language sentences can be given, thereby resulting in logically per-spicuous representations of the truth conditions of those sentences. The logicalforms representing these truth conditions would then determine which sentencesfollow validly from other sentences as premises, i.e., they would then determine

16Geach [1957], p.102. See Spade [1996], chapter 4, for a more detailed list of the “common”and “proper accidents” of nouns and verbs of Latin, but where only the “common accidents”are part of the grammar of Mental.

17Ibid.18Ibid., p.103.19Ibid., pp.103f.20See Trentman [1970] and Spade [1996], chapter 4.21See van Heijenoort 1967 for an account of the distinction between logic as language and

logic as calculus.221970, p.589.231996, p.110.

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the conditions for valid reasoning in terms of the recursive operations of logicalsyntax.24

The kind of analyses of natural language sentences intended here are whatthe terminist logicians called “expounding” (exponere) or “exposing” (exposi-tio). Thus, according to Calvin Normore, “expositio is a natural result of therecognition that the surface grammar of a sentence is not always a reliable guideeither to its truth conditions or its inferential connections to other sentences,”which suggests that the analysis (or exposition) of a natural language sentence(of Latin, English, etc.) should not result in another sentence of that languagebut in a logical form of an ideal language based on logical grammar.25 Thatis why the proper province of logic for Buridan, according to Normore, is “thearticulation of truth conditions for grammatically complex sentences,” i.e., theprocess of making “the logical form of sentences explicit.”26

What is needed in terminist logic, but up until now has not been given, isa representation of Mental as a logistic system based on the view of logic as aconceptualist theory of predication. Such a system should provide a perspicu-ous representation of the truth conditions of our speech and mental acts, andthereby, in terms of the recursive operations of logical syntax, the validity ofour arguments as well (as determined by the deductive level of the theory).Moreover, as a conceptualist theory of our speech and mental acts, the systemshould also provide (on the initial level) logical forms that perspicuously repre-sent the cognitive structure of those acts, including in particular the referentialand predicable concepts underlying them. It is these concepts that in one wayor another correspond to what the terminist logicians called supposition. Thesystem we have described for conceptual realism, we believe, can be used in justthis way, even if some features of the system may seem to be in conflict withcertain terminist theses.

8.2 Ockham’s Early Theory of Ficta

The ontology of intensional objects we have described in the previous chaptermight seem to be in conflict with the nominalism commonly associated withOckham and the terminist logicians. Ockham, for example, clearly rejectedthe Platonist interpretation of nominalized predicates; but that was because heassociated it with a Platonist or realist theory of predication. On this theory,a person is said to be wise, for example, because he exemplifies the quality orproperty denoted by ‘wisdom’. That is, a predication of the form ‘x is wise’is explained on the basis of a supposedly more basic sentence of the form ‘x

24The recursive operations of logical syntax will generate some logical forms that do notrepresent propositions of Mental, however, but which are needed for the deductive machineryof the ideal language by which to prove the validity of arguments—and for the generation ofthose forms that do represent the propositions of Mental as well. (See §8.6 below for more onthis point.)

25Normore 1985, p.192.26Ibid.

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exemplifies wisdom’, which means that signification in the sense in which ‘wise’signifies wise individuals is not a basic semantic notion after all.27

This is not how intensional objects are understood in conceptual realism,however, where predication is explained in terms of the mutual saturation ofa referential and predicable concept in a speech or mental act. Intensionalobjects, as we have said, are products of language and culture that do not pre-exist the evolution of consciousness, and, as such, can in no sense be the basisof a theory of predication. Something like a modal-moderate realist theory ofpredication is part of conceptual natural realism, but the natural kinds andproperties that are part of that theory are not objects, and therefore they arenot the intensional objects denoted by nominalized predicates. Natural kinds,for example, are unsaturated causal structures that are the basis of causal laws;and they become saturated only as the nexuses of states of affairs. A terministlogician, such as Ockham (reconstructed somewhat as described below), wouldhave rejected the (empirical) posits of conceptual natural realism, althoughanother terminist logician, such as Burley, might well have accepted them.28

Ockham, however, could have accepted the intensional objects of conceptualintensional realism, just as he once accepted ficta as intentional objects.

Ockham, in other words, did accept something like our account of intensionalobjects in his early view of concepts as ficta.29 On this view, which is some-times called the fictum theory, concepts are the intentional objects of acts ofintellection (e.g., judgments). Ficta were not regarded as independently “real”entities, but were said to have only an “intentional being” (esse objectivum), ac-cording to which “their being is their being cognized.”30 Ficta included not only“universals,” such as humanity and triangularity, but also logical objects, suchas propositions (as abstract objects), and fictitious objects, such as chimerasand goatstags, and also impossible objects, such as the round square.31 Theseare just the sort of objects that are accounted for in conceptual realism as in-tensional objects, which suggests that the latter might not really be so alien to

27See Loux [1974], p. 6.28See chapter twelve for a description of conceptual natural realism. The same logistic

formulation of natural realism seems to apply to Burley, incidentally, in that Burley acceptedthe thesis that some concepts have a natural kind, or a natural property., corresponding tothem.

29Spade 1996, p.154, and Normore 1990, p.59, for a description of Ockham’s fictum theory.30Spade [1996], p.156. Marilyn McCord Adams prefers to speak of Ockham’s early view

of concepts as the “objective existence theory.” She objects to calling such objects ficta,apparently because sometimes the intentional objects thought about are real and not fictitiousthings ([1977], pp.151f). Our concern here, however, is not with intent ional objects as such,whether real or fictitious, but with Ockham’s early theory of ficta as a way of seeing whythe intensional objects of conceptual realism can be accommodated in a reconstructed versionof terminist logic. After all, there are real objects corresponding to some of our intensionalobjects, even if they are not the same as those intensional objects.

31See Adams [1977], p.147. T.K. Scott, [1966a], p.16, notes that Gregory of Rimini arguedfor propositions in the modern sense of abstract objects, or what he called enunciables (enun-tiabile), which were denoted by infinitive or gerundive expressions that amounted in effectto nominalized sentences. These enunciables, like the abstract objects of conceptual realism,were taken as real, but not existent, objects.

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Ockham’s ontology after all, even if he later changed his mind about ficta.32

Ockham described his earlier view of concepts as intentional objects in hisReportatio, where he made a distinction between two kinds of mental language.One was Mental as we have described it so far, and “the other was a lan-guage whose terms were the mental representations of the spoken expressionsof natural language itself,” i.e., a mental language that is “posterior to spokenlanguage,” and therefore based on spoken language.33 This view of ficta is notunlike our view of intensional objects in conceptual realism, where conceptualnominalization and the object ification, or reification, of these objects is “poste-rior” to linguistic nominalization both in the historical development of languageand in the conceptual development of individuals in their acquisition and use oflanguage.

Ockham did give up his early theory of concepts as ficta in favor of his latertheory of concepts as mental acts (intellectiones). But that does not mean thatthe role ficta played in explaining how we can think about “unreal” objectscan now be explained by concepts as mental acts; and, in fact, we maintainthey cannot, and that something like ficta—namely, the intensional objects ofconceptual realism—are needed to fulfill this role. Ontologically, ficta and theintensional objects of conceptual realism are similar in the way they dependupon concepts; for just as ficta, as intentional objects, had their being in beingcognized in the mental acts that Ockham later identified with concepts, so toointensional objects similarly have their being in the concepts whose intensionsthey are. Ockham was right in his rejection of concepts as ficta, but wrong inthen rejecting ficta altogether. They have a role to play in mathematics andthe semantics of fiction, and stories and theories in general as we explained inthe previous chapter; and, perhaps even more importantly, as we will see, inthe semantics of those concepts that intensional verbs (such as ‘seek’, ‘promise’,‘owe’, etc.) stand for, as well as in our conceptualist theory of predication forconcepts based on relations in general, including the copula. Our proposal here isto take the intensional objects of conceptual realism as a “logical reconstruction”of Ockham’s early theory of ficta as intentional objects.

8.3 Ockham’s Later Theory of Concepts

Ockham’s later theory of concepts, which is sometimes called the intellectio ormental-act theory, does not identify concepts with intentional objects, but withmental acts themselves, i.e., with actual mental occurrences.34 The common-name concept man, for example, “is the very act of thinking of men”35; that is,as a mental occurrence, that very act signifies all men. This theory is similar tocontemporary nominalism according to which, e.g., an actual spoken linguistictoken of the word ‘man’ is said be true of all men, but as a matter of convention

32For an analysis of the round square as a fictum, see §7.10 of the previous chapter.33Normore 1990, p.59.34Spade, [1996], p. 155, and Adams [1977], p. 145.35Spade [1996], p. 155.

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only, and not because there is a “natural likeness” between the word and men.One problem with the mental-act theory is how the same concept, e.g., man,

can be common to all humans as a term of Mental. That is, if a person’s conceptman is just a mental act (event) of that person, and one person’s mental act isnever the same as another’s, or of the same person at another time, then howcan the same concept man be common to different people, or to the same personat different times? The answer that was given, apparently, is that although oneperson’s concept man is “numerically different” from another person’s conceptman—i.e. their mental acts of thinking of men are different mental events—nevertheless, the two concepts are “exact duplicates of one another.”36 But inwhat sense can the mental acts of two or more people, or of the same personat different times, be exact duplicates of one another? Is it because there is anassumed “natural likeness” between concepts and the things they signify, i.e.,that one person’s concept man will then have a “natural likeness” with anotherperson’s concept man (or with the same person’s concept man at a differenttime)? If so, then, for reasons already given against the supposed “naturallikeness” between concepts and the things they signify, this is an answer notacceptable today, or at least not for conceptual realism as we understand ithere.

Ockham does suggest an alternative—namely, that concepts as mental actsare qualities of the mind, and in particular qualities that “exist” only whena mind is exercising the mental act in question, as in moderate realism (butrestricted to qualities that inhere only in minds37). Different mental acts ofthinking of men are then just different instances of a mind’s having the samequality.38 This version of the mental act theory is sometimes called the qualitytheory of concepts, according to which “the concept is a real quality inheringin the mind just like any other real property.”39 This theory might explain howthe same concept can be exercised in two actual mental acts—namely, by beingthe same mental quality inhering in the mind or minds whose acts they are—but it doesn’t account for concepts that are in fact never exercised and that wenevertheless “tacitly know” or have in our conceptual repertoire—e.g., conceptsof very large numbers or of things that we could, but in fact never, think orspeak about. Also, it is not clear how a concept as a quality inhering in a mindonly when it is exercised can explain how the mind can exercise that concept,nor how it might inform the act with a referential or predicable nature.

Our proposal is to “reconstruct,” or replace, Ockham’s theory of conceptsas mental qualities with the theory of concepts as cognitive capacities thatwe have described in the previous chapter for conceptual realism. Conceptsin this sense do not have an “existence” independently of the more generalcapacity that humans have for language and thought, and yet, as capacities that

36Ibid., p. 93.37Why a moderate realism restricted to mental qualities should be acceptable, but not a

moderate realism that applies to the “external” world as well, is an issue we leave to othersto explain—if it can be explained at all.

38Spade [1996], p. 155, and Adams [1977], p. 145.39Spade [1996], p. 155.

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are intersubjectively realizable, they are objective in at least as strong a senseas Ockham’s notion of a “natural likeness” between concepts and what theysignify—but without the problems the latter notion raises. Also, as the rule-following capacities underlying our use of predicate and referential expressions innatural languages, concepts have by their very nature the function of informinga speech or mental act with a predicable or referential nature; and, of course,they are the very same capacities that are exercised in the production of thosespeech and mental acts. Finally, the unsaturated nature of a concept explainsits non-occurrent, or quasi-dispositional status—that is, its status as a capacitythat could, but need not, be exercised in an appropriate context, or, that mightin fact never be exercised at all.

8.4 Personal Supposition and Reference

Reference in conceptual realism is a pragmatic notion that applies only whenreferential concepts are exercised in speech or mental acts. Reference in termin-ist logic is also a pragmatic notion, but applies to the way categorematic termsare said to supposit for the things they signify when used in a speech or mentalact. Both systems distinguish reference to concepts from reference to things,but only conceptual realism is explicit in distinguishing reference to concepts interms of predicate quantifiers. Reference to concepts in terminist logic, which iscalled simple supposition, does not explicitly involve predicate quantifiers, butthis might be a matter only of surface grammar.40 In any case, our concernhere will be with reference to things, which in terminist logic is called personalsupposition.

Personal supposition in terminist logic is not the same as reference to thingsin conceptual realism, because (as we explain in the following section) cate-gorematic terms can have personal supposition either as subjects or predicatesof categorical propositions, whereas referential concepts in conceptual realismcan never function as predicable concepts, nor can predicable concepts functionas referential concepts. Nevertheless, except for the so-called merely confusedpersonal supposition of predicates containing an intensional verb or modal oper-ator (as discussed in §7 below), the personal supposition of terms in categoricalpropositions does coincide with a combined notion of activated and deactivatedreference in conceptual realism, where the deactivated reference is involved inthe truth conditions determined by a predicable concept. Both systems, more-over, give a uniform account of general and singular reference to things.

As already noted in the previous chapter, referential concepts in conceptualrealism, like predicable concepts, are unsaturated cognitive structures; but thestructures are not the same. Rather, like the way that quantifier phrases have astructure that is complementary to predicate expressions, or the way that nounphrases are complementary to verb phrases, referential concepts and predicableconcepts are cognitive structures that are complementary to one another. This

40There is also another type of supposition, material supposition, in which a term standsfor itself or other spoken or written signs. We will not deal with this type of supposition here.

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complementarity is such that when they are exercised together in a speech ormental act each saturates the other; and just as the predicable concept is whatinforms that act with a predicable nature, so too the referential concept iswhat informs the act with a referential nature. An affirmative assertion thatis analyzable in terms of a noun phrase and a verb phrase (regardless of thecomplexity of either) is semantically analyzable, for example, in terms of anovert joint application of a referential concept with a predicable concept; andthe assertion itself, as a speech act, is the result of the mutual saturation of theircomplementary structures in that act. It is just this sort of mutual saturationof complementary cognitive structures that constitutes the nexus of predicationin conceptualism. It is also what accounts for the unity of a speech or mentalact, i.e. of an assertion or judgment, a problem that Ockham, who anticipatedF.H. Bradley’s infinite regress argument, was unable to resolve.41 Ockham, forexample, assumed that a judgment that every man is an animal was literallymade up of a universal quantifier, the concept man, the mental copula is, andthe concept animal.42 But then what unifies these mental terms into a singleunified mental act? A fifth mental term that “tied” these items together wouldneed a sixth to “tie” it with the others, which in turn would need a seventh, andso on ad infinitum. That is not how a judgment or assertion is understood inconceptual realism, where concepts, as unsaturated cognitive structures, are notobjects, and therefore cannot be actual constituents of a mental act (event).43

Referential concepts, as we have explained, are what the quantifier phrasesof our logistic system stand for when the latter are affixed to the symboliccounterparts of names, where both proper and common names are understoodto have such counterparts, just as they do in Mental, the language of thoughtof terminist logic. A proper name is distinguished in the system from commonnames by a meaning postulate to the effect that at most one thing can bereferred to by that name, and that the name refers to the same thing in everypossible world in which it refers to anything at all. But names, whether properor common are different from predicate expressions, as Geach has pointed out,because they can be used in “simple acts of naming” outside the context ofa sentence.44 Naming is not the same as referring, it should be emphasized,because the latter is an act that does not occur outside the (implicit if notexplicit) context of a sentence used in a speech act, i.e., independently of anassociated act of predicating.

41See Spade [1996], chapter 4, §3.42Ibid., p. 123. Spade points out that not all terminists agreed with Ockham, and Buridan

as well, on this view of judgments or mental propositions as complexes of syncategorematicand categorematic mental terms. Gregory of Rimini and Peter of Ailly, in particular, criticizedthe view, and argued instead that judgments, or mental propositions, unlike the assertions ofspoken language, were “structureless mental acts” that occur, as it were, all at once. Thisview is similar to the notion of a judgment or assertion in conceptual realism, and might wellbe “reconstructed” in terms of the latter.

43On our account of predication as the mutual saturation of a referential and predicableconcept, there cannot be even a first step toward Bradley’s infinite regress.

44[1980], p.52.

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8.5 The Identity Theory of the Copula

As noted in the previous chapter the direct object of a relational predicate isdeactivated when that predicate stands for the predicable concept of a speechor mental act. This applies no less to the copula in its use to express an identitythan it does to transitive verbs such as ‘seek’ and ‘find’. But, because quantifierphrases occurring within a complex predicate do not stand for the referentialconcepts they stand for when used as grammatical subjects, we need to dis-tinguish a predicable concept based upon the copula from one based on strictidentity. We introduce a new symbol, ‘Is’, that we will use for this purpose,and note that, like the transitive verb ‘find’, the copula Is is extensional in itsrange as well as in its domain, except that in this case the copula becomes astrict identity. The schematic meaning postulate for Is then is as follows:

[λxIs(x, [QyS])] = [λx(Qy)(x = y)].

With ∃ as a special case of the schematic determiner Q, we have

[λxIs(x, [∃yS])] = [λx(∃y)(x = y)]

as a particular meaning postulate or conceptual truth regarding the copula Is.By means of this notation, we can perspicuously represent the cognitive struc-ture of an assertion of, e.g., ‘Socrates is a man’ as follows (assuming ‘Socrates’is being used with existential presupposition):

(∃xSocrates)[λxIs(x, [∃yMan])](x),

which, by λ-conversion and the above meaning postulate, is equivalent to:

(∃xSocrates)(∃yMan)(x = y).

This last formula, however, unlike the one above, does not represent the struc-ture of a speech or mental act, although it does represent the same truth con-ditions.

Something like this kind of analysis was involved in the so-called two-name,or identity, theory of the copula in terminist logic. Apparently, Ockham andother terminists thought that every affirmative categorical proposition amountedto asserting an identity between the personal suppositions of the subject andthe predicate terms of the proposition, as, e.g., the suppositions of the names‘Socrates’ and ‘man’ in an assertion of ‘Socrates is a man’. Negative judgments,on the other hand, amounted to a denial of such an identity.45 As a result,the identity theory of the copula came to be developed as a theory of the truthconditions of categorical propositions, a theory that is now referred to as thedoctrine of supposition proper .46

45See Spade [1996], p. 133, for a discussion of this “old and venerable theory” of theintellect’s “composing and dividing” of concepts in making affirmative or negative judgments,where dividing is “composing negatively.”

46See Scott [1966], p. 30, and Spade [1996], chapter 8, §iii.

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The personal suppositions of a proper or common name are the thing(s)signified by that name, but, unlike signification, supposition is understood tobe relativized to the propositional context of a speech or mental act where aquantifier might occur with the name. That is why the theory of supposition ofterminist logic is really a theory of the truth conditions of categorical proposi-tions as linguistic or mental acts. These truth conditions, as we have said, aredetermined by the identity theory of the copula together with the quantifiersthat occur with the names.47 Thus, for example, an assertion of ‘Some man isa thief’, which on our analysis has the form

(∃xMan)[λxIs(x, [∃yThief ])](x),

is equivalent, by λ-conversion and the above meaning postulate, to

(∃xMan)(∃yThief)(x = y),

which indicates that the truth conditions of this assertion amount to the identityof some supposition of the term ‘man’ with a supposition of the term ‘thief’,where each supposition, as a result of the quantifiers attached to the terms,amounts to a restriction on what the terms signify.48 Similarly, an assertion of‘Every man is an animal’, which on our analysis has the form,

(∀xMan)[λxIs(x, [∃yAnimal])](x),

is equivalent, by λ-conversion and the above meaning postulate, to

(∀xMan)(∃yAnimal)(x = y),

which indicates that the truth conditions of the assertion involve an identitybetween each supposition of the categorical term ‘man’ and some suppositionof the categorical term ‘animal’, where, again each supposition is a restriction,as determined by the quantifiers attached to each term, of what they signify.49

The same kind of analysis also applies to categorical propositions expressedby means of a predicate adjective with the ‘is’ of predication instead of the cop-ula. An assertion of ‘Every swan is white’, for example, which in our framework

47Tense or modal modifications of the copula will “ampliate” the personal supposition of theterms, and in that way modify the truth conditions of the speech or mental act in question.(See, e.g., Scott 1966, p. 33, and Spade 1966, chapter 10.) For simplicity of presentation, werestrict ourselves here to present tense uses of the copula.

48One way to construe the personal suppositions of ‘man’ and ‘thief’ here as a form ofreference (as is frequently claimed in the literature) is by noting that the assertion that someman is a thief is equivalent to an assertion that some man and some thief are identical,

(∃xMan ∧ ∃yThief)[λxy(x = y)](x, y),

i.e., where a conjunctive referential concept is used involving both ‘man’ and ‘thief’.49To “reconstruct” these suppositions as a form of reference, we can again use a conjunctive

referential concept to assert that each man and some animal are such that they are identical:

(∀xMan ∧ ∃yAnimal)[λxy(x = y)](x, y),

which is equivalent to our original assertion.

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is symbolized as (∀xSwan)White(x), is not interpreted by the terminists as anidentity between swans and white, or whiteness, or whitenesses (whatever any ofthese might be taken to be as objects). Rather, the predicate adjective ‘white’is interpreted as an attributive adjective, so that to say a thing is white is to saythat it is a white thing.50 Predicate adjectives, in other words, were analyzed bythe terminists as attributive adjectives applied to the common name ‘thing’.51

In conceptual realism, however, the common name ‘white thing’ is interpretedas the complex common name ‘thing that is white’, which is symbolized in oursystem as ‘Thing/White(x)’ (or as ‘Object/White(x)’). Thus, whereas the ter-minist logician would interpret ‘Every swan is white’ as ‘Every swan is a whitething’, we can reconstruct the terminists’ analysis as ‘Every swan is a thing thatis white’, which can be symbolized as follows:

(∀xSwan)[λxIs(x, [∃yThing/White(y)])](x).

This formula, by λ-conversion and the above meaning postulate, has the sametruth conditions as

(∀xSwan)(∃yThing/White(y))(x = y),

which, in terms of supposition theory, is to say that each supposition of thecommon name ‘swan’ is identical with a supposition of the (complex) commonname ‘thing that is white’, or, more simply, with the common name ‘whitething’.

Negative categorical sentences such as ‘No raven is white’ are interpreted asdenials or negations, as we have already said. That is, to assert that no ravenis white is to deny that some raven is white: ¬(∃xRaven)White(x), which isprovably equivalent to denying that some raven is a white thing; i.e.,

¬(∃xRaven)White(x) ↔ ¬(∃xRaven)[λxIs(x, [∃yThing/White(y)])](x)

is provable in our system. But denying that some raven is a white thing isequivalent, by λ-conversion and the above meaning postulate, to

¬(∃xRaven)(∃yThing/White(y))(x = y),

which in terms of the theory of supposition describes the truth conditions as adenial that some supposition of the common name ‘raven’ is identical with asupposition of the complex common name ‘thing that is white’, or more simply,with the common name ‘white thing’.

Finally, the logical form of a negative particular categorical sentence such as‘Some swan is not white’, which is symbolized for us as

(∃xSwan)[λx¬White(x)](x),50See Normore [1985], p. 194.51It is not clear if in some cases it is a common name subordinate to ‘thing’. E.g. in

asserting ‘Socrates is wise’, are we asserting that Socrates is a wise man (or person), or thatSocrates is a wise thing?

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8.6. ASCENDING AND DESCENDING 183

is understood in terminist logic as the equivalent statement that some swan isnot a white thing’, which is symbolized as

(∃xSwan)[λx¬Is(x, [∃yThing/White(x)])](x).

This last formula, by λ-conversion and the above meaning postulate, is equiva-lent to

(∃xSwan)(∀yThing/White(x))(x �= y),

the truth conditions for which are that some supposition of the common name‘swan’ is not identical with any supposition of the complex common name ‘thingthat is white’.

8.6 Ascending and Descending

Supposition theory is not one theory but two. The first, supposition theoryproper, is a theory of the truth conditions of categorical propositions as de-scribed in the preceding section. The second, called the doctrine of the modes ofsupposition, has to do with how many things a categorematic term supposits forin a given speech or mental act.52 This doctrine, despite the reference to “modesof supposition,” is not a theory about different “ways of referring.” Rather, the“modes” are just different types and subtypes of personal supposition. The twobasic types are discrete and common supposition, and the purpose of the theoryis to explain, or “reduce,” the latter in terms of the former.53 Common sup-position is divided into determinate and confused supposition as subtypes, andthe latter is further divided into the sub-subtypes of confused and distributivesupposition and merely confused supposition.54

Modes of Supposition��

Discrete Common��

Determinate Confused

A term is said to have discrete supposition in a categorical sentence only if itis either a proper name, a demonstrative pronoun (such as ‘this’ and ‘that’) or acommon name preceded by a demonstrative pronoun (such as ‘this man’, ‘thathorse’, etc.). Terms that have discrete supposition are said to be discrete terms,and categorical propositions in which they occur as the grammatical subject aresaid to be “singular propositions.”55

52Paul Spade, in 1996, p. 294, was the first to propose this interpretation—though he doesnot explain or develop it, as we do here, in terms of the principle of descent (described in thissection). T.K. Scott,in 1966, was the first to distinguish the two doctrines, but he claims thatthe second has to do with the elimination of quantifiers (pp. 36f).

53Spade 1996, p. 277.54Spade 1996, chapter 9.55Spade [1996], p. 276.

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The explanation, or “reduction,” of the types (and subtypes) of commonsupposition in terms of discrete supposition is given in terms of a “descent” to,and sometimes an “ascent” from, singular propositions. A proper name suchas ‘Socrates’ will have discrete supposition only when Socrates exists, however,which means that even an assertion of ‘Socrates is Socrates’ will be false whenSocrates does not exist.56 The situation is similar but not quite the same inconceptual realism, where we distinguish between using a proper name withexistential presupposition and using it without such a presupposition. Thus if,in asserting ‘Socrates is Socrates’, we are referentially using the name ‘Socrates’with existential presupposition, the symbolic counterpart of this assertion is

(∃xSocrates)[λxIs(x, [∃ySocrates])](x),

in which case the assertion is false if it is asserted at a time when Socrates doesnot exist—or so we may assume in conceptual realism. But if we are referentiallyusing ‘Socrates’ without existential presupposition (which is usually not thecase), the symbolic counterpart of our assertion is

(∀xSocrates)[λxIs(x, [∃ySocrates])](x),

which is not false but vacuously true when Socrates does not exist.57

This last point, of course, has to do with universal sentences not having“existential import,” contrary to the way they were interpreted by medieval lo-gicians. That is, in conceptual realism, and in modern logic in general, universal

56Scott [1966], p. 41.57A proper name, such as ‘Socrates’, of a concrete, as opposed to an abstract, object can

be stipulated to be “existence-entailing” in the sense that if the name can be used to refer toanything, then that thing exists (as a concrete object):

(∀xSocrates)E!(x), (PN-E!)

where E! stands for concrete existence, as opposed to being (the value of an individual vari-able bound by ∃). Note that the symbolic counterpart of ‘Socrates exists’, where the name‘Socrates’ is used only with existential presupposition, is (∃xSocrates)E!(x). Denying thatSocrates exists, i.e., ¬(∃xSocrates)E!(x), is then equivalent to (∀xSocrates)¬E!(x), fromwhich, together with the above stipulation about the name ‘Socrates’, it follows that Socratesdoes not have being when he does not exist, e.g., ¬(∃xSocrates)(x = x). But, because(∃xSocrates)[λxIs(x, [∃ySocrates])](x) is provably equivalent to this last formula withoutthe negation, it then follows that it too is false when Socrates does not exist.

It may be preferable, however, to allow that Socrates can exist, and in that sense havebeing, even when he does not exist (as the terminist logicians also seemed to have assumed).In that case, instead of (PN-E!), we would assume

(∀xSocrates)♦E!(x). (PN-♦E!)

Then, assuming that whatever can exist (in the concrete sense) has being, i.e.,

(∀x)[♦E!(x) → (∃y)(x = y)],

it would follow, given (PN-♦E!(x)), that ‘Socrates is Socrates’ is true even at a time (or world)when Socrates did not exist (in the concrete sense). But, in either case, it should be noted,because to be an abstract object is to be a thing that cannot exist (in the sense of concreteexistence), we still have

Abstract =df [λx¬♦E!(x)].

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conditionals—and therefore all sentences beginning with a universal quantifierphrase—are true, and not false as interpreted by the terminists, when their an-tecedents (subject terms) are vacuous.58 The utility of this view, both for scienceand natural language, has been more than justified over the last century; andwe assume that its acceptance in our reconstruction of terminist logic is but aminor modification, and improvement, of this system—just as the acceptance ofa logic free of existential presuppositions for proper names and definite descrip-tions is a minor modification, and improvement, of so-called standard first-orderlogic. We need only stipulate when a name S is assumed to supposit something,i.e. when something is an S, if existential import is needed to validate a descentor ascent, as it will be in some cases.

Latin lacks a definite article, incidentally, which means that the terministsdid not consider definite descriptions in their analyses at all.59 The existentialpresuppositions of demonstrative phrases is another matter, however, becausesuch demonstrative phrases are central to the descent to (and, in some cases,ascent from) singular propositions. The term ‘man’, for example, has determi-nate supposition in ‘Socrates is a man’, which means that the descent (at a timewhen Socrates exists) to a certain disjunction of singulars, ‘Socrates is this manor Socrates is that man or ... or Socrates is that man’, is valid—and so toois the ascent from the singulars to ‘Socrates is a man’ (again, at a time whenSocrates exists), and therefore from their disjunction.

Because the demonstrative ‘that’ can be used a number of times in a disjunc-tion, or conjunction, of singular propositions, we must, in our logical language,distinguish each use from the others.60 For convenience, we will use ‘That1’,‘That2’,..., ‘Thatn’, etc. (for each positive integer n), as variable-binding opera-tors that operate on a name (complex or simple), resulting thereby in a quantifierphrase that can then be applied to a formula. We will also read ‘That1’ as theEnglish ‘this’. Thus, where S is a common name (complex or simple), e.g.,‘Man/Snubnosed(x)’, then ‘(That2xMan/Snubnosed(x))’, read as ‘that man

58Note that by the meaning postulate (MP1) of §7.3 of the previous chapter,

(∀xSocrates)[λxIs(x, [∃ySocrates])](x) ↔(∀x)[(∃ySocrates)(x = y) → Is(x, [∃ySocrates])],

is provable. The universal conditional on the right-hand side of this biconditional is vacuouslytrue if ¬(∃ySocrates)(x = y) is true for any value of x as bound by ∀, which it is whenSocrates does not exist, or so we may assume already noted.

59This is not to say that uniqueness cannot be expressed in Latin by means other thandefinite descriptions.

60Note that more than one referential concept can be exercised in a disjunction, or con-junction, of singular propositions. The detective who says, while pointing to two differentmen, ‘This man is the killer or that man is the killer’ is exercising two different referentialconcepts, expressed by ‘this man’ and ‘that man’, in his speech act (whereas, of course, asthe direct object, ‘the killer’ is deactivated in both disjuncts). Similarly, when a school coachsays, while pointing to certain boys in his class, ‘That boy is on team A and ... and that boyis on team A’, he using the referential expression ‘that boy’ to refer to a certain number ofdifferent boys in his conjunctive statement. Conjunctive and disjunctive statements are notbasic statements, needless to say, and hence are not subject to the restriction that only onereferential, and one predicable, concept can be exercised in them.

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who is snubnosed’ (or as ‘that snubnosed man’), is a quantifier phrase of oursymbolic language; and, as such, the phrase stands for a referential concept.The symbolic analysis of an assertion of ‘That man who is snubnosed is wise’,which involves the mutual saturation of a referential and a predicable concept,can now be given a logically perspicuous representation as follows:

(That2xMan/Snubnosed(x))Wise(x).

We also interpret the use of ‘this’ and ‘that’ that occur without a common name,as in ‘This is a dog’, ‘That is a man’, etc., similar to the way we interpret theobjectual quantifiers (∀x) and (∃x), i.e. as implicitly containing the commonname ‘thing’ (or ‘object’), as in ‘This thing is a dog’, ‘That thing is a man’, etc.

Now our point about the existential presupposition of a demonstrative phraseis that when a speaker says, e.g., ‘That man is sitting’, he is presupposing thatthe thing he is indicating is a man, i.e., that ‘That man is a man’ is true. Butif the speaker is pointing to a manikin that he has mistaken for a man, thenhis purported reference has failed, and both the speaker’s assertion and thesentence ‘That man is a man’ are false in such a context. A speaker’s use of ademonstrative phrase, we maintain, is equivalent to, if not synonymous with, ause with existential presupposition of a definite description; in particular, thata use of a demonstrative phrase, e.g., ‘that S’, where S is a complex or simplecommon name, is equivalent to using with existential presupposition the definitedescription ‘the S that I am indicating’.61 A sentence of the form ‘The S is anS’, where the definite description is used with existential presupposition—as,e.g., in the implicit premise of Descartes’s ontological argument, ‘The perfectbeing is a perfect being’—is not a valid thesis; and, because of the equivalencebetween demonstrative phrases and definite descriptions used with existentialpresuppositions, neither are sentences of the form ‘That S is an S’.62

Determinate supposition, we have said, means a descent to a certain dis-junction of singulars. More specifically, a common name S has determinatesupposition in a (categorical) proposition P if S occurs in P as part of a quan-tifier phrase and the descent from P to a disjunction of singular propositions(Q1 ∨ ... ∨ Qn), where Qi, for 1 ≤ i ≤ n, is obtained from P by replacing thequantifier phrase containing S by ‘thati S’, is valid. To ensure validity, thedisjunction must be exhaustive of all the S there are (when P is asserted).

61We assume here that there can be no cases of correctly using a demonstrative phrase, suchas ‘that man’, without existential presupposition—the way there can be cases of correctly usinga definite description without existential presupposition.

The truth conditions for a sentence of the form ‘Thati S is F ’ are the same as ‘There isexactly one S that I am (now) indicating and it is F ’, which can be symbolized as follows(where ‘G(z)’ abbreviates ‘I am indicating z’):

(ThatiyS)F (y) ↔ (∃yS)[(∀zS)(G(z) ↔ z = y) ∧ F (y)].

62‘That S is an S’ will be true in any context in which ‘Something is that S’ is true; i.e.when (∃x)(ThatiyS)(x = y) is true.

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Thus, in the determinate supposition of ‘man’ in ‘Socrates is a man’ (assertedwhen Socrates exists), which is symbolized as

(∃xSocrates)[λxIs(x, [∃yMan])](x),

the descent, to be valid, cannot be to any disjunction of the form

(∃xSocrates)[λxIs(x, [That1yMan])](x) ∨ ... ∨ (∃xSocrates)[λxIs(x, [ThatnyMan])](x),

but only to a disjunction that is exhaustive of all of the men there are, i.e.of all the men who exist at the time of the assertion of ‘Socrates is a man’.In other words, even after having indicated a number of men by means of ademonstrative phrase, we may still not have indicated the man that in fact isSocrates if we have not exhausted all of the men there are. Implicit in such adescent, accordingly, is the assumption that we are indicating all the men thereare; that is, that

A thing is a man if, and only if, either it is that1 man or ... or it is thatn man.

is true (at the time of assertion). A similar assumption applies to any commonname, S, of concrete physical objects; that is, for some natural number n, theterminist logicians assumed that at any given time there are exactly n manythings that are S.63 As a generalized version of the above thesis for the commonname ‘man’, we will call this assumption, for any common name S (complexor simple) of concrete things, and any natural number n, the principle ofdescent for n many S, or simply PDn(S). The principle, which in the casewhen there are n many S says that something is an S if, and only if, it is that1S or that2 S, or ... or thatn S, can be symbolized as follows64:

(∀x)[Is(x, [∃yS]) ↔ Is(x, [That1yS]) ∨ ... ∨ Is(x, [ThatnyS])].

This thesis, by the meaning postulate for Is, is provably equivalent to

(∀x)[(∃yS)(x = y) ↔ (That1yS)(x = y) ∨ ... ∨ (ThatnyS)(x = y)].

It is important to note here that the thesis of descent amounts to an explicitanswer to the question of how many S there (now) are in terms of the identitytheory of the copula, which as we indicated earlier is what the doctrine of themodes of supposition is really about.65

63The restriction must be to concrete physical objects, because the thesis will be false forsuch abstract objects as the natural numbers, and perhaps also for concrete events (whichare not physical objects). Unlike the system of conceptual realism, terminist logic gave noexplanation of how such abstract objects as the natural numbers are to be accounted for.

64The principle is taken to apply at any moment of time considered as the present. Relatedprinciples for ampliated terms are obtained by applying tense and modal operators to PDn(S).(We ignore the assumption that ThatiyS �= ThatjyS, for i, j ≤ n where i �= j, incidentally,because it is not needed for the inferences noted here.)

65When n is 0, we take the right-hand side of the biconditional within the scope of (∀x) tobe the formula (x �= x), from which it follows that nothing is S, which is as it should be whenn is 0.

Note, incidentally, that when something is an S, i.e. (∃x)Is(x, [∃yS]), then, by PDn(S)and distribution of (∃x) over ∨, each disjunct will have the form (∃x)Is(x, [ThatiyS]), which,in effect, stipulates that the existential presupposition of the demontrative phrase is fulfilled,i.e. that something is thati S.

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The principle of descent for ‘man’, i.e., PDn(Man), also validates the ascent(when Socrates exists) from any one of the singulars ‘Socrates is this man’, ...,‘Socrates is thatn man’ to the original proposition ‘Socrates is a man’ in which‘man’ is said to have determinate supposition. For this kind of proposition, inother words, we can validly ascend when and only when we can validly descend.

Determinate supposition applies to common names occurring as subjectterms as well as to common names occurring as predicate terms, as in ourexample of ‘Socrates is a man’. Thus, by PDn(Man), we can validly descendfrom ‘Some man is running’ to ‘This man is running or that2 man is runningor ... or thatn man is running’; and of course we can ascend from any oneof these disjuncts—and therefore from the disjunction as well—to the sentence‘Some man is running’. Similarly, by PDn(Man), we can validly descend from‘A man is not running’ to ‘This man is not running or that2 man is not runningor ... or thatn man is not running’; and, again, we can similarly ascend fromany, or all, of these disjuncts to ‘A man is not running’, or to ‘Some man isnot running’, both of which are symbolized the same way in our system. Withdeterminate supposition, in other words, we can validly descend on the basis ofthe principle of descent when, and only when we can validly ascend on the basisof that principle.66

Confused and distributive supposition is more problematic than determinatesupposition, however, because, according to Ockham, a common name will haveconfused and distributive supposition in a categorical proposition only whenone can descend to a conjunction of singulars, but cannot ascend from any onesingular in the conjunction.67 The problem is that such a descent is valid onlywhen each conjunct can be truthfully asserted if the original premise is true,and hence only when the existential presupposition of the demonstrative phrasein that conjunct is fulfilled, i.e., only when (∃x)Is(x, [ThatiyS]) is true for eachi such that 1 ≤ i ≤ n, where S is the common name in question and there areexactly n many S. This, as it turns out, is just the issue of existential import,but as applied to demonstrative phrases in particular. Such presuppositionswere implicit in what Ockham and the terminist logicians assumed for this typeof supposition.

The common name ‘man’ in ‘Every man is an animal’, for example, will haveconfused and distributive supposition by the principle of descent PDn(Man),but only if the existential presuppositions in question are fulfilled.68 If these

66The argument for the general claim is based on specific examples, to be sure; but that isbecause the examples can be easily schematized and shown to hold in general.

67There is a problem with this characterization when applied to the predicate of a negativeparticular proposition, however, because, although the descent is to a conjunction, the con-juncts will not be singular propositions. But then each conjunct can in turn be “reduced”by determinate supposition to a disjunction of singulars. Thus, on our characterization, acommon name S has confused and distributive supposition in a proposition P if S occurs in Pas part of a quantifier phrase and the descent from P to a conjunction (Q1 ∧ ...∧Qn) is valid,where each Qi (whether singular or otherwise) is obtained from P by replacing the quantifierphrase containing S by ‘thati S’.

68All that follows from PDn(Man) and (∀xMan)Is(x, [∃yAnimal]), using (MP1) of chap-ter 6, is the conjunction that [if anything is that1 man, then it is an animal] and ... and [if

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presuppositions are fulfilled, then, by PDn(Man), one can validly descend from‘Every man is an animal’ to the conjunction ‘This man is an animal and that2man is an animal and ... and thatn man is an animal’, i.e. to

(That1xMan)[λxIs(x, [∃yAnimal])](x)∧ ...∧ (ThatnxMan)[λxIs(x, [∃yAnimal])](x).

One cannot validly ascend, as Ockham says, from any one of these singulars tothe universal ‘Every man is an animal’; but, clearly, given PDn(Man), one canvalidly ascend to the universal from the whole conjunction.

Confused and distributive supposition was assumed by the terminists to ap-ply not only to the subject term of a universal affirmative, but to the subjectterm of a universal negative categorical proposition as well. But, again, thedescent will be valid only if the existential presuppositions of the demonstrativephrases in the conjunction in question are fulfilled. Consider, for example, thesame common name ‘man’, only now occurring as the subject of a universalnegative categorical, such as ‘No man is running’. The confused and distribu-tive supposition of ‘man’ in this proposition means that, by PDn(Man), onecan descend from this sentence to ‘This man is not running and that2 man isnot running and ... and thatn man is not running’ (with the negation in eachconjunct internal to the predicate). But, as in our previous example, the de-scent will be valid only when (∃x)Is(x, [ThatiyS]) is true for each i such that1 ≤ i ≤ n.

In our reconstruction, we note first that ‘No man is running’ is understood asdenying that some man is running, which is symbolized as ¬(∃xMan)Running(x), but which is provably equivalent to (∀xMan)¬Running(x). By this lastsentence and PDn(Man), it follows that anything that is that1 man or ... thatn

man is not running, i.e.

(∀x)[(That1yMan)(x = y) ∨ ... ∨ (ThatnyMan)(x = y) → ¬Running(x)],

and from this and (∃x)Is(x, [ThatiyMan]), for 1 ≤ i ≤ n, the desired conjunc-tion,

(That1yMan)[λy¬Running(x)](x) ∧ ... ∧ (ThatnyMan)[λy¬Running(x)](x)

follows.69 The same argument can be made in reverse order, moreover, whichmeans that, given PDn(Man), we can validly ascend from the conjunction tothe universal negative sentence. In other words, with the confused and distribu-tive supposition of a common name occurring as the subject term of a universal

anything is thatn man, then it is an animal], i.e.

(∀x)[Is(x, [That1yMan]) → Is(x, [∃yAnimal])] ∧ ... ∧(∀x)[Is(x, [ThatnyMan]) → Is(x, [∃yAnimal])].

Hence, if all universals were assumed to have existential import, then the existential presup-positions of these demonstrative phrases would be fulfilled. That is why we have said thatthis is just the issue of existential import, but as applied to demonstrative phrases.

69Without the assumption that (∃x)Is(x, [ThatiyS]) is true, for 1 ≤ i ≤ n, all that wouldfollow by PDn(Man) is the conjunction ‘[If anything is that1 man, then it is not running]and ... and [if anything is thatn man, then it is not running]’.

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affirmative or a universal negative proposition, one can validly descend to aconjunction when and only when one can also validly ascend from such a con-junction to the universal in question.

There are cases of confused and distributive supposition where the issue ofexistential import is not relevant, i.e., where the existential presuppositions ofdemonstrative phrases need not be fulfilled. In particular, a common nameoccurring as (part of) the predicate of any negative categorical propositionwill have confused and distributive supposition, regardless of the issue of ex-istential import. The common name ‘runner’ in ‘No man is a runner’, for ex-ample, unlike the common name ‘man’, will have confused and distributivesupposition independently of whether or not the existential presupposition ofany of the demonstrative phrases in question is fulfilled.70 In other words, byPDm(Runner), one can descend validly to the conjunction ‘No man is that1runner and ... and no man is thatm runner’ without any further assumptionsabout existential import.71 As a denial, ‘No man is a runner’ is symbolizedas ¬(∃xMan)[λxIs(x, [∃yRunner])](x), which, by quantifier negation and λ-conversion, is equivalent to (∀xMan)¬Is(x, [∃yRunner]). This last formula, byPDm(Runner) and elementary transformations, implies

(∀xMan)¬Is(x, [That1yRunner]) ∧ ... ∧ (∀xMan)¬Is(x, [ThatmyRunner]),

which, by quantifier negation, is equivalent to the conjunction

¬(∃xMan)[λxIs(x, [That1yRunner])](x) ∧ ... ∧∧ ¬(∃xMan)[λxIs(x, [ThatmyRunner])](x).

This argument can also be given in reverse order, so that one can validly ascend,by PDm(Runner), from the conjunction to the sentence ‘No man is a runner’.In other words, with confused and distributive supposition, one can validlydescend to a conjunction from a universal negative categorical proposition whenand only when one can validly ascend from the conjunction to that propositionby the same principle—regardless whether the common name occurs as thesubject or the predicate of the proposition, except that when it is the subject,the existential presuppositions of the demonstrative phrases in the conjunctionmust be fulfilled.

The way up is not always the same as the way down, however. Consider,for example, the common name ‘runner’ in the negative particular proposition‘Some man is not a runner’ (where the negation is internal to the predicate),which is symbolized as (∃xMan)[λx¬Is(x, [∃yRunner])](x). Here, the commonname ‘runner’ has confused and distributive supposition, which means that one

70Our examples come from Spade [1996], chapter 9. Ockham and other terminists assumedthat ‘is running’ can be construed as ‘is a runner’. This construal is dubious; but we will acceptit here as part of our reconstruction of terminist logic. A separate, alternative treatment canbe given for the present participle in conceptual realism in terms of the logic of events.

71The conjuncts in this case are not singular propositions, it might be noted; but, byPDn(Man), each conjunct can be expanded into a conjunction of singulars of the form‘Thatm+i man is not thati runner’, with the negation internal to the predicate.

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can descend, by PDm(Runner), to the conjunction ‘Some man is not this runnerand ... and some man is not thatm runner’.72 Note that, by PDm(Runner),(∃xMan)[λx¬Is(x, [∃yRunner])](x) implies

(∃xMan)¬[Is(x, [That1yRunner]) ∨ ... ∨ Is(x, [ThatmyRunner])],

which, by elementary transformations, implies (but is not implied by)

(∃xMan)¬Is(x, [That1yRunner]) ∧ ... ∧ (∃xMan)¬Is(x, [ThatmyRunner]),

which validates the descent to the conjunction ‘Some man is not this runnerand ... and some man is not thatm runner. But the reverse order of thisargument is not also valid, because, unlike the inference from (∃xMan)(ϕ ∧ψ) to (∃xMan)ϕ ∧ (∃xMan)ψ, the inference from (∃xMan)ϕ ∧ (∃xMan)ψ to(∃xMan)(ϕ∧ψ) is not valid. Thus, despite Heraclitus, the way up is not alwaysthe same as the way down.

8.7 How Confused is Merely Confused

Merely confused supposition has been the one type of supposition that has beencontroversial in terminist logic. It is the one type, for example, that does notallow for a valid descent to either a conjunction or disjunction of singular propo-sitions. Ockham’s main characterization is that a common name has merelyconfused supposition in a categorical proposition if one can validly descend to a“disjoint predicate.”73 The common name ‘animal’, for example, has merely con-fused supposition in the universal affirmative ‘Every man is an animal’, which,as already noted, is symbolized as (∀xMan)[λxIs(x, [∃yAnimal])](x). That is,by the principle of descent, PDk(Animal), λ-conversion, and elementary trans-formations, we can validly descend from this proposition to ‘Every man is [thisanimal or that2 animal ... or thatk animal]’, which is symbolized as follows:

(∀xMan)[λx(Is(x, [That1yAnimal]) ∨ ... ∨ Is(x, [ThatkyAnimal]))](x).

One cannot, of course, validly distribute (after λ-conversion) the universal quan-tifier (∀xMan) over the disjunction

[Is(x, [That1yAnimal]) ∨ ... ∨ Is(x, [ThatkyAnimal])]

to get ‘Every man is this animal or ... or every man is thatk animal’. So, in thiscase no further valid “reduction” to singulars is possible.

72Once again, the conjuncts are not singular propositions, but, by PDn(Man), each con-junct can be expanded into a disjunction so that the final result is a conjunction of disjunctionsof singular propositions.

73Spade 1996, p. 284.

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Some authors have found this “reduction” to a disjunctive predicate “veryodd,” as though the resulting sentence has problematic truth conditions.74 Butthere are many sentences in science and natural language with disjunctive predi-cates that are clearly unproblematic. The sentence of arithmetic, ‘Every integeris odd or even’, symbolized as

(∀xInteger)[λx(Odd(x) ∨ Even(x))](x),

is perfectly clear as to its truth conditions, for example, even though it is notfurther “reducible” to ‘Every integer is odd or every integer is even’. Similarly,‘Every person is either male or female’ seems perfectly clear in its truth condi-tions, even though it is not “reducible” to ‘Every person is male or every personis female’.

Ockham also thinks that merely confused supposition applies to such sen-tences as ‘John promises Simon a horse’.75 But then, merely confused suppo-sition must also apply to ‘John gives Simon a horse’, because this sentence hasthe same logical form as ‘John promises Simon a horse’. On our analysis, thelogical form of these sentences as judgments or speech acts (where ‘John’ is usedwith existential presupposition) is given as:

(∃xJohn)[λxPromise(x, [∃ySimon], [∃zHorse])](x),

and(∃xJohn)[λxGive(x, [∃ySimon], [∃zHorse])](x).

Now, because ‘gives’ is an extensional verb with respect to all of its arguments,the following identity is a conceptually valid thesis of our system as a result ofthe meaning postulate for Give:

[λxGive(x, [∃ySimon], [∃zHorse])] = [λx(∃zHorse)Give(x, [∃ySimon], z)].

But then, by the principle of descent, PDj(Horse), we can validly descend tothe disjunction, ‘John gives Simon this horse or ... or John gives Simon thatj

horse’, in symbols:76

(∃xJohn)[λxGive(x, [∃ySimon], [That1zHorse])](x) ∨ ...∨ (∃xJohn)[λxGive(x, [∃ySimon], [ThatjzHorse])](x).

74This is Paul Spade’s view in Spade1996, p. 284. Note, however, that in our analysiswe have distributed the copula over ‘this animal or ... or thisk animal’. Perhaps Spade hassomething like

Is(x, [That1yAnimal ∨ ... ∨ ThatkyAnimal])in mind as the problematic “disjunctive predicate,” where the copula has not been distributed.If so, then he has a point, because this expression is not well-formed.

75It is not clear that Ockham thinks of an assertion of a sentence like this as a categoricalproposition. If it is a categorical, then, apparently, it is to be rephrased with a copula, e.g.,as ‘John is a man who promises Simon a horse’. But then the disjunction has to do withthe different demonstrative phrases, ‘this man who promises Simon a horse or ... or thatn

man who promises Simon a horse’, in which case the descent is by determinate supposition,and therefore unproblematic. Of course, we then still have to explain the supposition of thecommon name ‘horse’ in the singulars ‘John is thati man who promises Simon a horse’.

76By the above identity, the meaning postulate for Is, and (MP2) of §6 above,

[λx(∃zHorse)Give(x, [∃ySimon], z)] = [λx(∃w)(Is(w, [∃zHorse]) ∧Give(x, [∃ySimon], w))]

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This descent, however, is not really by merely confused supposition, which, ac-cording to Ockham, does not allow descent to a disjunction. Rather, the descentseems to be the same as that already described for determinate supposition.77

Nothing like this follows for ‘John promises Simon a horse’, however; and thereason is that, unlike ‘give’, ‘promise’ is not extensional in its third argumentposition, i.e., ‘promise’ is an intensional verb with respect to its direct objectargument position. Yet Ockham maintained that ‘horse’ has merely confusedsupposition in this sentence, and that the descent to a disjunctive predicate, asin ‘John promises Simon this horse or ... promises Simon thatj horse’, is valid.That is, according to Ockham, the descent from ‘John promises Simon a horse’,as symbolized above, to

(∃xJohn)[λx(Promise(x, [∃ySimon], [That1zHorse]) ∨ ... ∨∨ Promise(x, [∃ySimon], [ThatjzHorse]))](x),

is supposed to be valid, when, in fact, it is not valid—as many commentatorshave repeatedly noted over the years.78 Unlike these other commentators, how-ever, we have a theoretical account in terms of deactivated referential expressionsthat explains why such a descent fails—and why it succeeds in sentences havingthe same logical form. Based on this account, our proposal is that a common (orproper) name that is part of a deactivated referential expression that cannot,as it were, be “activated” in a given propositional context is a name for whichno “mode” of supposition should be said to apply in the context in question.

8.8 Summary and Concluding Remarks

• The framework of conceptual realism provides a logically ideal languagewithin which to reconstruct the medieval terminist logic of the 14th century.

• The terminist notion of a concept, which shifted from Ockham’s earlyview of a concept as an intentional object (the fictum theory) to his later view

is provable; and from this and PDj(Horse),

[λx(∃zHorse)Give(x, [∃ySimon], z)] = [λx(∃w)([Is(w, [That1zHorse]) ∨ ... ∨∨Is(w, [ThatjzHorse])] ∧Give(x, [∃ySimon], w))]

follows. From this last identity and the distribution of a conjunction over a disjunction, thedisjunction in question follows.

77Essentially the same argument would show that ‘horse’ has merely confused supposition,and not determinate supposition, however, in ‘Every man gives Simon a horse’. That is, thedescent, by PDj(Horse), to ‘Every man [gives Simon this horse or ... or gives Simon thatj

horse]’ is valid, whereas a “descent” to ‘Every man gives Simon this horse or ... or every mangives Simon thatj horse’ is not valid.

78Note that, as with the distribution of the copula over ‘this animal or ... or thatk animal’,we have distributed ‘promise’ over ‘this horse or ... or thatj horse’. Otherwise, the so-called“disjoint predicate”, as in

[λxPromise(x, [That1yHorse ∨ ... ∨ ThatjyHorse])]is not well-formed. If this is what Ockham intended, then Spade is right to “think this appealto disjoint terms is ... a mark of desperation” (Spade 1996, p. 284).

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of a concept as a mental act (the intellectio theory), is reconstructed in thisframework in terms of the notion of a concept as an unsaturated cognitivestructure.

• Referential and predicable concepts are unsaturated cognitive structuresthat mutually saturate each other in mental acts, analogous to the way thatquantifier phrases and predicate expressions mutually saturate each other inlanguage.

• Intentional objects (ficta) are not rejected but are reconstructed as theobjectified intensional contents of concepts, i.e., as intensional objects obtainedthrough the process of nominalization—and in that sense as products of theevolution of language and thought.

• Intensional objects are an essential part of the theory of predication ofconceptual realism. In particular, the truth conditions determined by predicableconcepts based on relations—including the relation the copula stands for—arecharacterized in part in terms of these object-ified intensional contents. It is bymeans of this conceptualist theory of predication that we are able to explainhow the identity theory of the copula, which was basic to terminist logic, appliesto categorical propositions.

• Reference in conceptual realism, based on the exercise and mutual satu-ration of referential and predicable concepts, is not the same as supposition interminist logic.

• Nevertheless, the various “modes” or types of personal supposition areaccounted for in a natural and intuitive way in terms of the theory of referenceof conceptual realism.

• Ockham’s application of merely confused supposition to common namesoccurring within the scope of an intensional verb is rejected, as it should be, butits rejection is grounded on the notion of a deactivated referential concept—adeactivation that, because of the intensionality of the context in question, cannotbe “activated,” the way it can be in extensional contexts.

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Chapter 9

On Geach Against GeneralReference

Theories of reference in the 20th Century have been almost exclusively theoriesof singular reference, i.e., theories of the use of proper names and definite de-scriptions to refer to single objects. General reference by means of quantifierphrases has usually been rejected, mainly because of a confusion of pragmaticswith semantics, i.e., a confusion of the referential use of quantifier phrases inspeech and mental acts with the truth conditions of sentences containing thosephrases.1

This confusion of pragmatics with semantics is in marked contrast with ourconceptualist theory of reference (as described in chapter seven) where singularand general reference are given a unified account. It is also in contrast withmedieval suppositio theories where a unified account was also given, but onlyin terms of categorical propositions. Bertrand Russell had a theory of generalreference in his 1903 Principles of Mathematics, but he later abandoned thattheory in his 1905 paper, “On Denoting”.

In his later 1905 theory, Russell took ordinary proper names to be eliminablein terms of definite descriptions, which were in turn eliminable contextually interms of quantifier phrases, and quantifier phrases were then said to be “re-ducible” to conjunctions and disjunctions of singular propositions. Thus, the1905 theory, according to Russell, “gives a reduction of all propositions in whichdenoting phrases [i.e., quantifier phrases and definite descriptions] occur in formsin which no such phrases occur.”2 Russell did allow for a category of “logicallyproper names,” however, i.e., expressions such as ‘this’ and ‘that’, each of whichhe said “applies directly to just one object, and does not in any way describethe object to which it applies.”3 Such a category of “logically proper names”

1This chapter is a revised and extended version of my 1998 paper, “Reference in ConceptualRealism,” in Synthese, vol. 14.

2Russell 1956, p. 45.3See, e.g., Russell’s 1914 paper “On the Nature of Acquaintance”, reprinted in Russell

1956.

195

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196 CHAPTER 9. ON GEACH AGAINST GENERAL REFERENCE

figured prominently in Russell’s logical atomism, where the idea of eliminatingall forms of general reference found its clearest paradigm. Indeed, this way ofreducing general reference to the singular reference of logically proper names, orwhat came to be called “objectual constants,” was laid out explicitly by RudolfCarnap in his state-description semantics, which he developed and applied evento quantified modal logic.4 In many ways, and however unwittingly, it is thisparadigm for reducing general reference to singular reference that is now partof the so-called “new theory of direct reference” in which there is only singularreference.5

Aside from the paradigm of logical atomism as a framework for eliminat-ing general reference, there were for many years no explicit arguments againsttheories of general reference, i.e., arguments that there could be only singularreference. This situation changed in 1962 when Peter Geach published his book,Reference and Generality, which was later revised and reprinted in 1980. In thisbook, Geach developed arguments that are supposed to apply to any theory ofgeneral reference, as well as some others that are designed to work specificallyagainst Russell’s 1903 theory and against the medieval suppositio theories.

Geach’s arguments do not work against our conceptualist theory of refer-ence, however, as we will explain in what follows. Nor do those arguments workagainst the medieval suppositio theories once they are interpreted and recon-structed within conceptual realism as we have done in the previous chapter.

9.1 Geach’s Negation Argument

The only “genuine” form of reference, according to Geach, is reference by meansof singular terms, and in particular in the use of proper names. One type ofargument that he gives against general reference is based on negation.

Consider, for example, an indicative sentence of English containing a propername ‘a’, and let ‘f( )’ represent the propositional context remaining whenthe name ‘a’ has been extracted from the sentence.6 The propositional context‘f( )’ is what Geach calls a predicable, which of course can be complex as wellas simple. Attaching a prime to ‘f ’, as in ‘f ′( )’, is then said to representa predicable contradictory to ‘f( )’. Geach does not explain what he meansby a predicable contradictory to ‘f( )’ other than that ‘f( )’ and ‘f ′( )’

4See Carnap 1946. Carnap showed that the thesis of the necessity of identity, the modalthesis of anti-essentialism, and what later came to be called the Barcan formula by some,but which really should be called the Carnap-Barcan formula, were all valid in his state-description semantics for quantified modal logic—long before these topics became popular inthe philosophical literature.

Carnap’s state-description semantics also amounted to one of the first substitution inter-pretation of quantifiers. In fact it was Carnap who first observed that a strong completenesstheorem even for modal free first-order formulas was not possible on the basis of such aninterpretation for an infinite domain (see Carnap, 1938, p. 1651).

5For a discussion and an account of the “new theory of direct reference,” see Humphreysand Fetzer, 1998.

6We are using Geach’s terminology here where by a “propositional context” we mean thecontext of an indicative sentence.

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are supposed to result in contradictory sentences when a “genuine” referentialexpression is put in place of ‘a’. This is important because it is not clear whatGeach’s argument is without begging the question about whether or not theonly “genuine” form of reference is by means of proper names.

Briefly, Geach’s argument is that when ‘f( )’ and ‘f ′( )’ are attached to“any proper name ‘a’ as subject, they will give us contradictory predications;but if ‘∗A’ takes the place of ‘a’ [where ‘A’ is a sortal common name and ‘∗A’ is aquantifier phrase of English], the propositions ‘f(∗A)’ and ‘f ′(∗A)’ will in generalnot be contradictories—both may be true or both false”.7 For example, ‘Someman is wise” and ‘Some man is not wise’ can both be true, whereas, accordingto Geach, ‘Socrates is wise’ and ‘Socrates is not wise’ cannot both be true. Thisshows, Geach claims, that unlike the proper name ‘Socrates’, the quantifier(noun) phrase ‘some man’ is only a “quasi subject”, not a “genuine subject”,and therefore cannot really be used as a “genuine” referential expression.8 Inother words, quantifier phrases, unlike proper names, cannot be used to stand forreferential concepts (or, in Geach’s terms, cannot be “genuine logical subjects”)because they do not in general yield contradictory sentences when applied tocontradictory predicables.

Now in formal terms the only proper interpretation for a predicate thatis contradictory to a given predicate [λxϕx] is the predicate [λx¬ϕx]. Then,assuming that ‘Socrates’ is being used with existential presuppositions, the sen-tences ‘Socrates is wise’ and ‘Socrates is not wise’ can be symbolized as follows:

(∃xSocrates)Wise(x)

and(∃xSocrates)[λx¬Wise(x)](x),

where in ‘Socrates is not wise’ the negation is internal to the predicate. Thesesentences are in fact contradictory, which is in accordance with what Geachclaims—but only because the name ‘Socrates’ is being used with existentialpresupposition in both. That is, ‘Socrates is not wise’ is equivalent to ‘It is notthe case that Socrates is wise’, which is the contradictory of ‘Socrates is wise’,only because

(∃xSocrates)(∀ySocrates)(x = y) →[¬(∃xSocrates)Wise(x) ↔ (∃xSocrates)[λx¬Wise(x)](x)]

is valid, where the antecedent says in effect that ‘Socrates’ names one and onlyone thing. Without this assumption it does not follow that ‘Socrates is notwise’ is equivalent to ‘It is not the case that Socrates is wise’, and hence that‘Socrates is not wise’ is the contradictory of ‘Socrates is wise’ as Geach claims.

Geach’s “criterion”, or “definition” for “genuine reference,” i.e., his claimthat a “genuine” referring expression will yield contradictory propositions when

7Geach 1980, p.84.8Geach speaks of ‘referring phrases’ where we speak of referential expressions. He adopts

this terminology, which he takes to be a “misnomer”, only for the purpose of describing thetheories of general reference that he claims to refute (cf., e.g., op.cit, p.73).

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applied to contradictory predicables, is not unqualifiedly true even for propernames in other words. For example, if A is a proper name, such as ‘Pega-sus’, that names nothing, and F ( ) is a monadic predicate, so that F ( ) and¬F ( ), or [λxF (x)] and [λx¬F (x)], are contradictory “predicables,” then whenA is used without existential presupposition, we can have both (∀xA)F (x) and(∀xA)¬F (x) true. In other words, the two assertions that A is F and that A isnot F can both be true in a logic that is free of existential presuppositions forobjectual terms.

If we replace the name ‘Socrates’ in ‘Socrates is wise’ and ‘Socrates is notwise’ by ‘Some man’, so as to obtain ‘Some man is wise’ and ‘Some man is notwise’, which can be symbolized as follows,

(∃xMan)Wise(x)

and(∃xMan)[λx¬Wise(x)](x)

then it is clear that both can be true, which is in accordance with what Geachclaims. In other words, ‘It is not the case that some man is wise’ is not equivalentto ‘Some man is not wise’. But does this show that ‘Some man’ is not beingused to refer to some man?

Geach does not justify or explain why yielding contradictory sentences whenapplied to contradictory predicables is a necessary condition for “genuine”reference—except, of course, for maintaining that this is what is true of propernames, but only, as we have noted, when the proper names are being used withexistential presuppositions. That referential expressions cannot be used as formsof “genuine” reference unless they function the same way as nonvacuous propernames is simply assumed, which begs the question at issue.

Now where a is an objectual variable that represents the kind of symbolGeach assumes a proper name to be, what Geach implicitly assumes is that[λxϕ](a) and ¬[λxϕ](a) are contradictories when in fact they are not, or, equiv-alently, that ¬[λxϕ](a) and [λx¬ϕ](a) say the same thing, when in fact they donot—or at least not in a logic that is free of existential presuppositions in theuse of a proper name. In other words, whereas

¬[λxϕ](a) ↔ (∀x)[x = a→ ¬ϕ],

and[λx¬ϕ](a) ↔ (∃x)[x = a ∧ ¬ϕ],

are valid in a logic free of existential presuppositions for objectual terms, we donot also have

¬[λxϕ](a) ↔ [λx¬ϕ](a),

or equivalently(∀x)[x = a→ ¬ϕ] ↔ (∃x)[x = a ∧ ¬ϕ]

as valid as well. It is not unqualifiedly true in such a logic that a will yieldcontradictory propositions when applied to contradictory predicables.

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Geach is apparently aware that his argument does not work against propernames that denote nothing; but instead of rejecting the argument he rejectsthe use of “empty proper names.”9 That response, however, only indicates howinadequate his theory of reference is for pragmatics, i.e., for a realistic theoryof speech and mental acts.

Now the condition for when a referring expression will yield contradictorypropositions when applied to contradictory predicables can be given even in freelogic, but it is a condition that applies to common names as well as to propernames. In particular, what is valid in a logic free of existential presuppositionsis that any proper or common name A, such as ‘Socrates’ or ‘moon of theEarth’, that denotes exactly one object will yield contradictory propositionswhen applied to contradictory predicables. In other words,

(∃xA)(∀yA)(x = y) → [¬(∃xA)ϕ ↔ (∃xA)¬ϕ] ∧ [¬(∀xA)ϕ ↔ (∀xA)¬ϕ]

is valid regardless whether or not A is a proper name or a common name. Butthere is nothing about this result that shows that the only “genuine” referentialexpressions are those of the form (∃xA), where A is a name, proper or common,for which the above antecedent condition is true. Geach simply begs the questionby assuming as a criterion for “genuine reference” a condition that only namesthat name exactly one thing satisfy. On such a criterion, of course there can beno such thing as general reference.

9.2 Disjunction and Conjunction Arguments

Geach gives a similar argument based on the observation that connectives “thatjoin propositions may be used to join predicables” to form complex predicateexpressions.10 Thus, for example, instead of making two separate assertions,such as

Sofia is sick

and

Sofia is home in bed

we could make an assertion using the complex predicate ‘sick and home in bed’,as in ‘Sofia is sick and home in bed’, which we can symbolize as:

(∃xSofia)[λx(Si ck(x) ∧Home-In-Bed(x))](x).

Similarly, instead of asserting a disjunction such as ‘Either Sofia is home orSofia is shopping’ we could assert ‘Sofia is either home or shopping’, symbolizedas

(∃xSofia)[λx(Home(x) ∨ Shopping(x))](x).9Ibid., p. 186.

10Ibid., p. 86.

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Now what Geach claims—or rather assumes without argument—is that “thevery meaning” such connectives as ‘and’ and ‘or’ have in a complex predicate isthe meaning they have as propositional connectives. That is, “by attaching acomplex predicable so formed to a logical subject [i.e. to what Geach considersa “genuine” referring expression] we get the same result as we should by firstattaching the several predicables to that subject, and then using the connectiveto join the propositions thus formed precisely as the respective predicables werejoined by that connective.”11

This claim is true when restricted to nonempty proper names, at least asfar as truth conditions are concerned. An assertion of ‘Sofia is home or shop-ping’, for example, has the same truth conditions (but not the same cognitivestructure) as an assertion of ‘Either Sofia is home or Sofia is shopping’. Indeed,where a is an objectual variable representing the kind of symbol Geach takes anonempty proper name to be,

[λx(ϕ ∨ ψ)](a) ↔ [λxϕ](a) ∨ [λxψ](a)

is true in conceptual realism.The same claim is not in general true when applied to a universal quantifier

phrase, on the other hand—nor, of course when a is an empty proper name.The sentence ‘Every integer is odd or even’, for example, is not equivalent to‘Every integer is odd or every integer is even’. Indeed,

(∀xA)[λx(ϕ ∨ ψ)](x) ↔ (∀xA)[λxϕ](x) ∨ (∀xA)[λxψ](x)

is not a valid schema in the logic of conceptual realism, whether A is a properor a common name. But this does not show that a universal quantifier phrasecannot be used as a “genuine” referential expression; and, in particular, thatthere is no reference to every integer in a speech act in which someone assertsthat every integer is odd or even. What it shows is that Geach’s claim is reallyan assumption, and hence that his argument begs the question at issue.

The equivalence does hold, however, if a proper or common name A can beused to name at most one object in a “simple act of naming”12; i.e.,

(∀xA)(∀yA)(x = y) → [(∀xA)[λx(ϕ ∨ ψ)](x) ↔ (∀xA[λxϕ](x) ∨ (∀xA)[λxψ](x)]

is valid in conceptual realism. And of course, we do have

(∃xA)[λx(ϕ ∨ ψ)](x) ↔ (∃xA)[λxϕ](x) ∨ (∃xA)[λxψ](x)

as valid, i.e., the distribution of (∃xA) over a disjunction is valid.The distribution of (∃xA) over a conjunction, on the other hand, is valid in

only one direction. But why does this show that we cannot use (∃xA) to referto an A? In other words, why should we conclude that the invalidity of

(∃xA)[λxϕ](x) ∧ (∃xA)[λxψ](x) → (∃xA)[λx(ϕ ∧ ψ)](x)11Ibid.12Geach 1980, p.53.

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shows that a quantifier phrase of the form (∃xA), where A is a common name(complex or simple), cannot be used as a “genuine” referential expression? Thefailure of a logical equivalence does not show this except by begging the questionthat only proper names can be “genuine” referential expressions.

It is noteworthy, moreover, that the antecedent of the above conditional, i.e.the conjunction,

(∃xA)[λxϕ](x) ∧ (∃xA)[λxψ](x),

does not represent a basic speech act that is analyzable in terms of a referentialand a predicable expression. What it can be used to represent is a speaker’sconjunction of two assertions in each of which the same referential concept isapplied. But to apply the same referential concept, especially one of the form(∃xA), in two conjoined assertions is not the same as to purport to refer tothe same object or objects in those assertions, unless, of course, the referentialconcept in question is based on the use of a proper name. We can assert, e.g.,that some republicans are honest and that some republicans are dishonest, butin doing so we do not purport to refer to the same republicans in both uses ofthe quantifier phrase ‘Some republicans’.

Geach’s implicit assumption is that if a quantifier phrase can be used as a“genuine” referential expression, then it must refer to one and only one object,and the same object(s), moreover, whenever it is so used. In other words, a“genuine” referential expression must refer the way a nonvacuous proper namerefers, which, of course, begs the question.

It is by begging the question and assuming that only propernames can be used as “genuine” referential expressionsthat Geach’s negation and complex-predicate argumentshave any plausibility.

9.3 Active Versus Deactivated Concepts

Geach does have a more interesting type of argument that does not beg thequestion, but which in our conceptualist theory involves the important distinc-tion we made in our last two chapters between active and deactivated referentialconcepts. In explaining this distinction, we noted that a referential concept, asa basic thesis of our theory, is never part of what informs a speech or mental actwith a predicable nature, but functions only as what informs such an act witha referential nature, i.e., as what accounts for the intentionality or aboutness ofthat act. Every basic assertion as expressed by a noun phrase and a verb phraseis the result, in other words, of applying just one referential concept and onepredicable concept.

What this means is that a complex predicate expression that contains aquantifier phrase cannot be applied in such a way as to presuppose an activeexercise of the referential concept that that quantifier phrase stands for. Thereferential concept that the quantifier phrase stands for has been “deactivated”,

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in other words, which means that the predicable concept expressed by the com-plex predicate that contains that quantifier phrase is formed not on the basis ofthe referential concept that the quantifier phrase stands for but on the basis ofits intensional content instead.

Now by the intensional content of a referential concept, as we explained inprevious chapters, we mean the intensional content of the predicable conceptbased on that referential concept. Thus, where A is a proper or common namesymbol, complex or simple, and Q is a quantifier symbol representing a deter-miner of natural language, the predicate expression that is determined by thequantifier phrase (QxA) was defined as follows13:

[QxA] =df [λF (QxA)F (x)].

This predicate expression can be nominalized, of course, in which case what itdenotes is the intensional content of the predicate, and thereby, indirectly, the in-tensional content of the referential (quantifier) expression (QxA). As explainedin our earlier lecture, we use [QxA] as an abbreviation of [λF (QxA)F (x)]. Also,it should be remembered that a referential (quantifier) expression that occurswithin an abstract singular term, i.e., within a nominalized complex predicate,has been deactivated and is not used in that occurrence to represent an activeexercise of the referential concept that the expression otherwise stands for as agrammatical subject.

The example we gave was

[Sofia]NP [seeks [a unicorn]]V P

↓ ↓ ↓(∃xSofia)[λxSeek(x, [∃yUnicorn])](x),

where the quantifier phrase ‘a unicorn’ that occurs as part the predicate ‘seeksa unicorn’ has been deactivated. The same quantifier is also deactivated in

SofiaNP [finds [a unicorn]]V P

↓ ↓ ↓(∃xSofia)[λxFind(x, [∃yUnicorn])](x).

But because the predicate Find is extensional in its second argument position,then the latter sentence implies

(∃yUnicorn)(∃xSofia)Finds(x, y).

The predicate Seek, on the other hand, is not extensional in its second argumentposition, which means that ‘Sofia seeks a unicorn’ does not imply that there is

13The application of the λ-operator to predicate variables is understood as an abbreviatednotation, which, in the monadic case, is indicated as follows:

[λFϕ] =df [λy(∃F )(y = F ∧ ϕ)],

where y does not occur free in ϕ.

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a unicorn. In other words, even though ‘Sofia seeks a unicorn’ and ‘Sofia findsa unicorn’ have the same logical form, nevertheless one implies that there is aunicorn, whereas the other does not. The difference, as we explained in ourprevious chapters, is that the following (instance of a) meaning postulate,

[λxFinds(x, [∃yA])] = [λx(∃yA)Finds(x, y)].

is assumed for Find, whereas no such similar meaning postulate can be assumedfor Seek.

This type of meaning postulate also applies to our use of the copula toexpress identity, as when we say that Sofia is an actress. Note that the predicableconcept expressed by ‘is an actress’ in this example cannot be represented by

[λx(∃yActress)(x = y)],

because the quantifier phrase (∃yActress) has not been deactivated. That is,this λ-abstract is not the appropriate way to express the cognitive structure ofthe speech act in question. What we need here is a symbolic counterpart ofthe copula, e.g., Is, as a two-place predicate constant. Thus, the appropriateanalysis of the speech act in question is:

[Sofia]NP [is an actress]V P

↙ ↘ ↘(∃xSofia) [λxIs(x, [∃yActress])]

↘ ↙ ↙(∃xSofia)[λxIs(x, [∃yActress])](x),

where the quantifier phrase (∃yActress) has been deactivated.Now of course this does not mean that we are asserting that Sofia is identical

with the intensional content of being an actress, just as in asserting that Sofiaseeks a unicorn we do not mean that Sofia seeks the intensional content of beinga unicorn. To get at the right truth conditions for this sort of assertion, we needto assume the following as a meaning postulate for the copula Is:

[λxIs(x, [∃yA])] = [λx(∃yA)(x = y)],

where A is a variable having complex or simple names, proper or common, assubstituends. Thus, because of this meaning postulate, the following

(∃xSofia)[λxIs(x, [∃yActress])](x) ↔ (∃xSofia)(∃yActress)(x = y)

is valid in the logic of conceptual realism.14

14Russell, incidentally, proposed a similar analysis in his 1903 Principles, where he assumedthat every proposition consists of a relation between “terms”, and that, e.g., the propositionexpressed by ‘Socrates is a man’ expresses a relation between Socrates and the denotingconcept a man. Presumably, the relation was not strict identity, but something like what weare representing here by Is. Of course, Russell was proposing a logical realist theory and not aconceptualist theory; and he had nothing like our distinction between active and deactivatedconcepts.

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9.4 Deactivation and Geach’s Arguments

In one of his arguments against general reference, Geach claims that “we cannotsuppose ‘some man’ to refer to some man in one single way,” because, if it werea “genuine” referring expression, then “we should have to distinguish severaltypes of reference—it is not easy to see how many”.15

Suppose, Geach argues, “we can say ‘some man’ refers to some man in astatement like this:

(1) Joan admires some man.

that is, a statement for which the question ‘which man?’ would be in order. Letus call this type of reference type-A. Then in a statement like the following one:

(2) Every girl admires some man.

‘some man’ must refer to some man in a different way, since the question ‘Whichman?’ is plainly silly”.16 Calling the type of reference indicated in (2) type-Breference, Geach goes on to argue that we must then distinguish further typesas well.

The problem with this argument is that in an assertion of either (1) or (2),the referential concept that the quantifier phrase ‘some man’ stands for has beendeactivated, i.e., the phrase is not being used to refer in either case. Of course,there is a difference between the two assertions in that (1) logically implies thatsome particular man is admired by Joan—assuming ‘Joan’ is being used withexistential presupposition in this context—whereas (2) does not logically implythat some particular man is admired by every girl. This can be easily seen to beso in the logical forms representing the cognitive structures of these assertions

(1′) (∃xJoan)[λxAdmire(x, [∃yMan])](x),

and

(2′) (∀xGirl)[λxAdmire(x, [∃yMan])](x).

Now it is natural to assume that ‘admire’ is extensional in this context inits second argument position.17 That is, we take

[λxAdmire(x, [QyA])] = [λx(QyA)Admire(x, y)]

to be a meaning postulate representing a conceptual truth in the context inquestion. Then, from an instance of this postulate it can be seen that (by λ-conversion and commutation of existential quantifier phrases), the statementthat some man is admired by Joan, which is analyzed as,

(∃yMan)[λyAdmire([∃xJoan], y)](y),

15Geach 1980, p. 32. Geach attributes this argument to Elizabeth Anscombe.16Ibid.17It is clear that Geach assumes this to be so in the context in question. In some contexts, it

would seem, ‘admire’ might function as an intensional verb—as, e.g., when we say of someonethat s/he admires Sherlock Holmes.

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or equivalently, not considering it as the form of an assertion,

(∃yMan)(∃xJoan)Admire(x, y),

follows validly from (1′), which indicates why the question ‘Which man?’ isappropriate in a context in which (1) is asserted.

Thesis: In general, wh-questions—i.e., ‘who’, ‘which’, ‘what’,‘when’ and ‘where’ questions—apply only to active refer-ential expressions, not to deactivated ones—or, as in thiscase, to those that could be activated as part of a statementthat follows validly from a given assertion.

Now what follows validly from (2′), on the other hand, is

(∀xGirl)(∃yMan)Admire(x, y),

and not(∃yMan)(∀xGirl)Admire(x, y),

which, in the form of an assertion, is equivalent to

(∃yMan)[λyAdmire([∀xGirl], y)](y),

that is, the statement that some man is admired by every girl. In other words,‘some man’ is not being used in (2) to refer to some particular man; nor does(2) imply a sentence in which one might refer to some particular man. That iswhy the question ‘Which man?’ is inappropriate in a context in which (2) isasserted.

It is simply false, on our account, to claim that there are two differenttypes of reference in assertions of (1) and (2). The referential concept that thequantifier phrase ‘some man’ stands for has been deactivated in both assertions,which means that the phrase is not being used in those sentences to refer, noless to refer in two different ways.

Another argument that Geach gives turns on his misconstruing a reflexivepronoun as “a pronoun of laziness,” i.e. as a pronoun that functions as a proxyfor its grammatical antecedent and that can be replaced by that expression“without changing the force of the proposition.”18 Thus, according to Geach,

“If [the quantifier phrase] ‘every man’ has reference to every man,and if a reflexive pronoun has the same reference as the subject ofthe verb, [then] how can ‘Every man sees every man’ be a differentstatement from ‘Every man sees himself’?”19

18Geach 1980, p. 151.19Ibid., p. 9.

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Now, it clear that

Every man sees every man.

and

Every man sees himself.

are different statements. But does this show that the quantifier phrase ‘everyman’ cannot be used to refer to every man, as Geach claims? Is it really clear inthis case that the reflexive pronoun ‘himself’ is functioning here as “a pronounof laziness,” and hence can be replaced by the quantifier phrase ‘every man’ sothat the result is an equivalent sentence, i.e., a sentence having the same forceas the original sentence?

In our theory the occurrence of the quantifier phrase ‘every man’ in the verbphrase ‘sees every man’ is not being used to refer to every man, but insteadstands for a deactivated referential concept. Let us compare an assertion of‘Every man sees every man’ with an assertion of ‘Gino sees Gino’. Assum-ing that ‘Gino’ is being used with existential presupposition, the logical formsrepresenting the cognitive structures of these two assertions are as follows:

(∀xMan)[λxSees(x, [∀yMan])](x),

and

(∃xGino)[λxSees(x, [∃yGino])](x),

where the occurrences of the referential expressions ‘every man’ and ‘Gino’ af-ter the transitive verb are deactivated and interpreted as standing for theirrespective intensional contents.

Note that unlike the above assertions, where the λ-abstracts represent dif-ferent predicable concepts, assertions of

Every man sees himself.

and

Gino sees himself.

involve an application of the same predicable concept, namely,

[λxSees(x, x)].

The logical forms representing the cognitive structures of these assertions, inother words, are as follows:

(∀xMan)[λxSees(x, x)](x),

and

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9.5. GEACH’S ARGUMENTS AGAINST COMPLEX NAMES 207

(∃xGino)[λxSees(x, x)](x).

The reflexive pronoun ‘himself’ is not functioning as “a pronoun of laziness” inthese assertions—even though it has “the same reference as the subject of theverb”20.

Now, if the relational concept of seeing, i.e.,

[λxySee(x, y)],

is extensional in its second argument position, then, because ‘Gino’ is a propername that is assumed to name exactly one object in the context in question, itfollows that ‘Gino see Gino’ and ‘Gino sees himself’ are equivalent, i.e.,

(∃xGino)[λxSees(x, [∃yGino])](x) ↔ (∃xGino)[λxSees(x, x)](x)

is provable.21 In other words, in the case of a proper name A, whereA is assumedto name exactly one object in the context in question, it is true that ‘A sees A’and ‘A sees her/himself’ are necessarily equivalent. This, of course, is not to saythat as assertions, or mental acts, ‘A sees A’ and ‘A sees her/himself’ have thesame cognitive structure, and in fact they have different cognitive structures asindicated by the above logical forms.

On the other hand, ‘Every man sees every man’ and ‘Every man sees himself’are not equivalent; but, contrary to Geach’s claim, this does not mean that theuse of ‘every man’ as the grammatical subject of an assertion of either of thesesentences does not refer to every man, even though its use as the direct objectof the verb does not stand for a referential concept.

Once again, Geach’s implicit assumption seems to be that a referential ex-pression is not a “genuine” referring expression, but only a “quasi subject”, ifit does not behave logically the way a nonempty proper name does.

9.5 Geach’s Arguments Against Complex Names

Some of Geach’s arguments are directed not only against referential expressionsof the form ‘every A’ and ‘some A’, but also against the view that there arecomplex names of the form ‘A that is F ’, and hence against complex referentialexpressions of the form ‘every A that is F ’ and ‘some A that is F ’, which, as al-ready noted in the previous chapters, we symbolize in our theory as (∀xA/F (x))and (∃xA/F (x)).

20That is, the variable x has the same value in the object (second) position of See(x, x) asit does in the subject (first) position.

21If ‘see’ is interpreted as an extensional transitive verb in a given context, then seeing inthat context does not imply knowing who or what it is that one sees. For example, Gino’sseeing Maria (in the extensional sense) does not imply that Gino knows that it is Maria hesees; and, similarly, Gino’s seeing Gino (as in a mirror or a photo) does not imply that Ginoknows that he sees himself. In some contexts ‘see’ might well be interpreted as an intensionalverb, where seeing implies knowing who or what one sees, and in that case, ‘Gino sees Gino’and ‘Gino sees himself’ would then be equivalent.

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One such argument that Geach gives against complex names is based on amedieval paralogism22:

Only an animal can bray; ergo, Socrates is an animal, if hecan bray.But any animal, if he can bray, is a donkey.Ergo, Socrates is a donkey.

Now Geach correctly observes that “we clearly cannot take ‘animal, if hecan bray’ as a complex term [i.e., as a complex name] that is a legitimatereading of ‘A’ in ‘Socrates is an A; any A is a donkey; ergo, Socrates is adonkey”23; but he does not explain the relevance of this observation, or howthis shows that a complex name like ‘animal that can bray’ is not “a genuinelogical unit,”24namely, a complex name.

Apparently, Geach is confusing the complex name ‘animal that can bray’ inthis argument with an expression that is not a complex name, namely, ‘animal,if he can bray’. Note that by the exportation rule

(∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ], (MP1)

mentioned in chapter seven §3, an assertion of ‘Every animal that can bray is adonkey’, which is analyzed as follows:

[Every animal that can bray]NP [is a donkey]V P

↓ ↓(∀xAnimal/Can-Bray(x)) [λxIs(x, [∃yDonkey])]

↓ ↓(∀xAnimal/Can-Bray(x))[λxIs(x, [∃yDonkey])](x)

is equivalent to an assertion of ‘Every animal, if he can bray, is a donkey’,analyzed as,

[Every animal]NP [if he can bray is a donkey]V P

↓ ↓(∀xAnimal) [λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x)

↓ ↓(∀xAnimal)[λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x)

In other words, the following biconditional is valid in the logic of conceptualrealism:

(∀xAnimal/Can-Bray(x))[λxIs(x, [∃yDonkey])](x) ↔(∀xAnimal)[λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x).

22Geach 1980, p. 143.23Ibid.24Ibid., p. 142.

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Geach confuses the grammatically correct claim that in the first assertionwe are referring to every animal that can bray with the grammatically incorrectclaim that in the second assertion we are referring to every animal, if he canbray. That is why Geach claims that “the phrase ‘animal that can bray’ is asystematically ambiguous one,”25 when in fact it is not.

9.6 Relative Pronouns as Referential Expressions

Geach does recognize that “we cannot count this as proved” and attempts to“confirm the suggestion of ambiguity by considering another sort of medievalexample.”26 This is the pair of contradictory sentences,

(3) Any man who owns a donkey feeds it.

(4) Some man who owns a donkey does not feed it.in which, on our account, ‘man who owns a donkey’ occurs as a complex name.

Now, according to Geach, if ‘man who owns a donkey’ is a complex name,then it is “replaceable by the single word ‘donkey-owner’,” in which case (3)and (4) would become “unintelligible”.27 Of course, this sort of “replacementargument” is fallacious in that it deprives the relative pronoun ‘it’ in (3) and(4) of an antecedent, as Geach himself seems to acknowledge. He then suggestsa supposedly “plausible rewording” of (3) and (4) in which ‘it’ is given anantecedent, namely,

(5) Any man who owns a donkey owns a donkey and feeds it.(6) Some man who owns a donkey owns a donkey and does

not feed it.But (5) and (6) are not equivalent to (3) and (4), as Geach himself notes,because, in particular, unlike (3) and (4), (5) and (6) are not contradictories inthat both would be true if each man who owned a donkey had two donkeys andfed only one of them.

Geach then rephrases (3) and (4) as(3′) Any man, if he owns a donkey, [then he] feeds it.

(4′) Some man owns a donkey and he does not feed it.which, by the export-import meaning postulates (MP1) and (MP2) for com-plex referential expressions (given in chapter seven, §3), are equivalent to (3)and (4). That is, as represented by appropriate instances of those meaningpostulates, (3) and (3′), and (4) and (4′), have the same truth conditions, eventhough the cognitive structures of the speech or mental acts they represent arenot the same.

By ignoring the distinction between logical forms that represent the cognitivestructure of our speech and mental acts on the initial level of analysis and the

25Ibid.26Ibid. We have changed the verb ‘beat’ in Geach’s example to ‘feed’, which in no way

affects his argument, or our criticism of it.27Ibid., p. 144.

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logical forms that represent the logical consequences of those acts on the secondlevel of analysis, Geach fallaciously concludes that “the complex term ‘A thatis P ’ is a sort of logical mirage. The structure of a proposition in which such acomplex term appears to occur can be readily seen only when we have replacedthe grammatically relative pronoun by a connective followed by a pronoun; whenthis is done, the apparent unity of the phrase disappears.”28

What is needed here for a proper analysis of (3) and (4) is an analysis of therole the relative pronoun ‘it’ has in in these kinds of sentences, which in the lit-erature have come to be called “donkey-sentences.” Our proposal is that relativepronouns in general, and the pronoun ‘it’ in particular, are referential expres-sions that are interpreted with respect to an antecedent referential expression.In particular, we maintain that the sentence (3), ‘Any man who owns a donkeyfeeds it’ is synonymous with, and in fact has the same cognitive structure as,the following sentence

(3′′) Any man who owns a donkey feeds that donkey.or, if one prefers, the same as

Any man who owns a donkey feeds it (i.e., that donkey).with the phrase ‘that donkey’ expressed, as it were, sotto voce.

Now because all referential expressions are analyzed in conceptual realism asquantifier phrases, what this means is that relative pronouns are to be logicallyanalyzed as quantifier phrases of the form ‘that A’, where A is the name, com-mon or proper, occurring in the antecedent referential phrase relative to whichthe pronoun is interpreted. What we need, accordingly, is a variable-binding‘that’-operator, T , that, when indexed by a variable, can be applied to a nameA, whether complex or simple, and result in a quantifier phrase, e.g., ‘T yA’,which is read as ‘that A’.

On this proposal, the cognitive structure of (3′′)—and, on our proposal,therefore of (3)—can now be analyzed as:

[Any man owns a donkey]NP[feeds that donkey]VP

↙ ↘(∀xMan/Own(x, [∃yDonkey])) [λxFeeds(x, [TyDonkey])]

↘ ↙(∀xMan/Own(x, [∃yDonkey]))[λxFeeds(x, [TyDonkey])](x)

The relative pronoun ‘it’ in (3), in other words, is a proxy for the pronominal ref-erential expression ‘that donkey’, which in this context stands for a deactivatedreferential concept relative to the deactivated antecedent referential conceptthat ‘a donkey’ stands for in the grammatical subject of (3).

Now, by the export-import meaning postulate (MP1) for complex referentialexpressions, the above analysis, which we will call (3cog), is equivalent to

(∀xMan)[λx(Own(x, [∃yDonkey]) → Feeds(x, [TyDonkey]))](x), (3′cog)

28Ibid., p. 145.

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which is easily seen to represent the cognitive structure of an assertion of (3′),i.e., the sentence ‘Any man, if he owns a donkey, [then he] feeds it’. But be-cause ‘own’ and ‘feed’ are extensional transitive verbs, the deactivated quantifierphrases ‘a donkey’ and ‘that donkey’ can be “reactivated,” in which case (3cog)and (3′cog) are equivalent to

(∀xMan)[(∃yDonkey)Own(x, y) → (TyDonkey)Feeds(x, y)],

which does not represent the cognitive structure of a speech or mental act, butdoes represent the truth conditions of an assertion of either (3) or (3′). We canobtain a logically more perspicuous representation of those truth conditions,moreover, by means of the following meaning postulate for the T -operator, i.e.,a postulate that makes clear that it is functioning as a pronoun relative to anantecedent referential expression:

[(∃yS)ϕy → (TyS)ψy] = [(∀yS)(ϕy → ψy)], (MPT2)

which, by Leibniz’s law implies the weaker equivalence,

[(∃yS)ϕy → (TyS)ψy] ↔ [(∀yS)(ϕy → ψy)].

Thus, by means of this postulate and the preceding formula, it follows that

(∀xMan)(∀yDonkey)[Owns(x, y) → Feeds(x, y)]

is equivalent to (3cog) and (3′cog), and this formula, it is clear, provides a logicallyperspicuous representation of their truth conditions, and hence of the truthconditions of (3) and (3′).

The meaning postulate (MPT2) for the ‘that’-operator explains why sen-tences like

If someone is married, then s/he (i.e., that person)has a spouse.

andIf a witness lied, then s/he (i.e., that witness)committed perjury.

have the truth conditions that they do, and are equivalent to

Anyone who is married has a spouse.and

Any witness who lied committed perjury.

As noted in section 7 of the previous chapter, the T -operator is designedto be used only on the initial level of analysis regarding the cognitive structureof our speech and mental acts. It is not designed to be used on the level ofdeductive transformations, where such rules as simplification, adjunction, andthe rewrite of bound variables might be applied. The idea is to restrict thestandard transformations to just Leibniz’s law as based on meaning postulatesuntil the occurrences of the T -operator have been eliminated.

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Turning now to a formal representation of the cognitive structure of (4), i.e.,the sentence ‘Some man who owns a donkey does not feed it’, let us note firstthat this sentence, on our proposal, has the same cognitive structure as

(4′′) Some man who owns a donkey does not feed that donkey.Now because the negation in the verb phrase ‘does not feed it’ is internal to thepredicate, we have the following as an analysis of (4′′), and therefore, on ourproposal, of (4) as well:

[Some man who owns a donkey]NP [does not feed that donkey]V P

↙ ↘(∃xMan/Own(x, [∃yDonkey])) [λx[λzw¬Feeds(z, w)](x, [TyDonkey])]

↘ ↙(∃xMan/Own(x, [∃yDonkey]))[λx[λzw¬Feeds(z, w)](x, [TyDonkey])](x)

This analysis of the cognitive structure of (4′′)—and hence, on our proposal, of(4) as well—can be simplified by applying the export-import meaning postulate(MP2) for complex names and the meaning postulates regarding the exten-sionality of ‘own’ and ‘feed’, and therefore of ‘does not feed’. In other words,by these meaning postulates, the above analysis of (4′′) and (4), which we willcall (4cog), is equivalent to29:

(∃xMan)[(∃yDonkey)Own(x, y) ∧ (TyDonkey)¬Feeds(x, y)].

Finally, the relevant meaning postulate for the T -operator in this case is thefollowing,30

[(∃yS)ϕy ∧ (TyS)ψy] = [(∃yS)(ϕy ∧ ψy)], (MPT1)

which, by Leibniz’s law implies

[(∃yS)ϕy ∧ (TyS)ψy] ↔ [(∃yS)(ϕy ∧ ψy)],

which, together with the preceding formula, shows that (4cog) is equivalent to,and therefore has the same truth conditions as,

(∃xMan)(∃yDonkey)[Own(x, y) ∧ ¬Feeds(x, y)].

This formula is easily seen to be a contradictory of the above logically perspic-uous representation of the truth conditions for (3). That is,

(∀xMan)(∀yDonkey)[Owns(x, y) → Feeds(x, y)]

29Here, we should keep in mind here that we cannot proceed from the initial level of analysisregarding cognitive structure to the deductive level until all applications of Leibniz’s law asbased on meaning postulates have been applied. In particular, λ-conversion is not to beapplied until after the meaning postulates for extensional verbs have been applied.

30An example of the use of this meaning postulate is one from Geach 1980, namely, ‘Someman broke the bank at Monte Carlo and that man died a pauper’, the truth conditions ofwhich are the same as ‘Some man broke the bank at Monte Carlo and died a pauper’.

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9.7. SUMMARY AND CONCLUDING REMARKS 213

and(∃xMan)(∃yDonkey)[Own(x, y) ∧ ¬Feeds(x, y)].

are contradictories in that one is equivalent to the negation of the other.We conclude, accordingly, that the sentences

(3) Any man who owns a donkey feeds it.

and(4) Some man who owns a donkey does not feed it.

do in fact have the truth conditions Geach says they have, even though theexpression ‘man who owns a donkey’ functions in both as a complex name,contrary to what Geach claims. In other words, the sentences (3) and (4) in noway support Geach’s claim that “the complex term ‘A that is P ’ is a sort oflogical mirage” and is not “a genuine logical unit”, and that such expressionsmust be expanded into forms where there are no complex names at all. Nor dothey show that there are inextricable difficulties with the conceptualist theoryof reference we have described here.

9.7 Summary and Concluding Remarks

• The conceptualist theory of reference we described in chapter seven notonly has all of the philosophically important features we listed there, but itprovides as well a general framework by which to refute the claim that there canbe only singular reference, and hence the claim that there can be no “genuine”form of general reference.

• The idea that the only genuine form of reference is singular reference hasbeen the dominant theory throughout the twentieth century, but that doctrineis based either on the type of arguments that Geach has given and that we haverefuted here, or it is based on a confusion of pragmatics with semantics, i.e.,that the analysis of the cognitive structure of our speech and mental acts is thesame as the analysis of their truth conditions.

• The truth conditions of sentences containing quantifier phrases are ofcourse reducible to the atomic components of those sentences, but that doesnot mean that those same quantifier phrases do not stand for referential con-cepts. Indeed, to the contrary, general reference is a basic feature of our speechand mental acts, which is why quantifier phrases occur as grammatical sub-jects, or noun phrases, in natural language—and occur as such, moreover, withas great, or greater frequency, than proper names do.

• The dominance of the doctrine that there can be only singular referenceexplains why the logical analysis of the cognitive structure of our speech andmental acts has been ignored in the analytic movement.

• By giving an analysis only of the truth conditions of our speech and mentalacts, the analytic movement has assumed that singular reference is the onlygenuine form of reference. As a result, the analytic movement ignored theproblem of giving a logical analysis of the cognitive structure of our speech and

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mental acts, because in order to do so it must give an account of general as wellas singular reference.

• What is needed is a theory of logical forms that can represent general aswell as singular reference, and in particular a theory such as we have constructedfor conceptual realism.

• Just as a unified account of general and singular reference was once givenby medieval logicians, but only for categorical propositions, conceptual real-ism provides a unified account of both general and singular reference for allpropositional forms combining a noun phrase with a verb phrase.

• It is by such a unified account that conceptual realism can give a logicalanalysis of the cognitive structure of our speech and mental acts, which is thestarting point for any formal ontology that is based initially on the structure ofthought. It is then by means of such a unified account that an analysis of theontological categories of reality can be given as well.

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Chapter 10

Lesniewski’s Ontology

Referential concepts in conceptual realism are based on a logic of proper andcommon names as parts of quantifier phrases. This conceptualist logic of namesis similar to Lesniewski’s logic of names in that the category of names inLesniewski’s system also contains common as well as proper names.1 Lesniewski’slogic is different, however, in that names do not occur as parts of quantifierphrases but are of the same category as objectual variables. Lesniewski de-scribed his logic of names as “ontology,” apparently because it was to be theinitial level of a theory of types, which Lesniewski called semantic categories.2

Lesniewski’s general framework also included mereology, which is a theory of therelation of part to whole, and protothetic, a quantificational logic over proposi-tions and n-ary truth-functions, for all positive integers n.

Lesniewski’s logic of names has been used for years as a framework in whichto interpret and reconstruct various doctrines of medieval logic.3 We have givenan alternative interpretation and reconstruction of medieval logic in terms ofthe framework of conceptual realism.4 It is relevant therefore to see how, orin what respect, Lesniewski’s logic of names is similar to our conceptualistlogic of names. In fact, as we will explain, Lesniewski’s logic of names canbe completely interpreted, and in that sense is reducible, to our conceptualistlogic of names.5

Lesniewski’s based his system of mereology, i.e., his logic of the relationshipbetween parts and wholes, on his logic of names, and though the exact form ofthis connection is not clear it has something to do with the notion of classes ina collective sense as opposed to a distributive sense. Our conceptualist logic ofnames, on the other hand, is the basis of a logic of classes as many, i.e., a logic

1This chapter is a development of material from my 2001 paper, “A Conceptualist Inter-pretation of Lesniewski’s Ontology,” History and Philosophy of Logic, vol. 22.

2See Lejewski 1958, p. 152, Slupecki 1955 and Iwanus 1973 for a description of Lesnieski’sgeneral framework as well as his logic of names.

3See, e.g., Henry 1972.4See chapter 8 and also Cocchiarella 2001 for the details of such a reconstruction.5See below and also Cocchiarella 2001 for a detailed proof that Lesniewski’s logic of names

is reducible to our conceptualist logic of names.

215

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216 CHAPTER 10. LESNIEWSKI’S ONTOLOGY

of classes that in some respects is similar to Lesniewski’s mereology, but in otherrespects it is also different. Unlike the connection between Lesniewski’s logic ofnames and his mereology, however, the connection between our logic of classesas many and our simple logic of names is both precise and a fundamental partof conceptual realism.

We will first briefly describe Lesniewski’s logic of names and then formu-late the simple logic of names that is a fragment of our broader, more com-prehensive formal ontology for conceptual realism. We will then explain howLesniewski’s system can be interpreted within our logic and how certain oddi-ties of Lesniewski’s system can be explained in terms of our logic where thoseoddities do not occur. We will then explain how the logic of classes as many isdeveloped as an extension of the simple logic of names.

The logic of names of Lesniewski’s general framework and of our frameworkof conceptual realism provide, incidentally, another illustration, or paradigm, ofhow different parts or aspects of a formal ontology can be developed indepen-dently of, or even prior to, the construction of a comprehensive system all atonce.

10.1 Lesniewski’s Logic of Names

In Lesniewski’s logic of names, as in our conceptualist logic, there is a distinctionbetween

1. shared, or common names, such as ‘man’, ‘horse’, ‘house’, etc., and eventhe ultimate superordinate common name ‘thing’, or ‘object’;

2. unshared names, i.e., names that name just one thing, such as propernames; and

3. vacuous names, i.e., names that name nothing.6

There is a categorial difference between names in Lesniewski’s logic andnames in our conceptualist logic, however. In Lesniewski’s logic names are of thesame category as the objectual variables, which means that they are legitimatesubstituends for those variables in first-order logic. In our conceptualist logic,names belong to a category of expressions to which quantifiers are applied andthat result in quantifier phrases such as ‘every raven’, ‘some man’, ‘every citizenover eighteen’, etc.

The one primitive of Lesniewski’s logic, aside from logical constants, is therelation symbol ‘ε’ for singular inclusion, which is read as the copula ‘is (a)’, asin ‘John is a teacher’, where both ‘John’ and ‘teacher’ are names.7 Using ‘a’,‘b’, ‘c’, etc., as objectual constants and variables for names, the basic formula of

6See Lewjeski’s 1958 for a detailed description of Lesniewski’s logic of names.7Apparently, it was �Lukasiewicz who prompted Lesniewski to develop his logic of names

when he expressed dissatisfaction with the way G. Peano used ‘∈’ for the copula in set theory.Cp. p. 414 of Rickey’s 1977.

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the logic is ‘a ε b’, where either shared, unshared, or vacuous names may occurin place of ‘a’ and ‘b’.

A statement of the form ‘a ε b’ is taken as true if, and onlyif ‘a’ names exactly one thing and that thing is also namedby ‘b’, though ‘b’ might name other things as well, as in ourexample of ‘John is a teacher’.

Identity is not a primitive logical concept of Lesniewski’s system, as it is inour conceptualist logic, but is defined instead as follows:

a = b =df a ε b ∧ b ε a.

That is, ‘a = b’ is true in Lesniewski’s logic if, and only if, ‘a’ and ‘b’ areunshared names that name the same thing. That seems like a plausible thesis,except that then, where ‘a’ is a shared or vacuous name, ‘a = a’ is false. Infact, because there are necessarily vacuous names, such as the complex commonname ‘thing that is both square and not square’, the following is provable inLesniewski’s logic:

(∃a)(a �= a),

which does not seem at all like a plausible thesis. Of course, this means that

(∀a)(a = a)

is not a valid thesis in Lesniewski’s system. Lesniewski does include a weaknotion of identity, which is defined as follows

a ◦ b =df (∀c)(c ε a↔ c ε b),

and which does not have these odd features. This notion, of course, means thata and b are co-extensive, not identical. But then, Lesniewski insisted on hislogic being extensional, and not intensional, in which case a ◦ b does amountto a kind of identity when a and b are either shared or unshared names. Ofcourse, in that case all vacuous names, such as ‘Pegasus’ and ‘Bucephalus’ areidentical in this weak sense. It also means that Lesniewski’s ontology is not anappropriate framework for tense and modal logic, or for intensional contexts ingeneral.8

Another valid thesis of Lesniewski’s logic is,

ϕ(c/a) → (∃a)ϕ(a),

which seems counter-intuitive when ‘c’ is a vacuous name. The following, forexample, would then be valid

¬(∃b)(b = Pegasus) → (∃a)¬(∃b)(b = a),

8One can intensionalize Lesniewski’s framework, of course, even though he himself wasagainst such a move.

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218 CHAPTER 10. LESNIEWSKI’S ONTOLOGY

and therefore, because the antecedent is true, then so is the consequent, whichsays that something is identical with nothing.

Perhaps these oddities can be explained by interpreting Lesniewski’s first-order quantifiers substitutionally rather than as referentially. That, however,is not how Lesniewski understood his logic of names, which, as we have said,he also called ontology.9 A logic that interprets the quantifiers of its basiclevel substitutionally, rather than referentially, would be an odd sort of formalontology. Does that mean that a name would be required for every object in theuniverse, including, e.g., every grain of sand and every microphysical particle?

We should also note that Lesniewski’s epsilon symbol, ‘ε’, for singular inclu-sion should not be confused with the epsilon symbol, ‘∈’, for membership in aset. In particular, whereas the following:

a ε b→ a ε a,

a ε b ∧ b ε c→ a ε c,

are both theorems of Lesniewski’s system, both are invalid for membership in aset.

Finally, the only nonlogical axiom of ontology—i.e., the only axiom in ad-dition to the logical axioms and inference rules of first-order predicate logicwithout identity—assumed by Lesniewski was the following:

(∀a)(∀b)[a ε b↔ (∃c)c ε a ∧ (∀c)(c ε a→ c ε b) ∧ (∀c)(∀d)(c ε a ∧ d ε a→ c ε d)].

This axiom alone does not suffice for the elementary logic of names, however,i.e., for Lesniewski’s logic of names as formulated independently of the type-theoretic part of Lesniewski’s framework. It has been shown, however, thatadding the following two axioms to the one above does suffice10:

(∀a)(∃b)(∀c)[c ε b↔ c ε c ∧ c /ε a], (Compl)

(∀a)(∀b)(∃c)(∀d)[d ε c↔ d ε a ∧ d ε b]. (Conj)

Expressed in terms of our conceptualist logic, where names are taken to ex-press name (or nominal) concepts, what these axioms stipulate is that thereis a complementary name concept corresponding to any given name concept,and, similarly, that a conjunctive name concept corresponds to any two nameconcepts with singular inclusion taken conjunctively.

9See, e.g., Lejewski 1958 and Kung and Canty 1979 for a discussion of this issue.10See Iwanus 1973 for a proof of this claim. Instead of the following axioms, Lesniewski

assumed a theory of definitions for constant names and name-forming functors. Lesniewski’stheory of definitions does not always satisfy the conditions for noncreativity, however; but itwas shown in Iwanus 1973 that with the addition of the following axioms then Lesniewski’stheory of definitions can be proved to be noncreative.

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10.2. THE SIMPLE LOGIC OF NAMES 219

10.2 The Simple Logic of Names

The simple logic of names, which, as we said, is an independent fragment of thefull logic of conceptual realism, can be described as a version of an identity logicthat is free of existential presuppositions regarding singular terms—i.e., freeobjectual variables and expressions that can be properly substituted for such.It contains both absolute and relative quantifier phrases, i.e., relative quantifierphrases such as (∀xA) and (∃xA), as well as absolute quantifier phrases suchas (∀x) and (∃y), which are read as (∀xObject) and (∃yObject), respectively.We will continue to use x, y, z, etc., with or without numerical subscripts, asobjectual variables, as we did in our second lecture.

We will now also use A,B,C, with or without numerical subscripts, as name,or “nominal”, variables. As explained in previous chapters, complex names areformed by adjoining (so-called “defining”) relative clauses to names, and we use‘/’, as in ‘A/ϕ’ to represent the adjunction of a formula ϕ to the name A (whichmay itself be complex). Thus, e.g., the quantifier phrase representing referenceto a house that is brown would be symbolized as (∃xHouse/Brown(x)).

We continue to take the universal quantifier, ∀, the (material) conditionalsign, →, the negation sign, ¬, and the identity sign, =, as primitive logicalconstants, and assume the others to be defined in the usual (abbreviatory) way.The absolute quantifier phrases (∀x) and (∃x) are read as ‘Every object ’ and‘Some object ’, or, equivalently, as ‘Everything’ and ‘Something’, respectively.That is, the absolute quantifiers are understood as implicitly containing the mostgeneral or ultimate common name ‘object’ (which we take to be synonymouswith ‘thing’). The quantifier phrases (∀A) and (∃A) are taken as referring toevery, or to some, name concept, respectively. Name constants are introducedin particular applications of the logic.11

Because complex names contain formulas as relative clauses, names andformulas are inductively defined simultaneously as follows12:

• (1) every name variable (or constant) is a name;

• (2) for all objectual variables x, y, (x = y) is a formula; and

• if ϕ, ψ are formulas, B is a name (complex or simple), and x and C arean objectual and a name variable respectively, then(3) ¬ϕ,(4) (ϕ→ ψ),(5) (∀x)ϕ,(6) (∀xB)ϕ, and(7) (∀C)ϕ are formulas, and(8) B/ϕ and(9) /ϕ are names, where /ϕ is read as ‘object that is ϕ’.

11The absolute quantifiers and the quantifiers for name concepts are understood to berelativized to a given universe of discourse in an applied form of the logic.

12We adopt the usual informal conventions for dropping parentheses and for sometimesusing brackets instead of parentheses.

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220 CHAPTER 10. LESNIEWSKI’S ONTOLOGY

We assume the usual definitions of bondage and freedom for objectual vari-ables and of the proper substitution of one objectual variable for another ina formula, and similarly we assume the definitions of bondage and freedom ofoccurrences of name variables in formulas, and the proper substitution in aformula ϕ of a name variable (or constant) B for free occurrences of a namevariable C.

Definition: A complex nameB/ξ is free for C in ϕ with respect to an objectualvariable x (as place holder) if(1) for each variable y such that (∀yC) occurs in ϕ and C is free at thatoccurrence, then y is free for x in B/ξ, and(2) no variable, name or objectual, other than x that is free inB/ξ becomesbound when a free occurrence of C in ϕ is replaced by an occurrence ofB/ξ(y/x).13

Note: If a name B (complex or simple) is free for C in ϕ with respect to avariable x, then the proper substitution of B for C in ϕ with respect to xis represented by ϕ(B[x]/C).

Among the rules, or meaning postulates, of our logic of names are four thatwere mentioned in our previous lecture. The first two connect relative quantifierphrases with absolute phrases, and the next two amount to export and importrules for quantifier phrases with complex names.

(∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ], (MP1)

(∃xA)ϕ ↔ (∃x)[(∃yA)(x = y) ∧ ϕ], (MP2)

(∀xB/ϕ)ψ ↔ (∀xB)[ϕ → ψ], (MP3)

(∃xB/ϕ)ψ ↔ (∃xB)[ϕ ∧ ψ]. (MP4)

Of course, strictly speaking, (MP2) and (MP4) are redundant because (∃xA)is taken as an abbreviation for ¬(∀xA)¬, whether A is simple or complex. Forthis reason, we will restate (MP1) and (MP3) as axioms 10 and 11. below.

The axioms of the simple logic of names are those of the free logic of identityplus the axioms for name quantifiers:

Axiom 1: All tautologous formulas;

Axiom 2: (∀x)[ϕ→ ψ] → [(∀x)ϕ → (∀x)ψ];

Axiom 3: (∀C)[ϕ→ ψ] → [(∀C)ϕ → (∀C)ψ];

Axiom 4: (∀C)ϕ → ϕ(B[x]/C), where B is free for C in ϕ with respect to x;

Axiom 5: χ→ (∀C)χ, where C is not free in χ;

Axiom 6: χ→ (∀x)χ, where x is not free in χ;

13The use of ‘/’ in ‘ξ(y/x)’ represents the result of properly substituting y for x in ξ, andshould not be confused with the use of ‘/’ to generate complex names.

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Axiom 7: (∀x)(∃y)(x = y), where x, y are different variables;

Axiom 8: x = x;

Axiom 9: x = y → (ϕ→ ψ), where ϕ, ψ are atomic formulas and ψis obtained from ϕ by replacing anoccurrence of y by x14

Axiom 10: (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ], where x, y are differentvariables;

Axiom 11: (∀xA/ψ)ϕ↔ (∀xA)[ψ → ϕ].

We assume as primitive inference rules modus ponens (MP) and universalgeneralization (UG) for absolute quantifiers indexed by either an objectual ora name variable. The rule of universal generalization for relative quantifiers,

if � ϕ, then � (∀xA)ϕ, (UGN )

is derivable by (UG) from Axiom 10. The usual laws for interchanging prov-ably equivalent formulas and for rewriting bound variables are easily seen tobe derivable as well. The universal instantiation law in free logic for objectualvariables,

(∃x)(x = y) → [(∀x)ϕ → ϕ(y/x)], (∃/UI)

where x, y are distinct variables and y is free for x in ϕ, is derivable by Leibniz’slaw (LL), i.e., Axiom 9, (UG), Axioms 2 and 6, and tautologous transfor-mations. The theorems that are counterparts to Axioms 10 and 11 for theexistential quantifiers, namely,

T1: � (∃xA)ϕ↔ (∃x)[(∃yA)(x = y) ∧ ϕ],

and

T2: � (∃xA/ψ)ϕ↔ (∃xA)[ψ ∧ ϕ]

are also derivable by elementary transformations and the definitions for ∧ and∃. Also, because absolute quantifiers are viewed as implicitly containing thecommon name ‘object’, we assume that Axiom 11 has the following schema asa special instance.

T3: � (∀x/ψ)ϕ↔ (∀x)[ψ → ϕ].

The following are some obvious theorems that are easily seen to be provable.

T4: (∀x)ϕ→ (∀xA)ϕ.

T5: (∀xA)ϕ → [(∃zA)(y = z) → ϕ(y/x)], where y is free for x in ϕ.

T6: (∃xA)(y = x) → (∃x)(y = x).

14As noted in our second lecture, the full version of Leibniz’s law is derivable from thisaxiom so long as no intensional operators are introduced into the logic.

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T7: (∀x)ϕ↔ (∀A)(∀xA)ϕ, where A is not free in ϕ.15

Finally, the following comprehension principle

(∀A)(∃B)(∀x)[(∃yB)(x = y) ↔ (∃yA)(x = y)] (CPN)

is immediately derivable from Axiom 1, or really from the contrapositive ofAxiom 1, which amounts to a form of existential generalization for name con-cepts. Stated as a schema derived from Axiom 4, (CPN ) can be described asfollows:

(∃B)(∀x)[(∃yB)(x = y) ↔ (∃yA/ϕ)(x = y)].

10.3 Consistency and Decidability

The simple logic of names that we have formulated in the previous section isboth consistent and decidable.16 This follows by noting that the logic is actuallyequiconsistent with second-order monadic predicate logic, which is known to beconsistent and decidable.17 We will not go through all of the details of showingthis here, but we give a general outline of the proof.18

First, let us note that by the rule (MP3) and a simple inductive argumentit can be shown that every formula of our conceptualist system in which acomplex name occurs is provably equivalent to a formula in which no complexname occurs.

Metatheorem 1: If ϕ is a formula of the simple logic of names—i.e., ofthe free first-order logic of identity extended to include name variables, quantifi-cation over such, and restricted quantifiers with respect to such—then there is aformula ψ in which no complex name occurs such that ϕ is provably equivalentto ψ in this logic, i.e., � ϕ↔ ψ.

Because of the above metatheorem, we can, in what follows, restrict ourselvesto formulas in which no complex name occurs.19 We assume a one-to-one corre-lation of the name variables A,B,C,D, etc., with one-place predicate variablesFA, FB, FC , FD, etc., and inductively define a translation function trs∗ from theformulas of our simple logic of names in which no complex name occurs intoformulas of second-order monadic predicate logic (with identity) as follows:

1. trs∗(x = y) = (x = y),15Proof: The left-to-right direction follows by T4, (UGN), quantifier laws. The right-

to-left direction follows by first universally instantiating A to thing identical to itself, i.e.,/(x = x), so that by axiom 4 we have � (∀A)(∀xA)ϕ → (∀x/x = x)ϕ, and, by T3, �(∀x/x = x)ϕ → (∀x)[x = x → ϕ], from which, by axiom 8, (UG), and axioms 2 and 1,� (∀x/x = x)ϕ → (∀x)ϕ; and from this the right-to-left direction of T7 follows.

16It is important to keep in mind here that the only relation symbol of the system is theidentity sign.

17See Church 1956, p. 303, exercise 52.4.18For all of the details, see Cocchiarella 2001a.19We also ignore name constants and concern ourselves only with formulas in which no

applied descriptive constants occur.

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10.3. CONSISTENCY AND DECIDABILITY 223

2. trs∗(¬ϕ) = ¬trs∗(ϕ),

3. trs∗(ϕ→ ψ) = [trs∗(ϕ) → trs∗(ψ)],

4. trs∗((∀x)ϕ) = (∀x)trs∗(ϕ),

5. trs∗((∀xA)ϕ) = (∀x)[FA(x) → ϕ],

6. trs∗((∀A)ϕ) = (∀FA)trs∗(ϕ).

It is clear that the translation under trs∗ of every theorem of our simplelogic of names in which no complex name occurs becomes a theorem of second-order monadic predicate logic. The restriction to formulas in which no complexnames occur can be dropped by allowing, for each formula ϕ in which a complexname does occur, the translation function trs∗ to assign trs∗(ψ) to ϕ, where ψis the first formula (in terms of some alphabetic ordering) in which no complexname occurs and such that � ϕ ↔ ψ. By extending trs∗ in this way, it thenfollows that every theorem of our conceptualist logic of names is translated intoa theorem of second-order monadic predicate logic, and hence, given the knownconsistency of the latter, that our conceptualist system is consistent.20

Metatheorem 2: If ϕ is a theorem of our present conceptualist logic,then trs∗(ϕ) is a theorem of second-order monadic predicate logic. Therefore,our simple conceptualist logic is consistent.

Now we can also show that our simple conceptualist logic of names is decid-able by noting that every theorem of second-order monadic predicate logic canbe translated into a theorem of our conceptualist logic, and hence that the onesystem is essentially equivalent to the other. It is well-known, of course, thatsecond-order monadic predicate logic is decidable.

Our proof that every theorem of second-order monadic predicate logic isa theorem of our conceptualist logic of names involves a translation function,trs′, which translates each formula of second-order monadic predicate logic intoa formula of our simple logic of names. In giving this translation function notethat monadic predicate variables can be put into a one-to-one correspondencewith name variables. We can, in other words, take each predicate variable tohave the form FA, where A is the name variable corresponding to that predicatevariable. The translation function, trs′, is then defined as follows:

1. trs′(x = y) = (x = y),

2. trs′(FA(x)) = (∃yA)(x = y), where y is the first objectual variable otherthan x,

3. trs′(¬ϕ) = ¬trs′(ϕ),

4. trs′(ϕ→ ψ) = (trs′(ϕ) → trs′(ψ)),

20For the consistency of second-order monadic predicate logic, see Church 1956, p. 303.

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224 CHAPTER 10. LESNIEWSKI’S ONTOLOGY

5. trs′((∀x)ϕ) = (∀x)trs′(ϕ),

6. trs′((∀FA)ϕ) = (∀A)trs′(ϕ).

It is easily seen that the trs′ translation of every axiom of second-ordermonadic predicate logic is a theorem of our simple logic of names and thatthe rules of inference, modus ponens and the rule of universal generalization,preserve theoremhood. From these observations we conclude that we have thefollowing metatheorem.

Metatheorem 3: If ϕ is a theorem of second-order monadic predicatelogic, then trs′(ϕ) is a theorem of our present conceptualist logic.

Finally, by metatheorem 1, to show that our conceptualist logic of namesis decidable, we need only show that the formulas in which no complex namesoccur are decidable. To show this we first prove the following metatheorem byinduction on these formulas.

Metatheorem 4: If ϕ is a formula of our conceptualist logic and nocomplex names occur in ϕ, then � ϕ↔ trs′(trs∗(ϕ)).21

It follows, accordingly, that if ϕ is a formula of our conceptualist logic ofnames in which no complex names occur, then to decide whether or not ϕ isa theorem of this logic it suffices to decide whether or not trs∗(ϕ) is a the-orem of second-order monadic predicate logic. If the latter is not a theoremof second-order monadic predicate logic, then, by metatheorem 2, ϕ is not atheorem of our conceptualist logic; and if trs∗(ϕ) is a theorem of second-ordermonadic predicate logic, then, by metatheorem 3, trs′(trs∗(ϕ)) is a theorem ofour conceptualist logic, and therefore, by metatheorem 4, so is ϕ. Hence, bymetatheorem 1, the decision problem for our conceptualist logic is reducible tothat of second-order monadic predicate logic.

Metatheorem 5: Our present conceptualist logic is both consistent anddecidable.

10.4 A Reduction of Lesniewski’s System

We now turn to a translation of Lesniewski’s logic of names, as briefly describedin section 2, into our conceptualist logic of names. We assume that the name

21Proof. As noted, we prove this metatheorem by induction on the formulas of our con-ceptualist logic in which no complex names occur. The case for atomic formulas, whichconsist only of identities, is of course immediate; and for negations and conditionals, againthe proof is immediate. Suppose the metatheorem holds for ϕ; then again it follows imme-diately that it holds for (∀A)ϕ. The only interesting case is for (∀xA)ϕ. But, by defini-tion of trs∗, trs∗((∀xA)ϕ) = (∀x)[FA(x) → trs∗(ϕ)], and therefore, by definition of trs′,trs′(trs∗((∀xA)ϕ)) = (∀x)[(∃yA)(x = y) → trs′(trs∗(ϕ))]; and therefore, by the inductivehypothesis and (MP1), � (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → trs′(trs∗(ϕ))], which completesour proof by induction.

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variables a, b, c, d (with or with out numerical subscripts) of Lesniewski’s logicare correlated one-to-one with the name variables A,B,C,D (with or withoutnumerical subscripts) of our conceptualist logic, i.e., that

A is correlated with a,B is correlated with b,C is correlated with c,etc.

Because the only atomic formulas of the system are of the form ‘a ε b’, the fol-lowing inductive definition of a translation function trs translates each formulaof Lesniewski’s logic into a formula of our conceptualist logic (with a replacedby A, b by B, etc.):

1. trs(a ε b) = (∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yB)(x = y),

2. trs(¬ϕ) = ¬trs(ϕ),

3. trs(ϕ→ ψ) = [trs(ϕ) → trs(ψ)],

4. trs((∀a)ϕ) = (∀A)trs(ϕ).

In regard to the translation of an atomic formula of the form a ε b, note thatthe first conjunct,

(∀xA)(∀yA)(x = y),

of the translation is interpreted as saying that at most one thing is A, andtherefore, because A is correlated with a, that at most one thing is a. Thesecond conjunct,

(∃xA)(∃yB)(x = y),

on the other hand, says that some A is a B, and therefore, that some a is ab. The two conjuncts together are then equivalent to saying that exactly onething is A, and hence a, and that thing is a B, i.e., a b, which is how Lesniewskiunderstood ‘a ε b’ as singular inclusion.

Note also that where ϕ is a logical axiom of the first-order logic of Lesniewski’ssystem, then trs(ϕ) is a theorem of our conceptualist logic.22 Modus ponensand (UG) also preserve validity under trs. Accordingly, to show that thisinterpretation amounts to a reduction of Lesniewski’s ontology, we need onlyprove that trs translates the axioms of Lesniewski’s logic into a theorem ofour present system. For example, both of the axioms, (Compl) and (Conj), ofLesniewski’s logic—one stipulating that every name has a complementary name,and the other that there is a name corresponding to the conjunction of singularinclusion in any two names—can be derived from the comprehension principle(CPN) of our conceptualist logic as follows.

22In fact, Axioms 1-3 are just the translations of the quantifier axioms assumed in thefirst-order theory for Lesniewski’s ontology. By definition of trs, moreover, it is obvious thatthe translation of a tautologous formula is also a tautologous formula.

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By (CPN) and Axiom 1, substituting for A the complex name form/¬(∃zA)(y = z) (which is read as ‘thing that is not an A’), we have

� (∃B)(∀x)[(∃yB)(x = y) ↔ (∃y/¬(∃zA)(y = z))(x = y)],

from which, by (MP4) and Leibniz’s law, it follows that

� (∃B)(∀x)[(∃yB)(x = y) ↔ ¬(∃zA)(x = z)],

which affirms the existence of a nominal concept that is the complement of A.Finally, by proofs similar to those given in section three, it can be shown on thebasis of this last theorem that the translation of (Compl), i.e. trs(Compl), isprovable in our conceptualist logic of identity.

Similarly, by (rewriting B to C in) (CPN) and Axiom 1, substituting nowfor A the complex name form /(∃zA)(y = z) ∧ (∃zB)(y = z) (which is read as‘thing that is both an A and a B’), we have

� (∃C)(∀x)[(∃yC)(x = y) ↔ (∃y/(∃zA)(y = z) ∧ (∃zB)(y = z))(x = y)],

which, by (MP4), reduces to

� (∃C)(∀x)[(∃yC)(x = y) ↔ (∃y)[(∃zA)(y = z) ∧ (∃zB)(y = z) ∧ y = x]],

and hence, by Leibniz’s law, to

� (∃C)(∀x)[(∃yC)(x = y) ↔ (∃zA)(x = z) ∧ (∃zB)(x = z)],

which affirms the existence of a nominal concept corresponding to the conjunc-tion of being both an A and a B. Again, by proofs similar to those in sectionthree, it can be shown on the basis of this last theorem that the translation of(Conj), i.e. trs(Conj), is provable in our conceptualist logic of identity.

The derivation of the translation of Lesniewski’s principal axiom, which isthe only one remaining, is relatively trivial, but long on details, and we will notgo into those detail here.23 In any case, we have the following metatheorem.

Metatheorem 6: If ϕ is a theorem of Lesniewski’s (first-order) logicof names, then trs(ϕ) is a theorem of our conceptualist simple logic of names.

Finally, let us turn to an explanation of the oddities of Lesniewski’s logic ofnames, i.e., an explanation in terms of our translation of Lesniewski’s logic intoour simple logic of names. First, in regard to the seemingly implausible thesis,

(∃a)(a �= a),

of Lesniewski’s logic, note that by Lesniewski’s definition of identity (and henceof nonidentity) (a �= a) is really short for ¬(a ε a ∧ a ε a), which is equivalentto ¬(a ε a). On our conceptualist interpretation, this formula translates into

¬[(∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yA)(x = y)],23See Cocchiarella 2001a for those details.

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which in effect says that it is not the case that exactly one thing is an A, athesis that is provable in our conceptualist logic when A is taken as a necessar-ily vacuous common name, such as ‘object that is not self-identical’, which issymbolized as ‘/(x �= x)’, or more fully as ‘Object/(x �= x)’. That is,

object that is not self-identical↓

/(x �= x)

In other words, the translation of Lesniewski’s thesis,

(∃a)(a �= a),

is equivalent in our conceptualist logic of names to

(∃A)¬[(∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yA)(x = y)],

which is provable in this logic.Note also that because (a = b) in Lesniewski’s system means (a ε b∧ b ε a),

then the translation of (a = b) into our conceptualist logic becomes

(∀xA)(∀yA)(x = y)∧ (∃xA)(∃yB)(x= y)∧ (∀xB)(∀yB)(x = y)∧ (∃xB)(∃yA)(x = y),

which in effect says that exactly one thing is A and that thing is B, and thatexactly one thing is B and that thing is A, a statement that is true when A andB are proper names, or unshared common names, of the same thing, and falseotherwise, which is exactly how Lesniewski understood the situation.

Now the form of existential generalization that we found odd in Lesniewski’s,namely,

ϕ(c/a) → (∃a)ϕ(a),

is translated into our conceptualist logic as:

ϕ(C/A) → (∃A)ϕ(A),

which, if C is free for A in ϕ, is provable in our simple logic of names, andyet, of course, from this it does not follow, as it does in Lesniewski’s logic, thatsomething is identical with nothing. Our earlier example of this oddity was theconditional

¬(∃b)(b = Pegasus) → (∃a)¬(∃b)(b = a).

Now what the antecedent ¬(∃b)(b = Pegasus) says under our conceptualistinterpretation is that there is no name concept B such that B names exactlyone thing and that thing is Pegasus, which of course is true, given that Pegasusdoes not exist. The consequent, (∃a)¬(∃b)(b = a), on the other hand, saysthat for some name concept A, there is no name concept B such that A namesexactly one thing and that thing is a B. That statement is in fact is true forany name A that names nothing, e.g., where A is the complex common name

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‘object that is not self-identical’. In other words, although the above formulaabout Pegasus seems odd as a formula in first-order logic, what it means underour conceptualist translation is not odd at all, but quite natural.

In regard to the following thesis of Lesniewski’s logic,

a ε b→ a ε a,

note that its translation into our conceptualist logic is,

(∀xA)(∀yA)(x = y)∧(∃xA)(∃yB)(x = y) → (∀xA)(∀yA)(x = y)∧(∃xA)(∃yA)(x = y),

which says that if exactly one thing is an A and that thing is a B, then exactlyone thing is an A and that thing is an A, which is unproblematic. Similarly,the translation, which we will avoid writing out in full here, of Lesniewski’sseemingly odd transitivity thesis,

a ε b ∧ b ε c→ a ε c,

says that if exactly one thing is an A and that thing is a B and that if exactlyone thing is a B—which therefore is the one thing that is an A—then exactlyone thing is A and that thing is C, which again is easily seen to be valid in ourconceptualist logic.

Finally, putting aside Lesniewski’s definition of identity, it is noteworthythat although ‘A = B’, unlike ‘x = y’, is not a well-formed formula of oursimple logic of names, nevertheless, it will be well-formed in our next chapterwhere we will extend the simple logic of names to include a logic of classes asmany. This extension involves a transformation of names as parts of quantifierphrases to objectual terms, i.e., terms that can be substituted for object vari-ables. In this extended framework, as we will see, when A and B are propernames, or unshared common names, of the same thing, then A = B will be trueindependently of Lesniewski’s definition of identity.

10.5 Pragmatic Uses of Proper and CommonNames

The apparent oddities of Lesniewski’s logic of names are the result of treat-ing both proper and common names as if they were “singular terms,” i.e., ex-pressions that can be substituends of object variables and occur as arguments(subjects) of predicates. That, in any case, is how they are understood inLesniewski’s elementary ontology as an applied first-order logic (without iden-tity). Of course, that is also how proper names, but not common names, areusually analyzed in modern logic, a practice we ourselves initially followed inour lecture on tense logic. But then, before the development of free logic whereproper names that denote nothing are allowed, it was sometimes also the prac-tice to transform proper names into monadic predicates. The proper name‘Socrates’, for example, became the monadic predicate ‘Socratizes’, which was

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true of exactly one thing, and the name ‘Pegasus’ became ‘Pegasizes’, which wastrue of nothing. In this way, the statement that Pegasus does not exist couldbe analyzed as saying that nothing Pegasizes.24

Common names, on the other hand, have usually been analyzed as, or reallytransformed into, monadic predicates in modern logic, both before and after thedevelopment of free logic. Of course, in our general framework of conceptual re-alism there are complex monadic predicates that are constructed on the basis ofboth proper and common names. Thus, where A is a name, proper or common,then

[λx(∃yA)(x = y)]

is a monadic predicate, read as ‘x is an A’ when A is a common name, and as‘x is A’ when A is a proper name.

Lesniewski’s logic of names is viewed as odd in modern logic, as we havesaid, because it takes common names to be more like proper names than likemonadic predicates, and in particular it represents them the way that “singularterms” are represented in modern logic. On this view, if common names wereto be put in the same syntactic category as proper names, then it should be bytaking both as monadic predicates.

Now in our conceptualist logic, proper names and common names are in thesame syntactic category, but it is not the category of monadic predicates, noris it the category of “singular terms”. Proper and common names belong to amore general category of names, and as such they are taken as parts of quantifierphrases, i.e., phrases that stand for referential concepts. This is not at all liketaking them as “singular terms,” the way they are in Lesniewski’s logic, though,as we will explain shortly, they can be transformed into “singular terms”, i.e.,terms that can be substituends of objectual variables and occupy the argumentpositions of predicates.

The important point is that unlike the view of names in Lesniewski’s logic,the occurrence of names as parts of quantifier phrases, i.e., of referential expres-sions, is an essential component of how the nexus of predication is understood inconceptual realism. In other words, having a single category of names contain-ing both proper and common names is a basic part of our theory of reference.This does not mean that we cannot distinguish a proper name from a commonname in our logic. In particular, a proper name, when introduced into an ap-plied formal language, brings with it a meaning postulate to the effect that thename can be used to refer to at most one thing. If the language also containstense and modal operators, then not only is it stipulated that a proper namecan be used to refer to at most one thing, but that it must be the same thing atany time in each possible world in which that thing exists. Common names, onthe other hand, are not introduced into an applied formal language with such ameaning postulate.

Now there are other uses of proper and common names as well. Both, forexample, can be used in simple acts of naming, as when a parent teaches a child

24See, e,g,, Quine 1960, p. 179.

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what a dog or a cat is by pointing to the animal and saying ‘dog’, or ‘cat’.25

A simple act of naming is not an assertion and does not involve the exerciseof either a predicable or a referential concept. Also, names, both proper andcommon, can be used in greetings, or in exclamations as when someone shouts‘Wolf!’ or ‘Fire!’, which again are not assertions and do not involve the exerciseof a referential act. A common name such as ‘poison’ is also used as a label,which again is not a referential act. Nor are referential acts involved in the useof name labels that people wear at conferences. These kinds of uses of names,especially proper names or sortal common names, i.e., names that have identitycriteria associated with their use, are conceptually prior to the referential use ofnames in sentences. But the analysis of these kinds of uses as well as the use ofnames in referential acts belongs to the discipline of pragmatics, and not thatof semantics, which deals exclusively with denotation and truth conditions.26

10.6 Classes as Many as the Extensions of Names

Now in addition to these pragmatic uses of names there are also “denotative”uses of names as well, as when we speak of mankind, or humankind, by whichwe mean the totality, or entire group, of humans taken collectively—but not inthe sense of a set or class as an abstract object.27 Thus, we say that Socratesis a member of mankind, as well as that Socrates is a man. Also, instead of‘mankind’, we can use the plural of ‘man’ and say that Socrates is one amongmen. These in fact are transformations of the name ‘man’ into an “objectualterm,” i.e., an expression that can occur as an argument of predicates. But itis not a “singular term” in the sense that it denotes a single entity, e.g., a setor a class as an abstract object.

Instead of using the phrase ‘singular term’, which suggeststhat we are dealing with a “single” entity, a better, or lessmisleading, phrase is ‘objectual term’, which we have usedinstead. An “object” such as mankind is not a “single”entity, but a plural object, i.e., a plurality taken collectively.

The transformation of ‘man’ into ‘mankind’, or ‘human’ into ‘humankind’,and ‘dog’ into ‘dogkind’, etc. is different from the nominalizing transformationof a predicate adjective, such as ‘human’, into an abstract noun—i.e., an ab-stract “singular term”—such as ‘humanity’, or into the gerundive phrase ‘beinghuman’, or into the related infinitive and gerundive phrases ‘to be a man’ and‘being a man’, all of which are represented in our logical syntax by a nominal-ization of the (complex) predicate phrase [λx(∃yMan)(x = y)]( ).

25See, e.g., Geach 1980, p. 52.26Pragmatics and semantics are two of the three principal parts of semiotics. The third is

syntax.27See, e.g., Sellars 1963, p. 253.

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• The transformation of a predicate, e.g., ‘is human’, into an abstract noun,‘humanity’, results in a genuine “singular term”, i.e., a term that purportsto denote a single object, albeit an abstract intensional one.

• But the transformation of ‘man’ into ‘mankind’ or ‘men’, or ‘dog’ into‘dogkind’ or ‘dogs’, does not result in a “nominal” expression that pur-ports to denote a single object; nor does it purport to denote an abstractintensional object.

• What such a noun as ‘mankind’, or ‘dogkind’, or either of the plurals‘men’ and ‘dogs’, purports to denote is a plural object, namely, men, ordogs, taken collectively as a group—but not as a set or a class as a singleobject. The expressions ‘mankind’ and ‘dogkind’, are indeed “nominal”expressions, i.e., nouns, and therefore, logically, they should be representedas “objectual terms,” but not as “singular” terms in the sense of nominalexpressions that denote single objects. What they denote are pluralities,i.e., plural objects.

Now a plural object, such as a group of things, is what Bertrand Russellonce called a class as many, as opposed to a class as one. Russell allowed that aclass as many could consist of just a single object, as when a common name hasjust one object in its extension, in which case the class as many is the same asthat one object. On the other hand, there is no class as many that is empty.28

There is more than one notion of a class, in other words, and in fact thereis even more than one notion of a class in the sense of the iterative conceptof a set, i.e., the concept of a set based on Cantor’s power-set theorem. Theiterative concept of a set can be developed, for example, either with an axiom offoundation or an axiom of anti-foundation.29 But in neither case can there be auniversal set, and yet there are set theories, such as Quine’s NF and the relatedset theory NFU, in which there is a universal set. The notion of a universalclass is part of the traditional notion of a class as the extension of a predicableconcept (Begriffsumfange), and, as we have noted in lecture four, this was howclasses were understood by Frege in his Grundgetsetze. Now our point here isthat classes in all of these senses are single objects, not plural objects, i.e., theyare each a class as one, a single abstract object.

It is not the notion of a class as one, i.e., as a single abstract object, that weare concerned with here, but the notion of a class as many, i.e., of a class as aplurality, or plural object. It is this notion that is implicitly understood as theextension of a common count noun, or what we have been calling a commonname. In the development of our analysis of this notion, we will also take a classas many consisting of just one object as the extension of a nonvacuous propername.

28See Russell 1903, §§69–70.29For a development of set theory with an anti-foundation axiom, see Aczel 1988.

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Membership in a class as many can be defined once names are allowed to be“nominalized” and occur as objectual terms. The definition is as follows:

x ∈ y =df (∃A)[y = A ∧ (∃zA)(x = z)],

where A is a name variable or constant. Note that in this definition, A occursas a “nominalized”, objectual term in the conjunct ‘y = A’ as well as part of thequantifier phrase in the second conjunct ‘(∃zA)(x = z)’. An obvious theoremof the logic of classes as many, which we will develop in the next lecture, is thefollowing,

x ∈ A↔ (∃zA)(x = z),

where ‘x ∈ A’ can be read as ‘x is an A’, or ‘x is one among the A’, or ‘x is amember of the class as many of A’, or simply as ‘x is a member of A’.

Now as understood by Russell, there are three important features of thenotion of a class as many as the extension of a common name. These are:

1. First, that a vacuous common name, i.e., a common name that namesnothing, has no extension, which is not the same as having an empty classas its extension. Thus, according to Russell, “there is no such thing asthe null class, though there are null class-concepts,” i.e., common-nameconcepts that have no extension.30

2. Secondly, the extension of a common name that names just one thing isjust that one thing. In other words, unlike the singleton sets of set theory,which are not identical with their single member, the class that is theextension of a common name that names just one thing is none other thanthat one thing.

3. The second feature is related to our third, namely, that unlike sets, classesas the extensions of names are literally made up of their members so thatwhen they have more than one member they are in some sense pluralities(Vielheiten), or “plural objects,” and not things that can themselves bemembers of classes.

4. Thus, according to Russell, “though terms [i.e., objects] may be said tobelong to ... [a] class, the class [as a plurality] must not be treated as itselfa single logical subject.”31

It is a class as many that is the extension, or denotatum, of a commonname, and, on our analysis, also of a proper name. On our analysis, the logicof classes as many is a direct and natural extension of the simple logic of namesdescribed earlier in this lecture. The idea is that names, both proper andcommon, can be transformed into, or “nominalized” as, objectual terms thatcan be substituends of objectual variables and occur as arguments of predicates.When so transformed, what a name denotes is its extension, which in the case

30Russell 1903, §69.31Russell 1903, §70.

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of a common name with more than one object in its extension is a plural object,which we will also call a group. The extension of a proper name, on the otherhand, is the object, if any, that the name denotes as a “singular” term.

The resulting logic of classes as many is not entirely unlike the analysis givenin Lesniewski’s logic of names, where names occur only as objectual terms. Infact we can even formulate counterparts in this logic to certain of the odditiesof Lesniewski’s logic. But there is also a difference in that the counterparts ofLesniewski’s problematic oddities are refutable, and the counterparts that arenot refutable do not appear as odd but as natural consequences of an ontologywith both single and plural objects.

10.7 Summary and Concluding Remarks

• The conceptualist logic of names is similar to Lesniewski’s logic of namesin that the category of names in Lesniewski’s system contains common as well asproper names, i.e., names can be either shared (common) or unshared (proper),or they can be vacuous.

• Lesniewski’s logic is different, however, in that names do not occur as partsof quantifier phrases but are of the same category as object variables.

• There are a number of oddities that are provable in Lesniewski’s logic ofnames, such as (∃a)(a �= a), which seems to say that some object is not identicalwith itself. Similarly, the law of existential generalization in Lesniewski’s logicseems to have such odd consequences as there being something that is identicalwith nothing.

• Both Lesniewski’s logic of names and the conceptualist logic of namescan be formulated as subsystems of their respective formal ontologies, and assubsystems both are consistent and decidable.

• Lesniewski’s logic of names is interpretable in terms of the conceptualistlogic of names, and the oddities of Lesniewski’s logic are seen as no longerodd when interpreted within the conceptualist logic. It is Lesniewski’s apparenttreatment of names as object terms that accounts for the oddities of Lesniewski’slogic. Names in conceptualism occur primarily as parts of quantifier phrases,which explains their referential role in speech and mental acts.

• But names, both proper and common, can be transformed into objectualterms in conceptual realism, and when so transformed, or “nominalized”, theyare interpreted as denoting classes as many, which are not the same as sets orclasses as ones. Classes as many are the extensions of names.

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Chapter 11

Plurals and the Logic ofClasses as Many

In chapter ten we formalized the simple logic of names that is an important partof the theory of reference in conceptual realism. The category of names, it willbe remembered, includes both proper and common, and complex and simple,names, all of which occur as parts of quantifiers phrases. Quantifier phrases, ofcourse, are what stand for the referential concepts of conceptual realism. Weexplained in that chapter how Lesniewski’s logic of names, which Lesniewskicalled ontology, can be interpreted and reduced to our conceptualist logic ofnames, and how in that reduction we can explain and account for the odditiesof Lesniewski’s logic.

We concluded chapter ten with observations about the “nominalization,” ortransformation, of names as parts of quantifier phrases into objectual terms.What a “nominalized” name denotes as an objectual term, we said, is the ex-tension of that name, i.e., of the concept that the name stands for in its role aspart of a quantifier phrase. The extension of a name is not a set, nor a classas a “single object”. Rather, the extension of a name is a class as many, i.e.,a class as a plurality that is literally made up of its members. We listed threeof the central features of classes as many as originally described by BertrandRussell in his 1903 Principles of Mathematics. These are, first, that a vacuousname—that is, a name that names nothing—has no extension, which is not thesame as having an empty class as its extension. In other words, there is noempty class as many. Secondly, the extension of a name that names just onething is none other than that one thing; that is, a class as many that has justone member is identical with that one member. In other words it is becausea class as many is literally many up of its members that it is nothing if it hasno members; and that is also why it is identical with its one member if it hasjust one member. Finally, that is also why a class as many that has more thanone member is merely a plurality, or plural object, which is to say that as a

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plurality it is not a “single object,” and therefore it cannot itself be a memberof any class as many.

11.1 The Logic of Classes as Many

We begin here where left off in chapter ten, namely, with the logic of classes asmany as an extension of the simple logic of names. We assume in this regardall of the axioms and theorems of the simple logic names given in that lecture.That logic consisted essentially of a free first-order logic of identity extended toinclude the category of names as parts of quantifiers, and where the quantifiers∀ and ∃ can be indexed by name variables as well as objectual variables.1

Now because names can be transformed into objectual terms we need avariable-binding operator that generates complex names the way that the λ-operator generates complex predicates.2 We will use the cap-notation withbrackets, [xA/...x...], for this purpose. Accordingly, where A is a name, properor common, complex or simple, we take [xA] to be a complex name, but one inwhich the variable x is bound. Thus, where A is a name and ϕ is a formula,[xA], [xA/ϕ], and [x/ϕ] are names in which all of the free occurrences of x inA and ϕ are bound. We read these expressions as follows:

[xA] is read as ‘the class (or group) of A’,[xA/ϕ] is read as ‘the class (or group) of A that are ϕ’, and[x/ϕ] is read as ‘the class (or group) of things that are ϕ’.

A formal language L is now understood as a set of predicate and name con-stants, instead of a set of predicates and objectual constants, as was originallydescribed in our lecture on tense and modal logic. There will be objectualconstants in a formal language as well, but they will be generated from thename constants by a “nominalizing” transformation. In our more comprehen-sive framework, which we are not concerned with here, objectual constants arealso generated from predicate constants by the nominalizing transformation de-scribed in our fourth lecture. We extend the simultaneous inductive definitionof names and formulas given in §3 of our previous lecture to include names ofthis complex form as well as follows: If L is a formal language, then:

• (1) Every name variable or name constant in L is a name of L;

• (2) if a, b are either objectual variables, name variables or name constantsin L, or names of L the form [xB], where x is an objectual variable andB is a name (complex or simple) of L, then (a = b) is a formula of L; and

• if ϕ, ψ are formulas of L, B is a name (complex or simple) of L, and x andC are an objectual and a name variable, respectively, then

1Much of the material in this chapter is based on Cocchiarella 2002.2It should be remembered that in free logic being a substituend of free objectual variables—

i.e., being an “objectual term”—is not the same as denoting a value of the bound objectualvariables. That is, in free logic some objectual terms may denote nothing.

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(3) ¬ϕ,(4) (ϕ→ ψ),(5) (∀x)ϕ,(6) (∀xB)ϕ, and(7) (∀C)ϕ are formulas of L, and(8) B/ϕ,(9) /ϕ, and(10) [xB] are names of L.

Note that by definition we now have formulas of the form (∀y[xA])ϕ, aswell as those of the form (∀xA)ϕ and (∀yA(y/x))ϕ as in §3 of our previouslecture. We reduce the first to the last of these forms by adding the followingaxiom schema to those already listed in §3 of our previous lecture, but nowunderstood to apply to our extended notions of name and formula:

Axiom 12: (∀y[xA])ϕ↔ (∀yA(y/x))ϕ, where y does not occur in A.

Because we are retaining the axioms and theorems of §3 of our previous lecture,our first axiom of the logic of classes as many, Axiom 12, begins where we leftoff, the last axiom of which was axiom 11, and the last theorem of which wasT7.

We might note, incidentally, that Axiom 12 is a conversion principle forcomplex names as parts of quantifier phrases. It is the analogue for complexnames of the form [x/A] of λ-conversion for complex predicates of the form[λxϕ].

Given our understanding of the existential quantifier as defined in terms ofnegation and the universal quantifier, this means we also have the following asa theorem (where y is free for x in A):

T8: � (∃y[xA])ϕ↔ (∃yA(y/x))ϕ.

Two other axioms about the occurrence of names as objectual terms are:

Axiom 13: (∃A)(A = [xB]), where B is a name and A is a name variablethat does not occur (free) in B; and

Axiom 14: A = [xA], where A is a simple name, i.e., a namevariable or constant.

Axiom 13, incidentally, is a comprehension principle for complex names,and as such is the analogue for complex names of the comprehension principle(CP∗

λ) for complex predicates. What it says is that every complex name of theform [xB] is a value of the bound name variables, and therefore stands for aname, or nominal, concept. Axiom 14 tells us that the name concept [xA] isnone other than the name concept A.

It is noteworthy that our earlier axiom 4 of §3 is now redundant and canbe derived by Leibniz’s law, (LL∗), from axiom 13. That is, by (LL∗), �C = [xB] → [ϕ → ϕ([xB]/C)], and therefore by (UG), axioms 3, 5, andtautologous transformations, � (∃C)(C = [xB]) → [(∀C)ϕ → ϕ([xB]/C)], andhence, by axiom 13,

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T9: � (∀C)ϕ→ ϕ([xB]/C).

Strictly speaking, T9 is actually slightly stronger than axiom 4 in that itincludes cases in which complex names occur as objectual terms, i.e., wheresome, or all, of the occurrences of C in ϕ may be as objectual terms, and hencewhere not all occurrences of [xB] in ϕ([xB]/C) can be replaced by B if B is acomplex name of the form A/ψ or of the form /ψ. If C occurs in ϕ only as partof a quantifier phrase, then ϕ([xB]/C) is equivalent to ϕ(B/C) by axiom 12.

We turn now to definitions of some of the concepts that are important inthe logic of classes.

Note that although we adopt the same symbols that are used inset theory to express membership, inclusion and proper inclusion, itshould be kept in mind that the present notion of class is not thatof set theory.

Definition 2 x ∈ y ↔ (∃A)[y = A ∧ (∃zA)(x = z)].

Definition 3 x ⊆ y ↔ (∀z)[z ∈ x→ z ∈ y].

Definition 4 x ⊂ y ↔ x ⊆ y ∧ y � x.

Note also that the argument for Russell’s paradox for classes doesnot lead to a contradiction within this system as described so far,nor will it do with the axioms yet to be listed.

Rather, what it shows is that the Russell class as many does not “exist” inthe sense of being the value of a bound objectual variable, which is not to saythat the name concept of the Russell class does not have its own conceptualmode of being as a value of the bound name variables. Indeed, as the followingdefinition indicates, the name, or nominal, concept of the Russell class can bedefined in purely logical terms.

Definition 5 Rus = [x/(∃A)(x = A ∧ x /∈ A)].

That the Russell class as many does not “exist” as an object, i.e., as avalue of the bound objectual variables, is important to note because it has beenclaimed that “the objective view” of plural objects, i.e., the view of them asobjects (such as classes as many), is refuted by Russell’s paradox.3 The factthat the Russell class does not “exist” in the logic of classes as many is statedin the following theorem. (Proofs will be given only as footnotes.)

T10: � ¬(∃x)(x = Rus).4

3See, e.g., Schein 1993, pages 5, 15, and 32-37.4Proof. By axiom 13 and identity logic, � (∃A)(Rus = A), and by definition 1, �

Rus ∈ Rus ↔ (∃A)[Rus = A∧ (∃xA)(x = Rus)], and therefore by Leibniz’s law, a quantifier-confinement law and tautologous transformations, � Rus ∈ Rus ↔ (∃xRus)(x = Rus). Butthen, by definition of Rus and T8, � (∃xRus)(x = Rus) ↔ (∃x/(∃A)(x = A ∧ x /∈ A))(x =Rus), and therefore, by T1, � (∃xRus)(x = Rus) ↔ (∃x)[(∃A)(x = A ∧ x /∈ A) ∧ x = Rus],from which, by Leibniz’s law, it follows that � (∃xRus)(x = Rus) ↔ (∃x)[Rus /∈ Rus ∧x = Rus]; and, accordingly, by quantifier-confinement laws, and tautologous transformations,� (∃x)(x = Rus) → (Rus ∈ Rus ↔ Rus /∈ Rus), from which we conclude that � ¬(∃x)(x =Rus).

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What Russell’s argument shows is that not every name concept has an ex-tension that can be “object”-ified in the sense of being the value of a boundobjectual variable.

Now the question arises as to whether or not we can specify a necessaryand sufficient condition for when a name concept has an extension that canbe “object”-ified, i.e., for when the extension of the concept can be provento “exist” as the value of a bound objectual variable. In fact, the answer isaffirmative. In other words, unlike the situation in set theory, such a conditioncan be specified for the notion of a class as many. An important part of thiscondition is Nelson Goodman’s notion of an “atom,” which, although it wasintended for a strictly nominalistic framework, we can utilize for our purposesand define as follows.5

Definition 6 Atom = [x/¬(∃y)(y ⊂ x)].

This notion of an atom has nothing to do with physical atoms, of course.Rather, it corresponds in our present system approximately to the notion of anurelement, or “individual,” in set theory. We say “approximately” because inour system atoms are identical with their singletons, and hence each atom will bea member of itself. This means that not only are ordinary physical objects atomsin this sense, but so are the propositions and intensional objects denoted bynominalized sentences and predicates in the fuller system of conceptual realism.Of course, the original meaning of ‘atom’ in ancient Greek philosophy was that ofbeing indivisible, which is exactly what was meant by ‘individual’ in medievalLatin. An atom, or individual, in other words, is a “single” object, which isapropos in that objects in our ontology are either single or plural. We willhenceforth use ‘atom’ and ‘individual’ in just this sense.

The following axiom (where y does not occur in A) specifies when and onlywhen a name concept A has an extension that can be “object”-ified (as a valueof the bound objectual variables).

Axiom 15: (∃y)(y = [xA]) ↔ (∃xA)(x = x) ∧ (∀xA)(∃zAtom)(x = z).

Stated informally, axiom 15 says that the extension of a name concept A canbe “object”-ified (as a value of the bound objectual variables) if, and only if,something is an A and every A is an atom.6 An immediate consequence of thisaxiom, and of T8 and T1, is the following theorem schema, which stipulatesexactly when an arbitrary condition ϕx has an extension that can be “object”-ified.

T11: � (∃y)(y = [x/ϕx]) ↔ (∃x)ϕx ∧ (∀x/ϕx)(∃zAtom)(x = z).

Note that where ϕx is the impossible condition (x �= x), it follows from T11that there can be no empty class, which, as already noted, is our first basic

5See Goodman 1956 for Goodman’s account of atoms in nominalism.6That something is an A is perspicuously symbolized by (∃y)(∃xA)(y = x). But because

(∃xA)(x = x) ↔ (∃y)(∃xA)(y = x) is provable, we will use (∃xA)(x = x) as a shorter way ofsaying the same thing.

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feature of the notion of a class as many. We define the empty-class concept asfollows and then note that its extension, by T11, cannot “exist” (as a value ofthe bound objectual variables), as well as that no object can belong to it.

Definition 7 Λ = [x/(x �= x)].

T12a: � ¬(∃x)(x = Λ).

T12b: � ¬(∃x)(x ∈ Λ).

Finally, our last axiom concerns the second basic feature of classes as many;namely, that every atom, or individual, is identical with its singleton. In termsof a name concept A, the axiom stipulates that if at most one thing is an Aand that whatever is an A is an atom, then whatever is an A is identical tothe extension of A, which in that case is a singleton if in fact anything is an A.Where y does not occur in A, the axiom is as follows.

Axiom 16: (∀xA)(∀yA)(x = y) ∧ (∀xA)(∃zAtom)(x = z) →(∀yA)(y = [xA]).

A more explicit statement of the thesis that an atom is identical with itssingleton is given in the following theorem.

T13: � (∃zAtom)(x = z) → x = [y/(y = x)].7

By T13, it follows that every atom is identical with the extension of somename concept, e.g., the concept of being that atom. Of course, non-atoms, i.e.,plural objects, are the extensions of name concepts as well (by the definitionsof Atom, ⊂, and ∈), and hence anything whatsoever is the extension of a nameconcept.

T14: � (∃zAtom)(x = z) → (∃A)(x = A).

T15: � ¬(∃zAtom)(x = z) → (∃A)(x = A).

T16: � (∃A)(x = A).

Note that if A is a proper name of an ordinary, physical object (and hencean atom), then, by the meaning postulate for proper names, the antecedent ofaxiom 16 is true, and therefore, by axioms 16 and 14, (∀yA)(y = A). Inother words, if A is a proper name of an atom, then F (A) ↔ (∀yA)F (y) is true,which in our conceptualist framework explains the role proper names have as“singular terms” (i.e., as substituends of free objectual variables) in free logic.That is, by Leibniz’s law, (UG), axioms 2 and 6, T4, a quantifier-confinementlaw,

(∀yA)(y = A) � F (A) ↔ (∀yA)F (y).

7Proof. Where A be the nominal concept thing-that-is-identical-to x, i.e. /(y = x), then,by axiom 11 and (LL∗), � (∀y/y = x)(∀w/w = x)(y = w), and, similarly, � (∃zAtom)(x =z) → (∀y/y = x)(∃zAtom)(y = z). Therefore, by axiom 16, � (∃zAtom)(x = z) → (∀y/y =x)(y = [y/(y = x]). But, by T6, � (∃zAtom)(x = z) → (∃z)(z = x), and therefore by (∃/UI),T3 and axiom 8, � (∃zAtom)(x = z) → x = [y/(y = x].

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Of course, if A is a non-vacuous proper name of an ordinary object, then(∃yA)(y = A) is true as well, and hence F (A) ↔ (∃yA)F (y) as true as well.That is,

(∃yA)(y = A) � F (A) ↔ (∃yA)F (y).

What these last results indicate is that the role proper names haveas “singular terms,”— i.e., as substituends of free objectual variablesthat purport to denote a “single” object — in standard free logic isreducible to, and fully explainable in terms of, the role proper nameshave in our logic of classes as many.

A consequence of T13, the definition of ∈, T8, and Leibniz’s law is thethesis that every atom is a member of itself. A similar argument, but withoutT13, shows that every object is a member of its singleton, which does not meanthat every “real” object, i.e., every value of the bound objectual variables, isidentical with its singleton.8

T17: � (∀xAtom)(x ∈ x).

T18: � (∀x)(x ∈ [z/(z = x)]).

Finally, we note that by definition of membership and Leibniz’s law an objectx belongs to the extension of a name concept A if, and only if, x is an A. Fromthis it follows that only atoms can belong to an “ object”-ified class as many,and hence that classes as many that are not atoms are not themselves membersof any (real) classes as many, which is our third basic feature of classes as many.

T19: � x ∈ A↔ (∃yA)(x = y).

T20a: � (∀x)[z ∈ x→ (∃wAtom)(z = w)].

T20b: � ¬(∃wAtom)(z = w) → ¬(∃x)(z ∈ x).9

11.2 Extensional Identity

The “nominalist’s dictum,” according to Nelson Goodman, is that “no two dis-tinct things can have the same atoms.”10 Such a dictum, it would seem, shouldapply to classes as many as traditionally understood, regardless whether or nota more comprehensive framework containing such classes is nominalistic or not.

8In other words, if x is a “real” class as many with more than one member, then x �=[z/(z = x)], even though x ∈ [z/(z = x)]. The latter, like x′s being a member of the universalclass, means only that x is identical with itself.

9Proof. By definition of ∈, � z ∈ x → (∃A)[x = A ∧ (∃wA)(z = w)], and therefore, byT6, � z ∈ x → (∃w)(z = w). By axiom 15, � (∃y)(A = y) → (∀zA)(∃wAtom)(z = w);and therefore, by axiom 10, T19, and (LL∗), � (∃y)(x = y) ∧ x = A → (∀z)[z ∈ A →(∃wAtom)(z = w)]. But then, by quantifier-confinement laws, T16, (LL∗), (∃/UI) andelementary transformations, � (∃y)(x = y) → [z ∈ x → (∃wAtom)(z = w)]. Therefore, by(UG) and axiom 7, � (∀x)[z ∈ x → (∃wAtom)(z = w)], which is T20a. T20b then followsby a quantifier-confinement law and tautologous transformations.

10Goodman 1956, p. 21.

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In fact, the dictum is provable here if we assume an axiom of extensionality forclasses.

But there is a problem with the axiom of extensionality. In particular, if thefull unrestricted version of Leibniz’s law is not modified, then having an axiomof extensionality would seem to commit us to a strictly extensional frameworkeven if it is not otherwise nominalistic. Name concepts that have the sameextension at a given moment in a given possible world would then, by Leibniz’slaw, be necessarily equivalent, and therefore have the same extension at all timesin every possible world, which is counter-intuitive. It is hoped, for example, thatthe extension of a common name concept such as ‘country that is democratic’can have more and more members in it over time. Common name concepts ofanimals, e.g., ‘buffalo’, certainly have different extensions over time. Some, as inthe case of names of plants and animals that have become extinct have changedtheir extensions radically from having millions of members to now having none.The idea that common name concepts cannot have different extensions overtime, no less in different possible worlds, is a consequence we do not want inour broader framework of conceptual realism.

On the other hand, classes as many are extensional objects, and an axiomof extensionality that applies at least to classes as many is the only naturalassumption to make in an ontology with classes as many, as in fact ours is. Allobjects, in other words, whether they are single or plural, are classes as many,and the idea that classes as many are not identical when they have the samemembers is difficult to reconcile with such an ontology.

Fortunately, there is an alternative, namely, that the full version of Leibniz’slaw as it applies to all contexts is to be restricted to atoms, i.e., single objects,or individuals in the ontological sense. The restricted version for extensionalcontexts can then still be applied to pluralities, i.e., classes as many that havemore than one member. Thus, in addition to the axiom of extensionality, wewill take the following as an new axiom schema of our general framework.

Axiom 17: (∃zAtom)(x = z) ∧ (∃zAtom)(y = z) → [x = y → (ϕ↔ ψ)],where ψ is obtained from ϕ by replacing one or morefree occurrences of x by free occurrences of y.

This axiom is redundant if we do not add any nonextensional contexts, e.g.,tense or modal operators, to the logic of classes as many. The reason is because,in a strictly extensional language, the full, unrestricted version of Leibniz’s lawis derivable from axiom 9.11 In other words, Axiom 9 remains in effect, butall we can prove from it is that Leibniz’s law holds for all extensional contexts.

This distinction between how Leibniz’s law applies to atoms and howit applies to classes as many in general is an ontological feature ofour logic in that it distinguishes the individuality of atoms from theplurality of groups.

11As given in §3 of our previous lecture, Axiom 9 is restricted to atomic formulas. The un-restricted version is then derivable by induction over the formulas of an extensional language.See, e.g., the proof given of (LL) in our second lecture.

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11.2. EXTENSIONAL IDENTITY 243

Indeed, unlike atoms, or individuals in the strict ontological sense, the iden-tity of groups, or pluralities, i.e., classes as many with more than one member,essentially reduces to the fact that they are made up of the same members,which does not justify the full, unrestricted ontological content of Leibniz’s law.

A related point about Axiom 17 is that it is an ontological thesis aboutthe values of object variables and not about the objectual terms, e.g., nameconstants, that might be substituted for object variables. The validity of in-stantiating objectual terms for the object variables in this axiom depends onthe contexts in which those variables occur and how “rigid” those objectualterms are with respect to those contexts. Proper names are assumed to berigid with respect to tense and modal contexts, for example, but not in generalwith respect to belief and other cognitive-modal contexts except under specialassumptions, such as knowing who, or what, the terms denote.

We now include the axiom of extensionality, which we will refer to hereafteras (ext), among the axioms.

Axiom 18 (ext): (∀z)[z ∈ x↔ z ∈ y] → x = y.

Goodman’s nominalistic dictum that things are identical if they have thesame atoms is now provable as the following theorem.

T21: � (∀x)(∀y)[(∀zAtom)(z ∈ x↔ z ∈ y) → x = y].12

Note that by T13 and the definition of ∈, whatever belongs to an atom isidentical with that atom, and therefore atoms are identical if, and only if, theythe have the same members.

T22: � (∀xAtom)[y ∈ x→ y = x].

T23: � (∀xAtom)(∀yAtom)[x = y ↔ (∀z)(z ∈ x↔ z ∈ y)].

Note also that by T21 (and other theorems) it follows that everything “real”,whether it is an atom or not, has an atom in it.T24: � (∀x)(∃zAtom)(z ∈ x).13

Another useful theorem is the following, which, together with T21, showsthat every non-atom must have at least two atoms as members. Of course,

12Proof. By T5, T20a, (UG), quantifier-confinement laws, and elementary transfor-mations, � (∀x)[(∀zAtom)(z ∈ x ↔ z ∈ y) → (∀z)(z ∈ x → z ∈ y)], and similarly� (∀y)[(∀zAtom)(z ∈ x ↔ z ∈ y) → (∀z)(z ∈ y → z ∈ x)], from which, given (ext),T21 follows.

13Proof. By T5 and T17, � (∃zAtom)(x = z) → (∃zAtom)(z ∈ x), and hence, bycontraposition and the definition of Atom, � ¬(∃zAtom)(z ∈ x) → ¬(∃z[x¬(∃y)(y ⊂ x)])(x =z); and therefore, by axioms 12, 11 and elementary transformations, � ¬(∃zAtom)(z ∈x) → (∀z)[x = z → (∃y)(y ⊂ z)], from which, by (LL∗) and a quantifier-confinement law, itfollows that � ¬(∃zAtom)(z ∈ x) → [(∃z)(x = z) → (∃y)(y ⊂ x)]; and therefore, by (UG)and axioms 2 and 7, � (∀x)[¬(∃zAtom)(z ∈ x) → (∃y)(y ⊂ x)]. Now, by definition of ⊂, �¬(∃zAtom)(z ∈ x)∧y ⊂ x→ ¬(∃zAtom)(z ∈ y), and therefore � ¬(∃zAtom)(z ∈ x)∧y ⊂ x→(∀zAtom)[z ∈ x ↔ z ∈ y], and, accordingly by (UG) and T21, � (∀x)(∀y)[¬(∃zAtom)(z ∈x) ∧ y ⊂ x → x =ex y]. But then, by definition of ⊂, � (∀x)(∀y)[¬(∃zAtom)(z ∈ x) →(y ⊂ x → x ⊆ y ∧ x � y)]; and therefore, by quantifier logic, � (∀x)[¬(∃zAtom)(z ∈ x) →¬(∃y)(y ⊂ x)]. Together with the above result, this shows that � (∀x)[¬(∃zAtom)(z ∈ x) →(∃y)(y ⊂ x) ∧ ¬(∃y)(y ⊂ x)], from which T24 follows by quantifier logic.

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conversely, any “real” class (as many) that has at least two members cannotbe an atom, because then each of those members is properly contained in thatclass.T25: � (∀x)(∀y)(y ⊂ x→ (∃zAtom)[z ∈ x ∧ z /∈ y]).14

T26: � (∀x)[¬(∃yAtom)(x = y) ↔ (∃z1/z1 ∈ x)(∃z2/z2 ∈ x)(z1 �= z2)].15

Two consequences of the extensionality axiom, (ext), are the strict identityof a class with the class of it members and the rewrite of bound variables forclass expressions.

T27a: � x = [z/(z ∈ x)].

T27b: � [xA] = [yA(y/x)], where y does not occur in A.16

11.3 The Universal Class

We have seen that, unlike the situation in set theory, the empty class as manydoes not “exist” (as a value of the bound objectual variables). But what aboutthe universal class? In ZF, Zermelo-Frankel set theory, there is no universal set,but in Quine’s set theory NF (New Foundations) and the related set theory,NFU (New Foundations with Urelements), there is a universal set. In ourpresent theory, the situation is more complicated. For example, if nothing exists,then of course the universal class does not exist. But, in addition, becausesomething exists only if an atom does, i.e., by T24 and (∃/UI),

T28: � (∃x)(x = x) → (∃xAtom)(x = x),

it follows that the universal class does not exist if there are no atoms, i.e.,individuals—which is unlike the situation in set theory where classes existwhether or not there are any urelements, i.e., individuals. As it turns out,we can also show that the universal class does not exist if there are at least twoatoms. If there is just one atom, however, the situation is more problematic.

14Proof. By quantifier logic and definition of ⊂, � y ⊂ x→ (∀zAtom)(z ∈ y → z ∈ x), andtherefore, by (UG) and T21, � (∀x)(∀y)(y ⊂ x → [(∀zAtom)(z ∈ x → z ∈ y) → x =ex y]).But then, by definition of ⊂ and =ex, � (∀x)(∀y)(y ⊂ x → [(∀zAtom)(z ∈ x → z ∈ y) → x ⊆y ∧ x � y]), and hence � (∀x)(∀y)(y ⊂ x→ (∃zAtom)[z ∈ x ∧ z /∈ y]).

15Proof. By T25, � (∀x)(∀y)(y ⊂ x → (∃z1Atom)[z1 ∈ x ∧ z1 /∈ y]), and by T24and (∃/UI), � (∃w)(y = w) → (∃z2Atom)(z2 ∈ y). But, by (LL∗) and definition of ⊂,� y ⊂ x ∧ z1 /∈ y ∧ z2 ∈ y → z2 ∈ x ∧ z1 �= z2, and therefore, by quantifier logic, � (∃w)(y =w) → (∀x)[y ⊂ x→ (∃z1Atom)(∃z2Atom)(z1 �= z2∧z1 ∈ x∧z2 ∈ x)]. Accordingly, by (UG),axiom 7, T1 and quantifier logic, � (∀x)[(∃y)(y ⊂ x) → (∃z1Atom/z1 ∈ x)(∃z2Atom/z2 ∈x))(z1 �= z2)]. But, by quantifier logic and definition of Atom, � (∀x)[¬(∃yAtom)(x = y) →(∃y)(y ⊂ x)], from which the left-right-direction of T26 follows. The converse direction is ofcourse trivial for the reason already noted.

16Proof. By (∃/UI), T2, and (LL∗), � (∃y)(z = y) → [z ∈ x → (∃y/y ∈ x)(z = y)], andtherefore, by T8 and T19, � (∃y)(z = y) → (z ∈ x→ z ∈ [z/(z ∈ x)]), and hence, by axiom7, � (∀z)(z ∈ x → z ∈ [z/(z ∈ x)]). For the converse direction, by T19, (LL∗), and T8,� z ∈ [z/(z ∈ x)] → (∃y/y ∈ x)(z = y); and hence � z ∈ [z/(z ∈ x)] → z ∈ x. Therefore, by(UG), � x = [z/(z ∈ x)]. The proof that � [xA] =ex [yA(y/x)] follows from the definition of∈ and the rewrite rule for relative quantifiers, and T27b then follows by (ext).

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First, let us define the universal class in the usual way, i.e., as the extension ofthe common name ‘thing that is self-identical’, and then note that whether or notthe name concept thing-that-is-self-identical, i.e., [x/(x = x)], can be “object”-ified (as a value of the bound objectual variables), nevertheless, everything“real” (in the sense of being the value of a bound objectual variable) is in it.

Definition 8 V = [x/(x = x)].

T29: � (∀x)(x ∈ V).17

Note: all that T29 really says is that everything is a thing that isself-identical.

Now, by definition of ∈, nothing can belong to the empty class, i.e., x /∈ Λ,and therefore, by Leibniz’s law, if anything at all exists, the universal class isnot the empty class.

T30: � (∃x)(x = x) → V �= Λ.

But it does not follow that the universal class “exists” if anything does. Indeed,as already noted above, we can show that if there are at least two atoms, thenthe universal class does not exist. First, let us note that if something exists(and hence, by T28, there is an atom), then the class of atoms exists, i.e., thenthe name concept Atom can be “object”-ified as a value of the bound objectualvariables.

T31: � (∃x)(x = x) → (∃y)(y = Atom).18

On the other hand, let us also note that

if there are at least two atoms, then the class of atoms is not itselfan atom.

T32: � (∃xAtom)(∃yAtom)(x �= y) → ¬(∃zAtom)(z = Atom).19

By means of T32, we can now show that

if there are at least two atoms, then the universal class does not“exist” (as a value of the objectual variables).

17Proof. By axiom 8, � (∀x)(∃y)(x = y) ↔ (∀x)(∃y)(y = y ∧ x = y), and therefore, byT2, � (∀x)(∃y)(x = y) ↔ (∀x)(∃y/y = y)(x = y), from which T29 follows by T8, T19 andthe definition of V .

18Proof. By axiom 15, � (∃xAtom)(x = x) ∧ (∀xAtom)(∃yAtom)(x = y) → (∃y)(y =Atom), from which, by T28 and quantifier logic, T31 follows.

19Proof. By definition of ∈, T8, and elementary transformations, � x �= y → x /∈ [z/(z =y)] ∧ y /∈ [z/(z = x)], and therefore, by T13 and (LL∗), � (∃zAtom)(x = z) ∧ (∃zAtom)(y =z) ∧ (x �= y) → x /∈ y ∧ y /∈ x. By T20a, � (∃zAtom)(x = z) → x ⊆ Atom, and, byT19, � (∃zAtom)(y = z) → y ∈ Atom. Therefore, by definition of ⊂, � (∃zAtom)(x = z) ∧(∃zAtom)(y = z)∧y /∈ x → x ⊂ Atom, and hence � (∃zAtom)(x = z)∧(∃zAtom)(y = z)∧(x �=y) → x ⊂ Atom. But, by definition of Atom, � (∀x)(∀y)[x ⊂ y → ¬(∃zAtom)(z = y)], andhence, by T31, T6, and (∃/UI), � (∃zAtom)(x = z)∧ x ⊂ Atom → ¬(∃zAtom)(z = Atom).Therefore, � (∃xAtom)(∃yAtom)(x �= y) → ¬(∃zAtom)(z = Atom).

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T33: � (∃xAtom)(∃yAtom)(x �= y) → ¬(∃x)(x = V).20

Finally, in regard to the question of whether or not the universal class existsif the universe consists of just one atom, i.e., just one individual, note thatif that were in fact the case, then, where A is a name of that one atom, theconjunction (∃zAtom)(z = A)∧ (∀zAtom)(z = A) would be true, and thereforethe one atom A would be extensionally identical with the class of atoms, i.e.,then, by T31, T21, and (ext), (A = Atom) would be true as well. Now, byT29 and T19, (∀zAtom)[z ∈ Atom ↔ z ∈ V] is provable, which, by T21might suggest that (Atom = V) and hence (A = V) are true as well. But inorder for T21 to apply in this case we need to know that V “exists,” i.e., that(∃x)(x = V) is true. So, even if there were just one atom, we still could notconclude that the universal class is extensionally identical with that one atom.

11.4 Intersection, Union, and Complementation

Let us turn now to the Boolean operations of intersection, union and comple-mentation for classes as many. We adopt the following standard definitions ofeach.

Definition 9 x ∪ y = [z/z ∈ x ∨ z ∈ y].

Definition 10 x ∩ y = [z/(z ∈ x ∧ z ∈ y)].

Definition 11 x = [z/z /∈ x].

The following theorems regarding membership in the union and intersectionof classes are consequences of T19 and T8. The proof of the theorem regardingmembership in the complement of a class is slightly more involved.T34: � (∀z)(z ∈ x ∪ y ↔ z ∈ x ∨ z ∈ y).

T35: � (∀z)(z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y).

T36: � (∀z)(z ∈ x↔ z /∈ x).21

Two immediate consequences of T36 and (ext) (together with T12b andT29) are that the empty class is identical with the complement of the universalclass, and that the universal class is identical with the complement of the emptyclass.

T37: � Λ = V.20Proof. Note that by T20a and (∃/UI), � (∃x)(x = V ) → (∀x)[x ∈ V → (∃yAtom)(x =

y)]. But, by axiom 8, (UG), and axioms 2 and 6, � (∃x)(x = V ) → (∃x)(x = x), and hence,by T31 and (∃/UI), � (∃x)(x = V ) → [Atom ∈ V → (∃yAtom)(y = Atom)]. But, by T31,T29, and (∃/UI), � (∃x)(x = V ) → Atom ∈ V , and hence, � (∃x)(x = V ) → (∃yAtom)(y =Atom). Accordingly, by T32, � (∃xAtom)(∃yAtom)(x �= y) → ¬(∃x)(x = V ).

21Proof. By definition of ∈, � z ∈ x ↔ (∃A)[x = A ∧ (∃yA)(z = y)], and therefore, by(LL∗) and T8, � z ∈ x → (∃y/y /∈ x)(z = y), and hence, by T2 and (LL∗), � z ∈ x→ z /∈ x.For the converse direction, note that by T2 and (LL∗), � (∃y)(z = y) ∧ z /∈ x → (∃y/y /∈x)(z = y), and therefore, by the definitions of ∈ and x, � (∃y)(z = y) → [z /∈ x → z ∈ x], andhence by (UG) axioms 2 and 7, and elementary logic, � (∀z)(z /∈ x → z ∈ x).

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T38: � V = Λ.

In regard to the conditions for the existence of unions and intersections, wefirst prove a theorem that is useful in their respective proofs.

T39: � (∀x)[(∃z)(z ∈ x) ∧ (∀z/z ∈ x)(∃wAtom)(z = w)].22

T40: � (∀x)(∀y)(∃z)(z = x ∪ y).23

The related theorem for intersection requires a qualification, because someintersections—e.g., of distinct atoms—are empty, and, the empty class as manydoes not exist. Clearly, the relevant qualification is that the classes being inter-sected have a member in common.

T41: � (∀x)(∀y)[(∃z)(z ∈ x ∧ z ∈ y) → (∃z)(z = x ∩ y)].24

In regard to the existence of the complement of a class as many, we firstnote that if some atom is not in x, and therefore, by T36, is in x, then theclass as many of atoms in x exists, i.e., then [zAtom/(z ∈ x)] exists (as a valueof the bound objectual variables). This result cannot be shown for x alone,however, because, e.g., where x = Λ, then, by T38, x = V, in which case xdoes not exist, or at least not if there exist two or more atoms. Also, in thatcase [zAtom/(z ∈ x)] = Atom, and therefore, by T28, [zAtom/(z ∈ x)] existseven though x does not.

T42: � (∃zAtom)(z /∈ x) → (∃y)(y = [zAtom/(z ∈ x)]).25

Note that we can show that an atom is in [zAtom/(z ∈ x)] if, and only if,it is in x, but we cannot use this result (T43 below) to prove that x existsif [zAtom/(z ∈ x)] exists. In particular, we cannot use T21 to prove x =[zAtom/(z ∈ x)] unless we already know that both classes exist. The followingtheorems indicate what does in fact hold about the complement of a given classx.

T43: � (∀zAtom)(z ∈ [zAtom/z /∈ x] ↔ z ∈ x).26

22Proof. By T6, (∃/UI), (LL∗), and T17, � (∃zAtom)(x = z) → (∃z)(z ∈ x), and, byT26, � (∀x)[¬(∃zAtom)(x = z) → (∃z)(z ∈ x)]; hence, � (∀x)(∃z)(z ∈ x). But then T39follows by T20a and quantifier logic.

23Proof. By T39 (twice), � (∀x)[(∃z)(z ∈ x) ∧ (∀z/z ∈ x)(∃wAtom)(z = w)] and� (∀y)[(∃z)(z ∈ y) ∧ (∀z/z ∈ y)(∃wAtom)(z = w)], and therefore, by quantifier logic,� (∀x)(∀y)[(∃z)(z ∈ x ∨ z ∈ y) ∧ (∀z/z ∈ x ∨ z ∈ y)(∃wAtom)(z = w)]. Accordingly, byT11, � (∀x)(∀y)(∃z1)(z1 = [z/(z ∈ x ∨ z ∈ y)]), from which T40 follows by definition ofunion.

24Proof. By T39 (twice) and elementary logic, � (∀x)(∀y)(∀z/z ∈ x∧z ∈ y)(∃wAtom)(z =w)], and therefore, by T11 and the definition of ∩, � (∀x)(∀y)[(∃z/z ∈ x∧ z ∈ y) → (∃z)(z =x ∩ y)].

25Proof. By axiom 15, � (∃zAtom)(z ∈ x) ∧ (∀zAtom/z ∈ x)(∃wAtom)(z = w) →(∃y)(y = [zAtom/(z ∈ x)]); but, by axiom 11 and quantifier logic, � (∀zAtom/z ∈x)(∃wAtom)(z = w), and therefore, by T36, � (∃zAtom)(z /∈ x) → (∃y)(y = [zAtom/(z ∈x)]).

26Proof. By T19, � (∃yAtom/y /∈ x)(z = y) → z ∈ [yAtom/y /∈ x]; and, by T19 andT36, � (∀zAtom)(z ∈ [yAtom/y /∈ x] → z ∈ x). For the converse direction, by T36 and T2,� (∀zAtom)[z ∈ x → (∃yAtom/y /∈ x)(z = y)], and therefore, by T19, � (∀zAtom)(z ∈ x →z ∈ [yAtom/y /∈ x]).

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T44: � (∃y)(y = [zAtom/z /∈ x]) ∧ (∃y)(y = x) → [zAtom/z /∈ x] = x.27

Finally, let us consider the situation when there are a countably infinitenumber of individuals, i.e., single objects, in the world. Does it then follow byCantor’s power-set theorem that there are an uncountably infinite number ofgroups of individuals, i.e., of classes as many of more than one member? Andif so, wouldn’t that show that there is more to the notion of a group than thatof a simple plurality? That is, if out of ℵ0 many individuals we can obtain2ℵ0 many groups of individuals, then doesn’t that show that there is somethingabstract about groups? Perhaps. But curiously, Cantor’s proof is not provablein the logic of classes as many when applied to the class of individuals, whichsuggest that there is nothing abstract about plural objects after all. Indeed, asthe extensions of concepts that proper and common-name concepts stand for,it would be surprising if their cardinality were to exceed that of the conceptswhose extensions they are.28

We should note that a set-theoretic semantics has been constructed for thelogic of classes as many, and with respect to that semantics it has been shownthat the logic is consistent.29

Metatheorem: The logic of classes as many as described here is consistent.

11.5 Lesniewskian Theses Revisited

As we explained in our previous lecture, Lesniewski’s logic of names is reducibleto our conceptualist logic of names. On our interpretation, the oddities ofLesniewski’s logic are seen to be a result of his representing names, both properand common, the way singular terms are represented in modern logic. Theproblem was not his view that proper and common names constitute togethera syntactic category of their own, because that is how names are viewed in ourconceptualist logic as well. But in our conceptualist logic proper and commonnames function as parts of quantifier phrases, i.e., expressions that stand forreferential concepts in our analysis of the nexus of predication.

But if Lesniewski’s logic of names is reducible to our conceptualist logic ofnames, then might not the oddities that arise in Lesniewski’s logic also arise

27Proof. By T43, T21, (∃/UI), and (ext).28Where the number of individuals is finite and greater than 1, i.e., where there are n many

single objects in the universe, for some positive integer n > 1, then, by a simple inductiveargument it can be shown that the number of classes as many of objects, single and plural,is 2n − 1, and hence that there are 2n − (n + 1) plural objects. Of course 2n − 1 > n, wheren > 1.

But where the number of single objects is ℵ0, we cannot show that the number of classesas many is 2ℵ0 − 1 (which is just 2ℵ0 ). The attempt to derive a contradiction by Cantor’sargument of assuming a 1-1 mapping f of all of the classes as many of individuals into the classof all individuals fails because the Cantor diagonal class of individuals x such that x /∈ f(x)must be known to exist (or equivalently have a member) in order to derive a contradiction. Ifit exists, then a contradiction follows, and what this shows is that the Cantor class as many,like the Russell class as many, doesn’t exist in the logic of classes as many (which is “free” ofexistential presuppositions).

29See Appendix 1.

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11.5. LESNIEWSKIAN THESES REVISITED 249

when names as parts of quantifier phrases are “nominalized” and occur as ob-jectual terms in the logic of classes as many the way they occur in Lesniewski’slogic of names? In other words, to what extent, if any, are there any theorems inour logic of classes as many that are counterparts of the theses of Lesniewski’slogic that struck us as odd or noteworthy? Here, by a counterpart we mean aformula that results by replacing the names in a thesis of Lesniewski’s logic bythe “nominalized”, or transformed, names of our logic of classes as many, andalso, of course, replacing Lesniewski’s epsilon ‘ε’ by our epsilon ‘∈’.

First, let us consider the validity of the principle of existential generalizationin Lesniewski’s logic, i.e.,

ϕ(c/a) → (∃a)ϕ(a).

This principle is odd, we noted, when a is a vacuous name such as ‘Pegasus’,because in that case it follows from the fact that nothing is identical with Pega-sus that something is identical with nothing, which is absurd. The counterpartof this thesis in our logic of classes as many is clearly invalid. For example, theempty class as many does not “exist” in our logic, but from that it does not fol-low that something exists that does not exist. Indeed, it is actually disprovable,as it should be. That is, the negation of

¬(∃x)(x = Λ) → (∃y)¬(∃x)(x = y)

is provable in our logic of classes as many.Another thesis of Lesniewski’s logic that is odd is the following:

(∃a)(a �= a).

Now, by (UG) and axiom 8, the negation of this thesis, namely (∀x)(x = x),is a theorem in our logic of classes as many. But of course stating the matterthis way assumes that identity in Lesniewski’s logic means identity simpliciter,which it doesn’t. Identity is defined in Lesniewski’s logic, in other words, andwhat the above thesis really means on Lesniewski’s definition is the following:

(∃a)¬(a ε a).

Now the real counterpart of this thesis in our logic of classes as many is:

(∃x)¬(x ∈ x).

By quantifier negation, what this formula says is that not every object belongsto itself, which because all atoms belong to themselves, means that not everyobject is an atom. That is not a theorem of our logic, but it would be true if infact there were at least two atoms, in which case there would then be a group,i.e., a class as many with more than one member, which, by definition, wouldnot be an atom, and therefore, by T20b, not a member of anything, no less ofitself. Thus, although the counterpart of the above Lesniewskian thesis is not atheorem, nevertheless it is not disprovable, and in fact it is true if there are atleast two atoms, i.e., individuals in the ontological sense.

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In regard to the Lesniewskian thesis,

a ε b→ a ε a.

we note first that the counterpart of this formula, namely,

z ∈ x→ z ∈ z,

is refutable if there is at least one plural object, i.e., one “real” object that isnot an atom. This is because every “real” object is a member of the universalclass (by T29 ), even though the universal class itself is not “real” if there are atleast two atoms (T33). In other words, where z is a plural object, e.g., the classas many of citizens of Italy, then even though z is a member of the universalclass, i.e., z ∈ V, nevertheless z /∈ z. That is, because z is a plural object, itis not an atom, and therefore (by T20b) z is not a member anything. Here, itshould be kept in mind that even though V is not a value of the bound objectualvariables, it is nevertheless a substituend of the free objectual variables. Hence,where z is a plural object, the following instance of the above formula,

z ∈ V → z ∈ z

is false.There is a theorem that is somewhat similar to the above counterpart of

Lesniewski’s thesis, namely,

� (∃x)(z ∈ x) → z ∈ z.

In other words, if z belongs to something “real”, i.e., a value of the boundobjectual variables, then z is an atom (by T20a) and therefore z belongs toitself (by T17). This theorem is similar to, but still not the same as, theLesniewskian thesis.

Another theorem that is similar to, but not the same as, a thesis of Lesniewski’slogic, is:

� (∀y)(∀z)[x ∈ y ∧ y ∈ z → x ∈ z].

This formula is provable because if x belongs to a “real” object y and y belongsto a “real” object z, then both x and y must be atoms (by T20a), in whichcase, y = [w/w = y] (by T13); and hence x = y (because x ∈ y), and thereforex ∈ z (because y ∈ z). This theorem is similar to the Lesniewskian thesis,

a ε b ∧ b ε c→ a ε c,

but, again, the strict counterpart of this Lesniewskian thesis, namely,

x ∈ y ∧ y ∈ z → x ∈ z

is not provable in our logic, and is refutable if there are at least two atoms.Thus, if there are two “real” atoms a and b, then y = [w/(w = a ∨ w = b)] isalso “real” (by axiom 15, or T40). But then y ∈ [w/(w = y)] (by T18, eventhough y �= [w/(w = y)]), and hence we would have a ∈ y and y ∈ [w/(w = y)],and yet a /∈ [w/(w = y)], because a �= b, and hence a �= y.30

30It should be kept in mind that only real atoms are identical with their singletons.

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11.6 Groups and the Semantics of Plurals

One way in which the notion of a group is important is its use in determiningthe truth conditions of sentences that are irreducibly plural, i.e., sentences notlogically equivalent to sentences that can be expressed without a plural referenceto a group or plural predication about a group. An example of such a sentenceis the so-called Geach-Kaplan sentence, ‘Some critics admire only each other’,or, equivalently, ‘Some critics are such that each of them admires only others ofthem’.

Now the plural reference in this sentence is not just plural but irreduciblyplural, and it cannot be logically analyzed by quantifying just over critics. Thereference in this case is really to a group of critics, i.e., a class as many of criticshaving more than one member. The reference, moreover, is not to a set ofcritics, i.e., to an abstract object that is not itself a part of the physical world,but to a group of critics that is no less a part of the physical world than arethe critics in the group. The difference between the group and its members isthat the group, as a plural object, is ontologically founded upon its members assingle objects.

The reference, moreover, is not to just any class as many of critics, and inparticular not to any class as many that consists of just one member. A singlecritic who admires no one would in effect be a class as many of critics havingexactly one member, and every member of this class would vacuously satisfiesthe condition that he admires only other members of the class. But it is counter-intuitive to claim that the sentence ‘Some critics admire only each other’ couldbe true only because there is a critic who admires no one. The sentence is trueif, and only if, there is a group of critics every member of which admires onlyother members of the group.

In order to formulate this sentence properly we need first to define the notionof a group, and then note that, by definition, a group will have at least twomembers, and hence that a group is a plural object.

Definition 12 Grp = [x/(∃y)(y ⊂ x)].

T45: � (∀xGrp)(∃z1/z1 ∈ x)(∃z2/z2 ∈ x)(z1 �= z2).We can now represent the semantics of the sentence ‘Some critics admire

only each other’ in terms of a group of critics instead of just a class as many ofcritics. This can be formulated as follows:

[Some critics]NP [admire only each other]V P

↙ ↘(∃xGrp/x ⊆ [yCritic]) (∀y/y ∈ x)(∀z)[Admire(y, z) → z ∈ x ∧ z �= y]

↘ ↙(∃xGrp/x ⊆ [yCritic])(∀y/y ∈ x)(∀z)[Admire(y, z) → z ∈ x ∧ z �= y].

Another example of an irreducibly plural reference is ‘Some people are play-ing cards’, where by ‘some people’ we do not mean that at least one person

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is playing cards, but that a group of people are playing cards, and that theyare doing it together and not separately. The truth conditions of this sentencecan be represented as follows where the argument of the predicate is irreduciblyplural.

[Some people]NP [are playing cards]V P

↓ ↓(∃xGrp/x ⊆ [yP erson]) Playing-Cards(x)

↓ ↓(∃xGrp/x ⊆ [yP erson])Playing-Cards(x)

Of course, in saying that a group of people are playing cards we mean thateach member of the group is playing cards, but also that the members of thegroup are playing cards with every other member of the group. That is,

(∀xGrp/x ⊆ [yP erson])[Playing-Cards(x) → (∀y/y ∈ x)Playing-Cards(y)]

and(∀xGrp/x ⊆ [yP erson])[Playing-Cards(x) →

(∀y/y ∈ x)(∀z/z ∈ x/z �= y)Playing-Cards-with(y, z)]

are understood to be consequences of what is meant in saying that a group isplaying cards. We understand the preposition ‘with’ in this last formula, to bean operator that modifies a monadic predicate and generates a binary predicateby adding one new argument position to the predicate being modified. Thus,applying this modifier to ‘x is playing cards’ we get ‘x is playing cards with y’.The added part, ‘with y’, represents a prepositional phrase of English.31 Theconverse, however, does not follow in either case. That is, we could have everymember of a group playing cards without the group playing cards together, andwe could even have every member playing cards with every other member inseparate games without all of them playing cards together in a single game.

Another type of referential expression that is irreducibly plural is the pluraluse of ‘the’, as in ‘the inhabitants of Rome’ and ‘the Greeks who fought atThermopylae’. On our reading these expressions are to be taken as referring tothe inhabitants of Rome as a group and similarly to the Greeks who fought atThermopylae as a group. In this way the plural use of ‘the’ can be reduced tothe singular ‘the’, i.e., to a definite description of a group.

The singular ‘the’, as we described it in our fifth lecture, is represented bya quantifier (as are all determiners), in particular, ∃1, where the truth condi-tions of an assertion of the form ‘The A is F ’ are spelled out in essentiallythe Russellian manner (when the definite description is used with existentialpresupposition).

Consider now the sentence ‘The Greeks who fought at Thermopylae areheroes’, which we take to be equivalent to ‘The group of Greeks who fought

31Note that as described here ‘with’ cannot be iterated so as to result in a three-placepredicate. We assume, in this regard, that ‘playing cards with-with’ is not grammatical orlogically well-formed.

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at Thermopylae are heroes’. Using F (x) for the verb phrases ‘x fought atThermopylae’, we can semantically represent the sentence as follows:

[The Greeks who fought at Thermopylae]NP [are heroes]V P

↙ ↘[The group of Greeks who fought at Thermopylae]NP [are heroes]V P

↓ ↓(∃1xGrp/x = [xGreek/F (x)]) (∀y/y ∈ x)(∃zHero)(y = z)

↘ ↙(∃1xGrp/x = [xGreek/F (x)])[λx(∀y/y ∈ x)(∃zHero)(y = z)](x)

The truth conditions of this sentence amount to there being (now, at the timeof the assertion) exactly one group of Greeks who fought at Thermopylae andevery member of that group is a hero, which captures the intended content ofthe sentence in question. We might also note that another standard formulationof the English sentence, namely, that the class as many of Greeks who foughtat Thermopylae is contained within the class as many of heroes,

[xGreek/F (x)] ⊆ [xHero],

is a consequence of the above formulation; and, in fact, if it is assumed that[xGreek/F (x)] has at least two members and that each of its members is anatom, i.e., an individual, which in fact is the case, then the two formulationsare equivalent to one another.

Another type of example is plural identity, as in:

Russell and Whitehead are the coauthors of PM.

Here, reference is to the group consisting of Russell and Whitehead, and whatis predicated of this group is that it is identical with the group consisting ofthose who coauthored PM (Principia Mathematica). In other words, where ‘A’and ‘B’ are name constants for ‘Russell’ and ‘Whitehead’, a plural subject ofthe form ‘A and B’ is analyzed as follows:

A and B↓

The group consisting of A and B↓

(∃1xGrp/x = [z/(z = A ∨ z = B)

Similarly, the analysis of the phrase ‘the coauthors of PM’ is to be analyzed asfollows:

the coauthors of PM↓

The group of those who coauthored PM↓

(∃1yGrp/y = [z/Coauthored(z, PM)])

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The plural identity of the two groups can then be symbolized as,

(∃1xGrp/x = [z/(z = A∨z = B)(∃1yGrp/y = [z/Coauthored(z, PM)])(x = y),

where it is the identity of two groups that is explicitly stated in the identitypredicate. A similar analysis applies to the sentence

The triangles that have equal sides are the triangles that have equal angles.

That is, where ‘A’ is a name constant for ‘triangle’ and ‘F ’ and ‘G’ are one-placepredicates for ‘has equal sides’ and ‘has equal angles’, respectively, then the twoplural definite descriptions can be represented as:

The triangles that have equal sides↓

The group of triangles that have equal sides↓

(∃1xGrp/x = [zA/F (z)])

and with a similar analysis for ‘the group of triangles that have equal angles’,the plural identity of the two groups can be symbolized as:

(∃1xGrp/x = [zA/F (z)])(∃1yGrp/y = [zA/G(x)])(x = y),

where, again, it is the identity of two groups that is stated in the identitypredicate. We should note, however, that given the axiom of extensionality, thissentence is provably equivalent to

A triangle has equal sides if, and only if, it has equal angles.

which can be symbolized as:

(∀xA)[F (x) ↔ G(x)].

In other words, strictly speaking, the truth conditions of this sentence does notinvolve an irreducibly plural reference to, or predication of, groups.

An example is an irreducibly plural predication is one where we predicatecardinal numbers of a group, as when we say that the Apostles are twelve.Here, the plural definite description, ‘the Apostles’ is understood to refer to theApostles as a group, which means that we can symbolize the plural descriptionas follows:

The Apostles↓

The group of Apostles↓

(∃1xGrp/x = [xApostle])

What is predicated of this group is that it has twelve members. The verb phrase‘x has twelve members’ can be symbolized as a complex predicate as follows,

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11.7. PLURAL REFERENCE AND PREDICATION 255

x has twelve members↓

[λx(∃12y)(y ∈ x)](x)

As is well-known, the numerical quantifier ∃12 is definable in first-order logicwith identity, which we will not go into here.32 The important point is that thisis really a plural predicate, i.e., it can be truthfully applied only to a plurality,namely a group with twelve members. The whole sentence can then be analyzedas follows:

[The Apostles]NP [are twelve]V P

↙ ↘(∃1xGrp/x = [xApostle]) [λx(∃12y)(y ∈ x)](x)

↘ ↙(∃1xGrp/x = [xApostle])[λx(∃12y)(y ∈ x)](x)

or, by λ-conversion, more simply as

(∃1xGrp/x = [xApostle])(∃12y)(y ∈ x).

11.7 Plural Reference and Predication

The logical analyses of plural reference and predication that we have describedso far are primarily analyses of the truth conditions, i.e., the semantics, of pluralreference and predication. They are not analyses of the cognitive structure ofplural reference and predication as part of our speech and mental acts.

The question is how can we account for the cognitive struc-ture of plural reference and predication in terms of the log-ical forms that we use to represent our speech and mentalacts.

What we propose is to formalize the pluralization of both common namesand monadic predicates. We do this by means of an operator that when appliedto a name results in the plural form of that name, and similarly when appliedto a monadic predicate results in the plural form of that predicate. We will usethe letter ‘P ’ as the symbol for this plural operator and we will represent itsapplication to a name A or predicate F by placing the letter ‘P ’ as a superscriptof the name or predicate, as in AP and FP .

Thus, we now extend the simultaneous inductive definition of the mean-ingful (well-formed) expressions of our conceptualist framework to include thefollowing clauses:

1. if A is a name variable or constant, then AP is a plural name variable orconstant ; respectively;

32See chapter 7, §9, for an analysis of numerical quantifiers.

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2. if A is a name, x is an object variable, and ϕx is a formula, then [xA/ϕx]P

and [x/ϕx]P are plural names ;

3. if A/ϕ(x) is a (complex) name, then (A/ϕ(x))P = AP /[λxϕ(x)]P (x) and[xA/ϕ(x)]P = [xAP /[λxϕ(x)]P (x)];

4. if F is a one-place predicate variable or constant, or of the form [λxϕ(x)],then FP is a one-place plural predicate;

5. if AP is a plural name, x is an object variable, and ϕ is a formula, then(∀xAP )ϕ and (∃xAP )ϕ are formulas.

In regard to clause (5), we read, e.g., ‘(∀xManP )’ as the plural phrase ‘allmen’ and ‘(∃xManP )’ as the plural phrase ‘some men’, and similarly ‘(∀xDogP )’as ‘all dogs’ and ‘(∃xDogP )’ as ‘some dogs’, etc. We note that a plural nameis not a name simpliciter and that unlike the latter there is no rule for the“nominalization” of plural names, i.e., their transformation into objectual terms.This is because a nominalized name (occurring as an argument of a predicate)can already be read as plural if its extension is plural, and we do not want toconfuse and identify a name simpliciter with its plural form.

Note also that only monadic predicates are pluralized. A two-place relationR can be pluralized in either its first- or second-argument position, or even both,by using a λ-abstract, as, e.g.,

[λxR(x, y)]P ,

[λyR(x, y)]P ,

[λx[λy[R(x, y)]P (y)](x)]P ,

respectively; and a similar observation applies to n-place predicates for n > 2.Thus, for example, we can represent an assertion of ‘Some people are playingcards with Sofia’ by pluralizing the first-argument position of the two-placepredicate ‘x is playing cards with y’ as follows:

[Some people]NP [are playing cards with Sofia]V P

↓ ↓(∃xPersonP )[λxP laying-Cards-with(x, Sofia)]P (x)

Semantically, of course, we understand the plural reference in this assertionto be to a group of people, a fact that is made explicit by assuming the followingas a meaning postulate for all (nonplural) names A whether simple or complex:

(∃xAP )ϕ(x) ↔ (∃xGrp/x ⊆ [yA])ϕ(x). (MPP1)

Of course, if a group of people are playing cards with Sofia, then it follows thateach person in the group is playing cards with Sofia, though, as already noted,the converse does not also hold. The one-direction implication from the plural

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to the singular can be described by assuming the following as part of the waythat the monadic-predicate modifier ‘with’ operates33:

(∀x)([λxP laying-Cards-with(x, Sofia)]P (x) →(∀y/y ∈ x)[λxP laying-Cards-with(x, Sofia)](y)).

An example where the second argument of a relation is plural is a con-sequence of sentence ‘Some men carry the piano downstairs’, i.e., where theconsequence is that each man in the group (qua individual) carries the pianodownstairs with the other men in the group (qua group or plural object). First,where F (x) is read as ‘x carries the piano downstairs’, we note that the sentence‘Some men carry the piano downstairs’ can be analyzed as,

[Some men]NP [carry the piano downstairs]V P

↘ ↙(∃xManP ) FP (x)

↘ ↙(∃xManP )FP (x)

which, by the above meaning postulate for plurals, (MPP1), means that somegroup of men carry the piano downstairs. The consequence then is that somegroup of men is such that every man in the group carries the piano downstairswith the other men in the group. To analyze this, we need to represent whatit means to refer to the men in the group other than a given man. For this weuse the plural definite description, ‘the men in x other than z’, which can besymbolized as follows,

the men in x other than z↓

(∃1y(Man/(y ∈ x ∧ y �= z))P )

Finally, that there is a group of men such that every man in the group carriesthe piano downstairs with the other men in the group can now be representedas follows:

(∃xGrp/x ⊆ [yMan])(∀z/z ∈ x)(∃1y(Man/(y ∈ x ∧ y = z))P )[λyF -with(z, y)]P (y),

which, by (MPP1) as applied to (∃1y(Man/(y ∈ x ∧ y �= z))P ), reduces to

(∃xGrp/x ⊆ [yMan])(∀z/z ∈ x)(∃1yGrp/y = [yMan/y ∈ x ∧ y �= z])[λyF -with(z, y)]P (y),

where the relation ‘z carries the piano downstairs with y’ is taken as plural inits second-argument position.

Finally, let us turn to how the universal plural ‘All A’, the cognitive structureof which is represented by (∀xAP ), is to be semantically analyzed. Let us notefirst that if (∀xAP ) were taken as the logical dual of (∃xAP ), the way (∀x)

33Note that because everything is a class as many, this condition applies even when x is anatom. In that case, of course, the condition is redundant.

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is dual to (∃x), then the postulate for universal plural reference would be asfollows:

(∀xAP )ϕ(x) ↔ (∀xGrp/x ⊆ [yA])ϕ(x). (MPP?)

Then, given that the cognitive structure of an assertion of ‘All men are mortal’can be represented as,

(∀xManP )MortalP (x),

it would follow that, semantically, the assertion amounts to predicating mortal-ity to every group of men,

(∀xGrp/x ⊆ [yMan])MortalP (x),

which is equivalent to saying without existential presupposition that the mem-bers of the entire group of men taken collectively are mortal:

(∀xGrp/x = [yMan])MortalP (x).

This formula, given that the class as many of men is in fact a group—i.e., hasmore than one member—is in conceptualist terms very close to what Russellclaimed in his 1903 Principles, namely, that the denoting phrase ‘All men’ inthe sentence ‘All men are mortal’ denotes the class as many of men, which infact happens to be a group.

But what if the class of men were to consist of exactly one man, as, e.g.,at the time in the story of Genesis when Adam was first created. Presumably,the sentence ‘All men are mortal’ is true at the time in question. But is it avacuous truth? In other words, is it true only because there is no group of menat that time but only a class as many of men having just one member?

Similarly, consider the sentence ‘All moons of the earth are made of greencheese’. Presumably, this sentence is false and not vacuously true because thereis no group of moons of earth but only a class as many with one member. Inother words, regardless of the implicit duality of ‘All A’ with the plural ‘SomeAP ’, we cannot accept the above rule (MPP?) as a meaning postulate forsentences of the form (∀xAP )ϕ(x).

Yet, there is something to Russell’s claim that the phrase ‘All men’ in thesentence ‘All men are mortal’ denotes the class as many of men and differsin this regard from what ‘Every man’ denotes in ‘Every man is mortal’. Inconceptualist terms, in other words, the referential concept that ‘Every man’stands for is not the same as the referential concept that ‘All men’ stands for; noris the predicable concept that ‘is mortal’ stands for the same as the predicableconcept that ‘are mortal’ stands for.34 A judgment that all men are mortalhas a different cognitive structure from a judgement that every man is mortal,even if semantically they have the same truth conditions. It is the difference inreferential and predicable concepts—i.e., the difference between (∀xManP ) and

34This difference in predicable concepts was missed by Russell and explains why his laterrejection of classes as many as what ‘All men’ denotes was based on a confusion betweensingular and plural predication. See Russell 1903, p. 70.

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(∀xMan), on the one hand, and MortalP ( ) and Mortal( ) on the other—thatexplains why the judgments are different.

Now the point of these observations is that instead of (MPP?) we canrepresent the difference between ‘All Ap’ and ‘EveryA’ by adopting the followingmeaning postulate, which takes ‘All AP ’ to refer not just to every group of Abut to every class as many of A, whereas ‘Every A’ refers to each and every Ataken singly:

(∀xAP )ϕ(x) ↔ (∀x/x ⊆ [yA])ϕ(x). (MPP2)

Despite this difference, however, it follows that ‘Every A is F ’ is logically equiv-alent to ‘All A are F ’, i.e.,

(∀xA)F (x) ↔ (∀xAp)F p(x)

is valid in our conceptualist logic.Let us note, incidentally, that the plural verb phrase ‘are mortal’, in symbols,

MortalP , is semantically reducible to its singular form. That is, mortality canbe predicated in the plural of a class as many if, and only if, every memberof the class is mortal. In other words, as part of the meaning of the predicate‘mortal’ we have the following as a meaning postulate:

MortalP (x) ↔ (∀y/y ∈ x)Mortal(y).

It would be convenient, no doubt, if every plural predicate were reducibleto its singular form the way MortalP is, but that is not the case, as we notedearlier with the predicate [λxP laying-Cards-with(x, Sofia)]P , which is pluralin its first argument position. Nor is it true of the complex predicate for carryingthe piano downstairs with the other members of a group, which is plural in itssecond argument position.

The fact is that just as some references are irreducibly plural, so too somepredications are irreducibly plural. Even though plural objects, i.e., groups, areontologically founded upon the single objects that are their members, neverthe-less plural objects are an irreducible part of the world as much as are singleobjects, i.e., individuals. What is needed for both our scientific and our com-monsense frameworks is a logic that can account for plural objects and pluralpredication, whether in thought or in the world, no less so than it can accountfor single objects and singular predication. That is the logic we have presentedhere as a special part of the more general framework of conceptual realism.

11.8 Cardinal Numbers and Plural Quantifiers

Our earlier analysis of the sentence ‘The Apostles are twelve’ took it to havethe same truth conditions as ‘The group of Apostles has twelve members’. Thisanalysis is not quite adequate, however, in that it depends on paraphrase anddoes not account for the logical connection between ‘The Apostles are twelve’and ‘The group of Apostles has twelve members’. As part of this account,

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moreover, we need to explain the source of the plural ‘are’ in ‘The Apostles aretwelve’.

Now it is clear that the plural verb phrase ‘are twelve’ is not the pluralform of the singular verb phrase ‘is twelve’, the way the phrase ‘are Apostles’is the plural of ‘is an Apostle’, and hence that its source cannot be explainedas a simple pluralization of the verb. The explanation that we propose heredepends on the claim that the most basic way in which we speak and thinkabout different numbers of things is in our use of numerical quantifier phrases,as when we say that there are twelve Apostles. This statement, we claim, canbe grammatically analyzed as

[twelve Apostles]NP [there are]V P ,

where the noun phrase ‘twelve Apostles’ is taken as the subject of the sentence,and ‘there are’, which is the plural of ‘there is’, is taken as the verb phrase. Itis our contention that the source of the plural ‘are’ in ‘The Apostles are twelve’is the ‘are’ in ‘There are twelve Apostles’, which we earlier paraphrased as ‘Thegroup of Apostles has twelve members’.

The question now is how do we logically analyze the plural predicate ‘thereare’ and then logically connect the three different sentences:

There are twelve Apostles,The group of Apostles has twelve members,

The Apostles are twelve’.

The answer, it turns out, provides another and, in our view, a better way torepresent cardinal numbers than that already given in chapter six.

How then we are to analyze the quantifier phrase ‘there are’ when it functionsas a plural predicate? Consider, as an example, the sentence, ‘There are liberalrepublican senators’, which we take to be synonymous with ‘There are senatorswho are republican and liberal’. This sentence, let us note, is the plural form of‘There is a senator who is republican and liberal’, which can be grammaticallystructured as,

[a senator who is republican and liberal]NP [there is]V P ,

and the plural form of the sentence can be similarly grammatically structuredas,

[senators who are republican and liberal]NP [there are]V P .

Now the logical analysis of the above singular noun phrase is,

a senator who is republican and liberal↓

(∃xSenator/Republican(x) ∧ Liberal(x)),

and the logical analysis of the singular verb phrase as a predicate is

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there is↓

[λx(∃y)(x = y)],

the infinitive of which can be read as ‘to be an x such that there is somethingx is’. The singular form of the sentence can accordingly be formalized as:

(∃xSenator/Republican(x)∧ Liberal(x))[λx(∃y)(x = y)](x).

Now what we propose is that the logical form assigned to the plural form of thesentence should be derived from the singular form by having both the referentialand the predicate expressions pluralized:

(∃xSenator/Republican(x) ∧ Liberal(x))P [λx(∃y)(x = y)]P (x),

which, by (MP1), gives us

(∃xGrp/x ⊆ [ySenator/Republican(y)∧ Liberal(y)])[λx(∃y)(x = y)]P (x)

as its semantic representation for the plural reference.In regard to the semantics of the plural predicate, we assume that a group

has plural being in the sense of ‘there are’ if, and only if, each of its membershas being in the sense of ‘there is’. That is, the being of a plurality reducesto—i.e., is equivalent to—the being of each of the members of that plurality:

(∀xGrp)([λx(∃y)(x = y)]P (x) ↔ (∀z/z ∈ x)[λx(∃y)(x = y)](z)).

The right-hand side of this biconditional is logically true and is a consequenceof the free-logic axiom (∀z)(∃y)(z = y), which means that the left-hand side ofthe biconditional is equivalent to ‘x = x’. This means that the above formulais equivalent to

(∃xGrp/x ⊆ [ySenator/Republican(y)∧ Liberal(y)])(x = x),

and hence, by the exportation thesis for complex quantifier phrases and deletionof the redundant identity conjunct, ‘x = x’, equivalent to

(∃xGrp)(x ⊆ [ySenator/Republican(y)∧ Liberal(y)]).

In other words, the plural form of the sentence semantically amounts to sayingthat there is a group of senators who are republican and liberal, which, intu-itively, is exactly what we understand the initial statement to say. Our generalproposal is that other sentences with a plural ‘there are’ should be analyzed in asimilar way, i.e., as being represented by the plural predicate [λx(∃y)(x = y)]P .

In particular, to return to the issue in question, the sentence ‘There aretwelve Apostles’, which grammatically and then logically can analyzed as fol-lows:

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There are twelve Apostles↓

[twelve Apostles]NP [there are]V P

↓ ↓(∃12xApostles) [λx(∃y)(x = y)]P (x)

↓ ↓(∃12xApostles)[λx(∃y)(x = y)]P (x).

This last formula, as noted above about the plural predicate, reduces to

(∃12xApostle)(x = x),

as well as to(∃12x)(∃yApostle)(x = y),

which says that twelve things are Apostles, which is another way of saying thatthere are twelve Apostles.35

Now as we suggested in chapter seven the most basic way in which we speakand think about different numbers of things is in our use of numerical quantifierphrases, as when we say that there are two authors of PM, three people playingcards, twelve Apostles, etc. These quantifier phrases can be nominalized inconceptual realism the way that we earlier nominalized the phrase, ‘a unicorn’,which means that they are first transformed into a (complex) predicate, whichin turn is then nominalized and transformed into an abstract singular term,the denotatum of which is an intensional object. This kind of transformationis based, as we explained in chapter six, on a pattern of double abstractioncorresponding to Frege’s double-correlation thesis—except that Frege attachedonly the common name ‘object’ to his quantifiers and took other common namesto be predicates. Frege, moreover, being an extensionalist, did not identify theobjects correlated with his quantifiers as intensional objects, but as value-ranges(Wertverlaufe). The general idea in any case is that where ∃k is a numericalquantifier, which when applied to a name A is read as ‘there are k many A’, thedouble-correlation thesis can be formulated as follows,

(∃F )(∀A)(∀G)[(∃kxA)G(x) ↔ F ([xA/G(x)])],

where instead of a value-range as the argument of the concept F correspondingto the numerical quantifier ∃k as Frege would have it, we have a class as many.In particular, taking the numerical quantifier ∃12 for ∃k, we have

(∃F )(∀A)(∀G)[(∃12xA)G(x) ↔ F ([xA/G(x)])].

Now, clearly the concept F that is posited here as provably such that

35The application of ∃12 to the singular ‘Apostle’ is appropriate here because it says ineffect that there are twelve individual Apostles, i.e., twelve individuals named by ‘Apostle’.If we used the plural, ‘ApostlesP ’, the result would amount to saying that there are twelvegroups of Apostles, which is not what is intended.

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11.8. CARDINAL NUMBERS AND PLURAL QUANTIFIERS 263

for all name concepts A and all predicable concepts G, thereare twelve A that are G if, and only if, the group of A thatare G falls under F

is none other than the predicable concept of having twelve members, which wesymbolized earlier as [λx(∃12y)(y ∈ x)]. In other words, by the above double-correlation thesis we have

(∀A)(∀G)[(∃12xA)G(x) ↔ [λx(∃12y)(y ∈ x)]([xA/G(x)])],

from which, by substituting ‘Apostle’ for ‘A’ and ‘x = x’ for ‘G(x)’, and applyingλ-conversion, we have

(∃12xApostle)(x = x) ↔ (∃12y)(y ∈ [xApostle/(x = x)]),

and therefore by canceling the redundant identity in [xApostle/(x = x)],

(∃12xApostle)(x = x) ↔ (∃12y)(y ∈ [xApostle]).

But, as noted above, the left-hand side of this biconditional is equivalent to ourformulation of ‘There are twelve Apostles’, namely, (∃12xApostle)[λx(∃y)(x =y)]P (x), which means that the biconditional is equivalent to

(∃12xApostle)[λx(∃y)(x = y)]P (x) ↔ (∃12

y)(y ∈ [xApostle]).

Now the definite description ‘the group of Apostles’, as symbolized by (∃1yGrp/y =[xApostle]), denotes the same group as is denoted by the abstract [xApostle],i.e.,

(∃1zGrp/z = [xApostle])(z = [xApostle])

is true and provable. From this identity and the last biconditional above it fol-lows that the cognitive structure of the statement that there are twelve Apostlesis logically equivalent to that of the statement that the group of Apostles hastwelve members, i.e.,

(∃12xApostle)[λx(∃y)(x = y)]P (x) ↔ (∃1zGrp/z = [xApostle])(∃12

y)(y ∈ z)

is provable, which is one of the logical connections we wanted to establish.The predicable concept that we predicate in ‘The Apostles are twelve’, ac-

cordingly, is equivalent to the concept of having twelve members, a concept thatevery group with twelve members falls under. Thus, as a predicable concept, acardinal number K can be defined as the predicable concept that those groupsthat have k many members fall under, and hence the cardinal number k itselfcan be identified with the “object-ified” correlate of the concept K. In otherwords, starting with the quantifier notion expressed by ‘There are k many A’,we obtain the predicable concept K under which a group falls if, and only if,it has k many members, and then, by “object-ifying”, or ”nominalizing”, thepredicable concept K, we obtain the number k as the abstract object correlated

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with the concept. It is in this way, by going through a double abstraction from areferential concept to a corresponding predicable concept and then, by the pro-cess of nominalization, to an object that we are able to grasp and understandthe role of numbers as abstract objects.

In particular, the predicable concept 12 can now be defined as the conceptunder which an object falls if, and only if, that object is a group having twelvemembers:

12 = [λy(∃A)(y = A ∧ (∃12xA)(x ∈ y))].

Qua object, the number 12 is then the objectified correlate of this predicableconcept. It is in this way that we are able to logically connect the sentence ‘TheApostles are twelve’ with ‘The group of Apostles has twelve members’. That is,we can now be symbolize ‘The Apostles are twelve’ as

(∃1xApostlesP )12(x),

which, by (MPP1) reduces to

(∃1xGrp/x = [yApostle])12(x),

which, by definition, λ-conversion and identity logic, reduces to

(∃1xGrp/x = [yApostle])[λx(∃12y)(y ∈ x)](x).

Thus, the three sentences, ‘The Apostles are twelve’, ‘The group of Apostleshas twelve members’, and ‘There are twelve Apostles’ are each seen to be logi-cally equivalent to one another. Each, moreover, is provably equivalent to thefollowing simplest version of all:

12([xApostle]).

11.9 Summary and Concluding Remarks

• The logic of classes as many is an extension of the simple logic of namesformalized in chapter ten.

• Names, both proper and common, are transformed (or “nominalized”) inthis logic into objectual terms, both simple and complex.

• Russell’s paradox for classes fails in the logic of classes as many. In par-ticular, the Russell class as many does not “exist” as an object, i.e., as a valueof the bound objectual variables of this logic.

• A particular axiom (namely, Axiom 15) of the logic of classes as many,specifies when and only when a name concept A has an extension that can be“object”-ified (as a value of the bound objectual variables).

• Nelson Goodman’s notion of an “atom” is adopted as a way to distinguishsingle objects from plural objects. Atoms, i.e., single objects, are “individuals”in the traditional ontological sense, as opposed to plural objects, i.e., classes asmany of two or more members.

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11.9. SUMMARY AND CONCLUDING REMARKS 265

• The empty class as many does not “exist” as a value of the bound objectualvariables in this logic (T12).

• It is provable in the logic of classes as many that each atom is identicalwith its singleton (T13).

• The logical role of “nominalized” proper names in the logic of classes asmany is the same as the role of proper names in free first-order logic. Thus, therole proper names have as “singular terms” in standard free logic is reducibleto, and fully explainable in terms of, the role proper names have in the logic ofclasses as many, and therefore in our conceptualist theory of reference.

• Although the full, unrestricted version of Leibniz’s law applies to all atoms,i.e., single objects, or individuals in the ontological sense, only a version re-stricted to extensional contexts applies to groups, i.e., classes as many of twoor more objects. This distinction between how Leibniz’s law applies to atomsand how it applies to groups is an ontological feature that distinguishes theindividuality of atoms from the plurality of groups.

• Nelson Goodman’s nominalistic dictum that things are identical if theyhave the same atoms is provable in the logic of classes as many.

• If either there are no atoms or there are at least two atoms, then theuniversal class does not “exist” (as a value of the objectual variables). It isindeterminate whether or not the universal class exists if there is just one atom.

• Cantor’s power-set theorem is provable only for finite classes in the logicof classes as many.

• The oddities of Lesniewski’s logic of names do not also hold in the logic ofclasses as many.

• Groups are important in determining the truth conditions of sentencesthat are irreducibly plural.

• The plural use of ‘the’ as in ‘the Greeks who fought at Thermopylae’ canbe reduced to the singular ‘the’ as a definite description of a group.

• Predication of cardinal numbers, as in ‘the Apostles are twelve’ can beanalyzed in a natural way in terms of groups in this logic.

• The logical analyses of plural reference and predication in terms of groupsin the logic of classes as many amounts only to an analysis of the truth conditionsof plural reference and predication, and not also to an analysis of the cognitivestructure of plural reference and predication in our speech and mental acts.

• The logic of classes as many is extended to included a plural operatorthat can be applied to names or to predicate expressions, and an analysis of thecognitive structure of plural reference and predication is given in terms of theplural operator.

• The predicable concept expresssed in predicating a cardinal number n canbe identified with the concept under which all and only groups with n membersfall. The intensional object that results from the nominalization of this conceptcan be identified with the number n as an abstract object.

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11.10 Appendix 1: A Set-Theoretic Semantics

A set-theoretical semantics can be constructed for the logic of classes as many asformalized here, and the system can be shown to be consistent with respect tothat construction. We will forego the proofs in what follows and just sketch outthe semantics for a “standard” model of the system without indices (possibleworlds, moments of time, contexts of use, etc.), and therefore one in which theaxiom of extensionality is valid. Extending this semantics to one that includesindices, and hence to one in which the axiom of extensionality is not valid, isunproblematic and can be done in the usual way.

By an (object) language we understand a (possibly empty) set of predicateand nominal constants. We will take ‘∈’, ‘⊆’, ‘⊂’, and ‘Atom’ defined in thelogic of classes as primitive logical constants with their definitions as additionalaxioms of the logic. This means that the notion of a formula must be extendedto include atomic formulas consisting of n-place predicate constants applied ton many singular terms, for n ∈ ω.36

Definition 13 L is a language iff L is a countable set of nominal constants(proper and common names) and predicate constants.

By a set of “atoms” we understand a set that does not have the empty setas a member and no member of which has a member in common with that set.The idea is that “atoms” are to function as urelements. Thus, where D is anynonempty set, the set {〈d,D〉 : d ∈ D} is a set of atoms even if D is not.

Definition 14 D is a set of “atoms” iff D is a set such that 0 /∈ D and foreach d ∈ D, d ∩D = 0.

A “standard” model for a language consists of a set, possibly empty, of“atoms”, i.e., objects considered as urelements with respect to the various setsthat make up the model, and an assignment of extensions to the predicate andnominal constants. The assignment to the constants is not drawn just from theset D of atoms, however, but from an extended set D+ defined as follows.

Definition 15 If D is a set (of atoms), then D+ = D ∪ {X ⊆ D : X �= 0 andfor all d ∈ D, X �= {d}}.

We define the denotation function with respect to a set D as follows.

Definition 16 If D is a set (empty or otherwise), then the denotation functionof D, in symbols, denD, is that function with D ∪ {X : X ⊆ D+} as domainand such that(1) for all d ∈ D, denD(d) = d, and(2) for all X ⊆ D+,

denD(X) ={d, if X = {d}, for some (atom) d ∈ D

X otherwise .

36We assume our metalanguage to be ZF set theory, and we take ω to be the set of naturalnumbers, where for each n ∈ ω, n is the set of natural numbers less than n. Thus 0 is theempty set.

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11.10. APPENDIX 1: A SET-THEORETIC SEMANTICS 267

We now define the notion of a “standard” nominal model.

Definition 17 A is a nominal L-model iff L is a language and there are a setD, possibly empty, of “atoms” and a relation R such that(1) A = 〈D,R〉, and(2) R is a function with L as domain and such that for each nominal constantA ∈ L, R(A) ⊆ D+, and for each n ∈ ω, and each n-place predicate constantπ ∈ L, R(π) is a set of n-tuples drawn from D+.

Note that because R(A) ⊆ D+, where A is a nominal constant, then namesof single atoms are assigned singletons of those atoms, and not the atoms them-selves. This is “corrected for” in the definition of the denotation function ina model, which, in the “standard” semantics, we identify with the denotationfunction of the domain of atoms in the model.

Definition 18 If L is a language and A = 〈D,R〉 is a nominal L-model, thenthe denotation function in A , in symbols, DenA = denD.

The following metatheorem indicates some of the useful features of the deno-tation function. Part (4) is the semantic analogue of the axiom of extensionalityin the object language, which, as already noted, is valid in this semantics.

Metatheorem 1: If A = 〈D,R〉 is a nominal model, then(1) if d1, d2 ∈ D+, then DenA(d1) = DenA(d2) iff d1 = d2;(2) if X,Y ⊆ D+ and X,Y /∈ D, then DenA(X) = DenA(Y ) iff X = Y ;(3) if d1 ∈ D, d2 ∈ D+ and for all d3 ∈ D+ [there is an X ⊆ D+ such thatd3 ∈ X & DenA(d2) = DenA(X) only if there is a Y ⊆ D+ such that d3 ∈ Y &DenA(d1) = DenA(Y )], then d1 = d2.(4) if d1, d2 ∈ D+: if for all d3 ∈ D+, (there is an X ⊆ D+ such that d3 ∈ X &DenA(d1) = DenA(X) iff there is an Y ⊆ D+ such that d3 ∈ Y & DenA(d2) =DenA(Y )), then d1 = d2.

An assignment in a model of values to variables assigns objects in the unionof the domain of atoms and the non-empty, nonsingleton subsets of that domainto the individual variables and subsets of the latter to the nominal variables.

Definition 19 If A = 〈D,R〉 is a nominal L-model, then a is an assignment(of values to variables) in A iff a is a function with the set of individual andnominal variables as domain and such that(1) for each individual variable x, a(x) ∈ D′, for some D′ ⊇ D+, and(2) for each nominal variable A, a(A) ⊆ D+.

Thus, an assignment in a nominal model assigns values to the individualvariables that are drawn from a set D′ that contains D+, i.e., D+ ⊆ D′, and itassigns to the nominal variables subsets of D+. The distinction between D+andD′ is required for the “free logic” aspect of the first-order part of the logic ofclasses as many; that is, D+ is the set of values of bound individual variablesand D′ is the set of values of free individual variables.

We next inductively define the extension of a name or formula in a model.

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Definition 20 If L is a language, A = 〈D,R〉 is a nominal L-model, a is anassignment in A, and ξ is a name or formula of L, then the extension of ξin A relative to a and an individual variable z (as place holder), in symbolsext(ξ,A, a, z), is defined recursively as follows:(1) if ξ is a nominal variable or constant in L, then

ext(ξ,A, a, z) ={

R(ξ), if ξ is a constant in La(ξ), if ξ is a nominal or individual variable ;

(2) if ξ is an identity formula (a = b), where a, b are singular terms, i.e., eitherindividual variables, nominal variables or constants in L, or names of L of theform [xB], then

ext(ξ,A, a, z) ={

1, if DenA(ext(a,A, a, z)) = DenA(ext(b,A, a, z))0 otherwise ;

(3) If ξ is a ∈ b, for some singular terms a, b of L, then ext(ξ,A, a, z) = 1 iffor some X ⊆ D+, ext(a,A, a, z) ∈ X & DenA(b) = DenA(X); and otherwiseext(ξ,A, a, z) = 0;

(4) If ξ is a ⊆ b, for some singular terms a, b of L, then ext(ξ,A, a, z) = 1 if forall d ∈ D+, there is an X ⊆ D+ such that d ∈ X & DenA(a) = DenA(X) onlyif there is a Y ⊆ D+ such that d ∈ Y and DenA(b) = DenA(Y ); and otherwiseext(ξ,A, a, z) = 0;

(5) If ξ is a ⊂ b, for some singular terms a, b of L, then ext(ξ,A, a, z) = 1 iffor all d ∈ D+, there is an X ⊆ D+ such that d ∈ X & DenA(a) = DenA(X)only if there is a Y ⊆ D+ such that d ∈ Y and DenA(b) = DenA(Y ), and yetit is not the case that for all d ∈ D+, there is an Y ⊆ D+ such that d ∈ Y& DenA(b) = DenA(Y ), only if there is an X ⊆ D+ such that d ∈ X andDenA(a) = DenA(X);

(6) if ξ is π(a1, ..., an), for some n-place predicate constant in L, thenextA(ξ,A, a, z) = 1 if 〈DenA(extA(a1,A, a, z), ..., DenA(extA(an,A, a, z))〉 ∈R(π); and otherwise extA(ξ,A, a, z) = 0;

(7) if ξ is Atom, then ext(ξ,A, a, z) = D;

(8) if ξ is ¬ϕ, for some formula ϕ of L, then

ext(ξ,A, a, z) ={

1, if ext(ϕ,A, a, z) = 00, otherwise ;

(9) if ξ is (ϕ→ ψ), for some formulas ϕ, ψ of L, then

ext(ξ,A, a, z) ={

0, if ext(ϕ,A, a, z) = 1 and ext(ψ,A, a, z) = 01, otherwise ;

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(10) if ξ is (∀x)ϕ, for some formula ϕ of L and individual variable x, then

ext(ξ,A, a, z) ={

1 if for all d ∈ D+, ext(ϕ,A, a(d/x), z) = 10, otherwise ;

(11) if ξ is (∀xB)ϕ, for some formula ϕ of L, individual variable x, and nameB, then (note the change in place-holder) ext(ξ,A, a, z) = 1, if for all d ∈ D+ ∩ext(B,A, a(d/x), x), ext(ϕ,A, a(d/x), x) = 1; and otherwise ext(ξ,A, a, z) = 0;

(12) if ξ is (∀Cϕ), for some formula ϕ of L and nominal variable C, then

ext(ξ,A, a, z) ={

1, if for all X ⊆ D+, ext(ϕ,A, a(X/C), z) = 10, otherwise ;

(13) if ξ is B/ϕ, for some name B and formula ϕ of L, then ext(ξ,A, a, z) ={d ∈ D+ : d ∈ ext(B,A, a(d/z), z) & ext(ϕ,A, a(d/z), z) = 1};(14) if ξ is /ϕ, for some formula ϕ of L, then ext(ξ,A, a, z) = {d ∈ D+ : d ∈ext(ϕ,A, a(d/z), z) = 1}; and

(15) if ξ is [xB], for some individual variable x and name B of L, thenext(ξ,A, a, z) = {d ∈ D+ : d ∈ ext(B,A, a(d/x), z)}.

Note that in the definiens of clause 11 of this definition the place-holdervariable z is replaced by the variable x. Also, although the place-holder in thedefiniens remains unchanged in clauses 13 and 14, the assignment is modifiedwith respect to that place-holder. In clause 15, the place-holder is left unchangedand the assignment is modified with respect to the bound variable.

Definition 21 If L is a language, ϕ is a formula of L, A is a nominal L-modeland a is an assignment in A, then(1) a satisfies ϕ in A iff ext(ϕ,A, a, z) = 1, for some individual variable z (asplace-holder); and(2) ϕ is true in A iff every assignment in A satisfies ϕ in A.

We define logical consequence and validity in the usual way.

Definition 22 If L is a language and Γ ∪ {ϕ} is a set of formulas of L, then(1) ϕ is a logical consequence of Γ, in symbols, Γ |= ϕ, iff for every L-modelA and every assignment a in A, if a satisfies every formula in Γ in A, then asatisfies ϕ in A; and(2) ϕ is valid if it is a logical consequence of the empty set.

The following soundness theorem leads directly to a consistency proof forthe logic of classes as many plus the axiom of extensionality.

Metatheorem 2: (Soundness) If ϕ is a theorem of the logic of classes asmany plus the axiom of extensionality, then ϕ is valid with respect to the abovesemantics.

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The consistency of the logic of classes as many (plus the axiom of extension-ality) follows from the fact that {〈n, ω〉 : n ∈ ω} is a set of atoms and thereforethat 〈{〈n, ω〉 : n ∈ ω}+, 0〉 is a nominal model, and hence that every theorem istrue in 〈{〈n, ω〉 : n ∈ ω}+, 0〉.

Metatheorem 3: The logic of classes as many with the axiom of exten-sionality added is consistent.

11.11 Appendix 2: Bell’s System M

The system M of classes as many in Bell 2000 is different from the logic de-scribed here. M is designed to show how proper (or ultimate) classes, whichdo not belong to other classes, can be taken to be classes as many (though Belltakes all sets to be classes as many and redefines ‘set’ in terms of certain individ-uals identified as “labeled” classes). Unlike the logic described here, M is notdesigned to provide a semantics for plural references in natural language, and itis not clear how one might use it for that purpose. Nevertheless, Bell’s systemM can be translated into the logic of classes as many presented here with theresult that, with the axiom of extensionality added to the latter, the translationof each axiom of M is a theorem of our present system (and hence that M iscontained in the latter). M is a two-sorted first-order logic with capital letters,A,B,C, etc., for classes as many and lower-case letters x, y, z, etc., as individ-ual variables. The logic of classes as many described here contains a two-sortedfirst-order (free) logic as a proper part, where the capital letters A,B,C, etc.for nominal concepts are nominalized and transformed into singular terms forclasses as many. As primitive symbols, M also contains ∈ for the member-ship relation, the identity sign (applied to classes terms or to individual termsseparately), a labeling functor λ (applied to class terms), a co-labeling functor∗ (applied to individual terms), a monadic predicate I (applied to individualterms, where I(a) is read ‘a is an identifier’), a monadic predicate S (appliedto class terms, where S(A) is read ‘A is a set’), and the abstraction operator,{x : ϕ(x)} (read as ‘the class defined by ϕ’). Our translation function τ identi-fies the capital and individual variables of M with the same letters in our logicof classes as many and interprets the labeling function λ as the nominalizationof a nominal expression A, i.e., τ (λA) = A (nominalized). Membership in M isidentified with membership in the logic of classes as many and class abstractsare similarly identified with one another We extend τ so that, in addition tothe correlation of the terms of M with singular terms of our logic of classesas many, each formula of M is translated into a formula of the logic of classesas many described here, with the translation of the co-labeling functor ∗ givencontextually, i.e., for formulas such as ϕ(x∗) in which it occurs:

1. τ(x) = x and τ (A) = A, for each individual variable x and nominalvariable A;

2. τ(λa) = τ (a), where a is a class term of M (other than of the form b∗);

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11.11. APPENDIX 2: BELL’S SYSTEM M 271

3. τ({x : ϕ(x)}) = [x/τ(ϕ(x))];

4. τ(a ∈ b) = (τ (a) ∈ τ(b));

5. τ(a = b) = (a = b), where a, b are either both class terms of M or bothterms for individuals;

6. τ(I(a)) = (∃z)(τ (a) = z), where z is the first individual variable notoccurring in a;

7. τ(S(a)) = (∃z)(τ (a) = z);

8. τ(ϕ(a∗)) = (∃A)[τ (a) = A ∧ τ (ϕ(A/a))], where a is a term of M for anindividual and A is a class variable of M that is free for a in ϕ;

9. τ(¬ϕ) = ¬τ (ϕ); and τ (ϕ→ ψ) = [τ (ϕ) → τ(ψ)];

10. τ(∀xϕ) = (∀x)τ (ϕ); and τ (∀Aϕ) = (∀A)τ (ϕ)

The translation of the axiom of extensionality of M is just a version ofthe axiom of extensionality in the present logic of classes as many. Also, thetranslation of each “labeling axiom” of M is an obvious theorem of our logic:

τ(S(A) ↔ I(λA)) = [(∃z)(A = z) ↔ (∃z)(A = x)],τ(I(x) ↔ S(x∗)) = [(∃z)(x = z) ↔ (∃A)[x = A ∧ (∃z)(A = z)]],τ(S(B)) → (λB∗) = B) = [(∃z)(B = z) → (∃A)[B = A ∧A = B]],τ(I(x) → λ(x∗) = x) = [(∃z)(x = z) → (∃A)[x = A ∧A = x]].

And finally the axiom of comprehension of M is also translated into a theoremof our logic:

τ(y ∈ {x : ϕ(x)} ↔ ϕ(y/x)) = [(∃z[x/τ(ϕ(x))])(y = z) ↔ τ(ϕ(y/x))].

As noted, the labeling function of M that associates each class with a labeledindividual corresponds to the nominalization transformation of our present logicof classes as many, and those cases where the labeled individuals are “sets” (asdefined in M) correspond to those where a nominal concept is object ified (i.e.,where its nominalization is a value of the bound individual variables).

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Chapter 12

The Logic of Natural Kinds

12.1 Conceptual Natural Realism

The three main theories of universals in the history of philosophy, we have noted,have been nominalism, realism, and conceptualism. In nominalism, there areno universals that predicates stand for; there is only predication in language. Inconceptualism, predication in thought is what underlies predication in language,and what predicates stand for are concepts as rule-following cognitive capacitiesunderlying our use of predicate expressions. What predicates stand for in realismare real universals that are the basis for predication in reality, i.e., for the eventsand states of affairs that obtain in the world.

We have distinguished two types of realism in previous chapters, namelyBertrand Russell’s and Gottlob Frege’s different versions of logical realism asmodern forms of Platonism, and several variants of natural realism, one ofwhich is logical atomism. Another variant of natural realism is a modern formof Aristotle’s theory of natural kinds, or what is usually called Aristotelianessentialism.

Now the relationship between conceptualism and realism is more complexthan the simple kind of opposition that each has to nominalism. Conceptualintensional realism, for example, is similar to logical realism with respect tooverall logical structure, and yet the two formal ontologies are different on suchfundamental issues as the nature of universals and the nexus of predication.

The relationship between conceptualism and natural realism, on the otherhand, is even more complex. They do not, for example, have the same overalllogical structure, and they also differ on the nature of universals and the nexus ofpredication. And yet, conceptualism and natural realism have been intimatelyconnected with one another throughout the history of philosophy—though notalways in an unproblematic way. One reason for this connection is that withoutsome form of natural realism associated with it, conceptualism becomes anontology restricted to the conceptual realm; and without an ontological groundin nature, conceptualism turns into a form of conceptual idealism, sometimes

273

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with, and sometimes without, a transcendental subjectivity.Conceptualism, as we understand it here, is not a form of idealism, however,

but is based on a socio-biological theory of the human capacity for language,culture and thought; and therefore it must presuppose some form of naturalrealism as the causal ground of that capacity. On the other hand, naturalrealism must in turn presuppose some form of conceptualism by which to explainour capacity for language and thought, and in particular our capacity to formtheories of the world and conjecture about natural properties and relations aspart of the causal order. Conceptualism and natural realism, in other words,presuppose each other as part of a more general ontology, which, in one form oranother, may be called conceptual natural realism.1

The connection between conceptualism and natural realism goes back atleast as far as Aristotle whose doctrine of moderate realism, i.e., the doctrinethat universals “exist” only in things in nature, is well-known for its oppositionto Platonism, the doctrine that universals exist as abstract entities indepen-dently of concrete objects. Peter Abelard, in his Glosses on Porphyry, alsodealt with the connection between the conceptual and natural orders of being.In particular, Abelard gave an account that is very much like Aristotle’s in beingboth conceptualist and realist. But in combining these positions, Abelard didnot sharply distinguish the universals that underlie predication in thought fromthose that underlie predication in reality. In particular, a universal, accordingto Abelard, seems to exist in a double way, first as a common likeness in things,and then as a concept that exists in the human intellect through the mind’spower to abstract from our perception of things by attending to the likeness inthem. What Abelard describes is a form of natural realism, where a propertyexists only in the causal or natural order and as a common likeness in things;and yet if those things were to cease to exist, the property would somehow stillexist in the human intellect as a concept.

Aristotle also seems to have described natural kinds and properties in thisdouble way, i.e. as having a mode of being both in things and then, through aninductive abstraction (epagoge), in the mind as well. But then it is possible tointerpret him otherwise, especially in his discussion in the Posterior Analyticsof how concepts such as being a chimera or being a goat-stag can be formedotherwise than by abstraction. Such an alternative interpretation was in factdeveloped by Aquinas in his distinction between the active intellect (intellectusagens) and the receptive intellect (intellectus possibilis), which are not reallytwo intellects but two kinds of powers or capacities of the intellect simpliciter.2

In fact, Aquinas apparently developed the central idea of conceptual naturalrealism, namely, that the problem of the “double existence” of universals is notan ontological problem but a problem of explaining how the same predicate canstand for, or signify, a concept in the mind on the one hand, and a naturalproperty in nature on the other, where the natural property corresponds to, or

1Some of the differences between these forms depends on whether a constructive or holisticconceptualism is assumed, and whether the natural realism is part of an Aristotelian essen-tialism or not.

2Cf. Kenny 1969.

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is represented by, the concept. The two are not really the same universal, inother words, and do not even have the same mode of being.3

Concepts cannot literally be the same as the natural properties and relationsthey purport to represent, in other words, and in fact some concepts—especiallythose for artifacts and social conventions—do not represent any natural prop-erties or relations at all.

12.2 The Problem with Moderate Realism

One reason why the universals of natural realism were confused with predicableconcepts is that both can be designated by predicates—or, more precisely, thata predicate that stands for a concept can also be taken to stand for a naturalproperty or relation that corresponds to that concept. A predicate can be takento stand for a natural property or relation, in other words, as well as for aconcept, but then the sense in which it stands for a property or a relation issecondary to the sense in which it stands for a concept. As Aquinas noted, thetraditional problem about universals “existing in a double way” was really amatter of there being two ways in which a predicate can signify a universal, oneway being primary in which the predicate stands for a concept, and the otherbeing secondary in which the predicate stands for natural property or relationthat corresponds to that concept.

The sense in which a predicate stands for a concept is primary becauseit is the concept that determines the functional role of the predicate and theconditions under which it can be correctly used in a speech act. It is only byassuming (as a result of empirical evidence) that the truth conditions determinedby the concept have a causal ground based on a natural property or relation thatwe can then say that the predicate also stands for a natural property or relation.In other words, even though the natural property or relation in question mayin fact be the causal basis for our forming the concept, and therefore is prior inthe order of being, nevertheless, the concept is prior in the order of conception.

The distinction between concepts in the order of conception and naturalproperties and relations in the order of being does not mean that there shouldalso be a distinction in the theory of logical forms of conceptual natural realismbetween predicates that stand for concepts and predicates that stand for anatural property or relation. The whole point of the double significance of apredicate is that the same predicate can stand for both a concept in the primarysense and a natural property or relation in the secondary sense. Thus, it is notthat the same universal can “exist in a double way,” as Abelard assumed, firstin nature and then in the mind, but rather that the same predicate can standin a double way for both a concept and a natural property or relation—thoughit stands first for a concept, and then perhaps also for a natural property orrelation—but only in the sense of an hypothesis about nature, which might notalways be explicit, but only implicit.

3See Basti 2004.

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Now, just as a predicate constant can be taken to stand in double way bothfor a concept and a natural property or relation, so too a predicate variablecan be taken in a double way to have both concepts and natural properties andrelations as its values. The difference between the universals in the one order andthe universals in the other is reflected not in a difference between two “types”of predicate constants and variables—where the one “type” stands for conceptsand the other stands for natural properties and relations—but in the kind ofreference that is made by means of predicate quantifiers, i.e., the quantifiers thatcan be affixed to predicate variables and that determine the conditions underwhich a predicate constant can be substituted for a predicate variable. In thisway, the difference is reflected not in a difference of types of predicate variablesto which predicate quantifiers can be affixed, but in a difference between thepredicate quantifiers themselves, i.e., in the types of referential concepts thequantifiers stand for.

What we need to add to the second-order conceptualist theory of logicalforms described in chapter four, accordingly, are special quantifiers, ∀n and ∃n,that can be applied to predicate variables, and that, when so applied, can beused to refer to natural properties and relations. We assume, of course, that anatural property or relation is an existence-entailing property or relation, i.e.,that only existing objects—where, by existence we mean concrete existence—can be characterized by a natural property or stand in a natural relation. Inother words, where the monadic predicate E! stands for the concept of concreteexistence, the following is assumed as an axiom4:

(∀nF j)(∀x1)...(∀xj)[F (x1, ..., xj) → E!(x1) ∧ ... ∧ E!(xj)]. (N ⊆ E!)

The concept of concrete existence, we have said, is a logical construct, andhence there is no presumption that there is a natural property correspondingto it. In fact, as constructed within conceptual realism, it is an impredicativeconcept, because it is constructed, or formed, in terms of a totality to which itbelongs. That is, as defined earlier, to exist is to fall under an existence-entailingconcept, or, in symbols,

E!(x) ↔ (∃eF )F (x).

Now, because natural properties can be realized only by things that actuallyexist in nature, it might be thought that we could give a more specific kindof analysis in natural realism, and, in particular, that we could define concreteexistence as having a natural property, and therefore as being a constituent ofa fact, i.e., a state of affairs that obtains in the world:

E!(x) =df (∃nF )F (x). (E?)4A logical realist who wanted to distinguish natural properties and relations from properties

and relations in general might assume the simpler axiom

(∀nF )(∃eG)(F = G)

instead. In conceptual realism, however, concepts are values of predicate variables bound by∀ and ∃, whereas natural properties and relations are values of predicates bound by ∀n and∃n, and in this framework concepts are not the same type of entity as natural properties andrelations.

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Such an analysis will not suffice, however, because it is possible that someexisting objects might not have a natural property, but only stand to otherexisting objects in a natural relation. The point is that standing in a naturalrelation to other existing objects does not constitute having a natural property.Natural relations do not in general generate natural properties, and to claimotherwise is to confuse the conceptual order, where monadic concepts can beconstructed from relational concepts, with the natural order. Properties andrelations can be posited to exist in the natural order only as hypotheses aboutnature. This is part of what we mean when we said that conceptualism andnatural realism do not have the same overall logical structure.

Now Aristotle’s moderate realism as a form of natural realism can be statedas follows:

(∀nF j)(∃ex1)...(∃exj)F (x1, ..., xj), (MR)

where the quantifier ∃e is as described in the logic of actual (concrete) objects(in chapters two and six).

What the thesis of moderate realism, (MR), says is thatnatural properties and relations “exist” only as componentsof facts, i.e., the states of affairs that obtain in the world.It is in that sense that we say that a natural property orrelation “exists” only in things.

Unfortunately, this is much too restrictive a view of natural realism as ascientifically acceptable ontology. At the moment of the Big Bang, for example,when the universe was first formed, there was mostly raw energy and only veryfew elementary particles. There were no atoms or molecules of any kind, or atleast certainly not of any complex kind, all of which came later in the evolutionof the universe. Consequently, many of the natural properties and relations thatwe now know to characterize atoms and compounds as physical complexes werenot at that time realized in any objects at all.

But of course that does not mean that they did not then have a real modeof being within nature’s causal matrix even at the beginning of the universe.Indeed, even today there may yet be some transuranic elements, and naturalproperties of such, that, as a matter of contingent fact, will never be realized innature by any objects at all, but which, nevertheless, as a matter of a naturalor causal possibility, could be realized by atoms that are generated, e.g., in asupernova, or in a very high energy accelerator. The being of such a naturalproperty or relation does not consist of its being a property of some transuranicatom at the moment of the Big Bang, nor even, for that matter, of its beingin re at some time or other in the history of the universe. Instead, its beingconsists of its being part of nature’s causal matrix right from the beginning oftime, and therefore of its possibly being realized in nature, i.e., of its possiblybeing in re.

The being of a natural property or relation is its possiblybeing in re, i.e., its being realizable in nature as a matterof a natural or causal possibility.

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Now one obvious consequence of this view of natural properties and relationsis that they are not intensional objects, nor are they objects of any kind at all.This is so because if natural properties and relations were objects, then, in orderto be even when they are not in things, they would have to be abstract objects.How could they be, in other words, when they are not in things, unless their be-ing is that of an abstract object in a Platonic realm of forms, in which case theywould have a mode of being that transcended the natural world and nature’scausal matrix. But natural properties and relations do not exist independentlyof the world and its causal matrix, even though they are not contained withinthe space-time causal manifold the way concrete objects are. Natural proper-ties and relations cannot be objects, in other words, and therefore their modeof being as possibly being in re must have a different explanation. As universalsthat correspond to concepts as unsaturated cognitive capacities, the most plau-sible explanation is that they too have an unsaturated nature, albeit one thatis only analogous to, and not the same as, the unsaturated predicative natureof concepts.

As components of the nexus of predication in reality, which we can compre-hend only by analogy with the nexus of predication in thought, natural proper-ties and relations are unsaturated causally determinate structures that becomesaturated in the states of affairs that obtain in nature, and that otherwise “ex-ist” only within nature’s causal matrix. Thus, even though natural propertiesand relations do not “exist in a double way,” one in nature and the other in theintellect, nevertheless, they have a mode of being as unsaturated causal struc-tures that is analogous to that of concepts as unsaturated cognitive capacities,and hence their unsaturatedness must be understood by analogy with the un-saturated nature of concepts. In terms of the theory of logical forms of a formalontology, where predicates signify both kinds of universals, this means that bothkinds of universals are values of predicate variables, albeit variables bound bydifferent quantifiers, namely ∀n and ∃n in the case of natural properties andrelations, and ∀ and ∃ in the case of predicable concepts.

Finally, we should note that just as predicable concepts do not exist inde-pendently of the general capacity humans have for language and thought, so toonatural properties and relations do not exist independently of nature’s causalmatrix. That is why, just as the laws of compositionality for concept-formationof predicable concepts, as characterized by the comprehension principle, (CP∗

λ),can be said to characterize the logical structure of the intellect as the basis of thehuman capacity for language and thought, so too the laws of nature regardingthe causal connections between natural properties and relations, especially asdetermined by natural kinds, can be said to characterize the causal structure ofthe world. Thus, just as concepts have their being within the matrix of thoughtand concept-formation, so too natural properties and relations have their beingwithin the matrix of the laws of nature.

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12.3 Modal Moderate Realism

What is needed in the formal ontology of natural realism is a modal logic fora causal or natural necessity, or a causal or natural possibility. By a naturalpossibility we mean what is possible in nature, i.e., what is not precluded by thelaws of nature. A natural necessity therefore is what must be so because of thelaws of nature. This suggests that S5 is the appropriate modal logic for naturalnecessity; or to express the matter in model-theoretic terms, that the possibleworlds in the multiverse that have same laws of nature constitute an equivalenceclass.5 Different equivalence classes of the possible worlds in the multiverse willthen represent different causal matrices as determined by the laws of nature thatare invariant across the worlds in those equivalence classes. As is well-known,necessity, when interpreted as invariance over each equivalence class of a setof equivalence classes of models (“possible worlds”)—i.e., where each modelin any one such equivalence class is accessible from every other model in thatequivalence class—results in a completeness theorem for S5 modal logic.6 Oneversion of such a multiverse is the concordance model discussed in chapter three,§6.1, where the relation of accessibility is universal, and hence where there isbut one equivalence class of possible worlds.

By a causal possibility, on the other hand, we mean what can be broughtabout in nature through causal mechanisms of whatever natural sort, physical,biological, etc. A causal necessity then is what must be so because of its causalground, i.e., what caused it to be so. This suggests that S4 is the appropriatemodal logic to adopt, because whereas causal relations are transitive they arenot also symmetric, and, as is well-known, S4 is the modal logic characterizedby a transitive accessibility relation between possible worlds.7 Of course, we stillassume that all of the causally accessible worlds will have the same laws of natureas our world, i.e., the causal relation does not take us to worlds that violate thelaws of our universe. The many-worlds interpretation of quantum mechanics,as described in chapter three, §6.2, provides an example of a multiverse thatvalidates S4 in this way.

We will not attempt to decide here whether the appropriate modal logic forconceptual natural realism is S4 or S5. Instead, we will leave that decision tothe different variants of this ontology that might be developed. These variantsmight differ not only in respect of which modal logic is adopted, but also inwhether the first-order logic of the variant is possibilist or actualist, and alsowhether it allows for only a constructive conceptualism or a more comprehensiveholistic conceptualism. However, because S4 is a proper part of S5, we will useS4 here without assuming that S5 is thereby precluded. In regard to notation,

5For an account of the different worlds in the multiverse, or megaverse, see Kaku 2005.6See Cocchiarella 1986, chapter III, §7, for an axiomatization and completeness theorem

of natural realism based on this interpretation.7A model-theoretic approach to a causal modality would be based on an extension of the

notion of a causally extended system of world lines as described in chapter two for the causaltenses. At each node of such a causally connected system we would have not only the actualspace-time points of our universe that can be reached by a light signal, but also space-timepoints that are causally possible at that node as well.

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we will use �c for causal necessity and ♦c for causal possibility.Now, instead of the ontological thesis of moderate realism (MR), we have

the following ontological thesis of modal moderate realism as a fundamentalprinciple of natural realism:

(∀nF j)♦c(∃e

x1)...(∃exj)F (x1, ..., xj). (MMR)

Natural properties and relations “exist” not as components of actual facts, inother words, as was stipulated in the thesis of moderate realism, but as thenexuses of possible states of affairs. It is in this sense that the being of anatural property or relation is its possibly being in re.

There is no general comprehension principle that is valid in natural real-ism, incidentally, the way that the comprehension principle (CP∗

λ) is valid forconceptual realism. Natural properties and relations are not formed, or con-structed, out of other properties and relations by logical operations. But thisdoes not mean that no natural property or relation can be specified in termsof a complex formula, i.e., a formula in which logical constants occur. What itdoes mean is that such a specification cannot be validated on logical groundsalone, but must be taken as a contingent hypothesis about the world.

In order to consider specifying natural properties and relations in termsof complex formulas, it is convenient to have some abbreviatory notation. Inparticular, we can adopt some useful abbreviatory notation that simulates nom-inalizing predicates as objectual terms. We adopt for this purpose the followingnotation, which simulates a kind of identity between natural properties or rela-tions8:

(F j ≡c Gj) =df �c(∀x1)�c...�c(∀xj)�c[F (x1, ..., xj) ↔ G(x1, ..., xj)]

We say that ≡c represents an “identity” between natural properties and rela-tions because, unlike concepts, natural properties and relations are “identical”when, as matter of causal necessity, they are coextensive. As part of the causalstructure of the world, natural properties and relations retain their “identity”as natural properties and relations across all causally accessible worlds.

As part of nature’s causal matrix, natural properties andrelations are “identical” when, as a matter of causal neces-sity, they are co-extensive.

Now the assumption that there is a natural property or relation correspond-ing to a given predicable concept that is represented by a complex formula ϕ,and hence by the λ-abstract [λx1...xjφ], can be formulated as follows9:

(∃nF j)�c ([λx1...xjφ] ≡c F ) .8Only the initial occurrence of �c is needed here if the presumed modal logic is S5. The

additional occurrences are needed if the logic is S4.9With S5, instead of S4, this axiom can be stated simply as

(∃nF j) ([λx1...xjφ] ≡c F ) .

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Here, we note again, that, unlike the comprehension principle of conceptualrealism, such an assumption is at best only a scientific hypothesis, and as suchmust in principle be subject to confirmation or falsification. In this regard, thereis no comprehension principle valid in natural realism other than the trivial onestipulating that every natural property or (j-ary) relation is a value of the bound(j-ary) predicate variables, i.e.10,

(∀nF j)�c(∃nGj)(F ≡c G).

12.4 Aristotelian Essentialism

Conceptual natural realism without natural kinds might be an adequate ontolog-ical framework for some philosophers of science; but to others, especially thosewho fall in the tradition of Aristotle and Aquinas, it is only part of a larger,more interesting ontology of Aristotelian essentialism. This is a framework thatis a part of cosmology as well as of ontology. It is part of cosmology because it isbased on natural kinds as causal structures, and it is part of ontology in that itdetermines two types of predication in reality, essential and accidental. Naturalkinds—whether in the form of species or genera, and whether of natural kindsof “things,” such as plants and animals, or natural kinds of “stuff”, such as thechemical substances gold, oxygen, iron, etc., or compound substances such aswater, salt, bronze, etc.—are the bases of essential predication, whereas predi-cable concepts and natural properties and relations are the bases of accidental,or contingent, predication.

The basic assumption of this extension of natural realism is that in additionto the natural properties and relations that may correspond to some, but notall, of our predicable concepts, there are also natural kinds that may correspondto some, but not all, of our common-name concepts—especially those that aresortals, i.e., name concepts that have identity criteria associated with their use.By a natural kind we understand here a type of causal structure, or mechanismin nature, that is the basis of the powers or capacities to act, behave, function,etc., in certain determinate ways that objects belonging to that natural kindhave. Natural kinds, in fact, are the causal structures, or mechanisms in nature,that underlie the causal modalities, and in particular they underlie the naturallaws regarding the different natural kinds of things there are, or can be, inthe world. In this ontology, natural kinds are an essential part of the internalhierarchical network of nature’s causal matrix, and in fact they constitute themore stable nodes of that hierarchical network.

Now a natural kind is not a natural property or a “conjunction of naturalproperties,” as David Armstrong and other philosophers have claimed.11 In-stead of being a “conjunction of natural properties,” a natural kind is a typeof unsaturated causal structure that, when saturated, is the causal ground of

10See Appendix 1 of this chapter for a complete axiom set for a strict actualist version ofnatural realism. Again the first �c can be dropped if the modal logic is S5.

11Cf. Armstrong 1978, chapter 15. In my first paper on natural kinds, Cocchiarella 1976, Idid take natural kinds to be properties, but later corrected that view in Cocchiarella 1996.

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the events and states of affairs containing the natural properties that are saidto “conjunctively define” that natural kind. Indeed, if a natural kind were aconjunction of natural properties, then we would need an explanation of whysome conjunctions result in a natural kind whereas others do not. Why, in otherwords, do not all “conjunctions of natural properties” result in a natural kindif some do?

If certain “conjunctions of natural properties” were to “produce,” “gener-ate,” or result in a natural kind, whereas others do not, then that would sug-gest that there is more to a natural kind than just a “conjunction of naturalproperties.” In fact, the ontological dependence is just the opposite of whatthe conjunction thesis maintains, because instead of a “conjunction of naturalproperties” being the causal ground of a natural kind, it is the natural kind thatis the causal ground of the natural properties in the “conjunction”. Moreover,there really are no “conjunctions of properties” in nature, but only causallyrelated groups of events or states of affairs having those properties as predi-cable components, which, of course, we could in principle described in termsof a conjunction of sentences. In other words, as a causal structure, a naturalkind has an ontological priority over the natural properties that are predicatedof the objects of that kind, a priority that is part of what Aristotle means indescribing natural kinds as secondary substances.12 Neither natural properties,nor so-called “conjunctions of natural properties,” on the other hand, can bedescribed as secondary substances.

Now as the causal ground of natural properties, a natural kind has a“substance-like” structure in that it is unsaturated in a way that is comple-mentary to the unsaturated predicative structures of natural properties andrelations. The nexus of predication in reality, in other words, is a kind of mu-tual saturation of a “substance-like” natural-kind structure, as realized by anobject (or primary substance) of that kind, with a natural property or relationas a predicative structure, the result being an event or state of affairs that ob-tains in reality. Of course, the fact that natural kinds are unsaturated causalstructures to begin with allows for there being natural kinds that in fact arenot realized in nature at a given time but that could be realized, or broughtabout, in appropriate environmental circumstances. The transuranic elementsof atomic numbers 113 and 115, for example, have only recently been realizedin nature, even though for just a few fractions of a second.13

A natural kind, as an unsaturated “substance-like” causalstructure, has its being in possibly being realized in things,and in that regard natural kinds can be realized in natureat different times in the evolution of the universe, or evenpossibly not at all.

It is because the being of a natural kind is its possibly being realized in na-ture, that Aristotle’s problem of the fixity of species can be resolved in modal

12Cf. Categories 2a11.13Cf. Science News, February 7, 2004, vol. 165, no. 6, p. 84.

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moderate realism.The ontological difference between natural kinds and natural properties and

relations is analogous to the conceptual difference between common-name con-cepts and predicable concepts and the way that referential concepts based onthe former may be saturated by the latter in speech and mental acts. Thus,just as a referential concept that is based upon a common-name concept can besaturated by a predicable concept in a speech or mental act, so too a naturalkind, as the causal structure of an object of that kind, can be saturated by thenatural properties and relations of that object, the result being a complex ofevents or states of affairs having that object as a constituent.

Accordingly, just as a predicate expression can signify both a predicableconcept and a natural property or relation, a common name can also signify orstand in a double way for both a concept and a natural kind as a causal structure.Similarly, name variables can also be given a double interpretation as well. Thus,just as the quantifiers ∀n and ∃n can be affixed to predicate variables and enableus to refer to natural properties and relations, so too we can introduce specialquantifiers ∀k and ∃k, which, when affixed to name variables, enable us to referto natural kinds. For convenience, we will assume that objectual quantifiersrange over all objects, single or plural, abstract or concrete, and actual or merelypossible in nature. We also assume all the distinctions we have made in previouschapters, including those that are about classes as many and membership in aclass as many. Thus, e.g., where A is a common name, then x ∈ A, i.e., xbelongs to A-kind, if, and only if, x is an A, i.e.,

x ∈ A↔ (∃yA)(x = y).

Similarly, for complex common names, e.g., A/F (y), we have

x ∈ [yA/F (y)] ↔ (∃yA/F (y))(x = y).

That is, x belongs to the A-kind that are F if, and only if, x is an A that is F .Now because names can be transformed into objectual terms, we can state

the fact that every natural kind A is not just contingently a natural kind, butthat as a node in the network of nature’s causal matrix it is necessarily so, i.e.,

(∀kA)�c(∃kB)(A = B). (K1)

Of course, that a common name A is co-extensive with a natural kind B, i.e.,

(∃kB)(A = B),

does not mean that A is itself a natural kind.14 For example, assuming thatthe common name Man stands for a natural kind, but that the common name‘featherless biped’, i.e., Biped/Featheless, does not, then even though all and

14The unrestricted form of Leibniz’s law applies, as we noted in chapter eleven, only tosingle objects. Where A and B are common names (or common variables), what A = Bmeans is only that A and B are co-extensive, i.e., (∀x)[x ∈ A↔ x ∈ B].

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only men are featherless bipeds (or so we will assume), i.e., even though it isnow true that

Man = [xBiped/featherless(x)],

nevertheless it does not follow that being a featherless biped is a natural kind.That Man is a natural kind, incidentally, can be formulated as

(∃kB)�c(Man = B).

So-called “real definitions” can be described in terms of this notation bymeans of a specification of the following form:

(∃kB)�c(B = [xA/ϕx]),

where A is a natural kind genus, and B is specified as a species of A the membersof which satisfy the condition ϕx. This would not be a “nominal definition,” i.e.,a matter of introducing a simple common name as an abbreviation of a morecomplex common name; instead, it would be an hypothesis about the world,namely that there is a natural kind corresponding to the complex commonname [xA/ϕx].

A “real definition” is not a definition after all; rather, it isan hypothesis that a complex common name [xA/ϕx] namesa natural kind, and in particular a species of a genus A.

Now there are a number of interesting laws of natural kinds that can beformulated in this formal ontology. For example, one such law is:

An object belongs to a natural kind only if being of thatkind is essential to it, i.e., only if it must belong to thatkind whenever it exists as a real, concrete object.

With E! as the predicate for concrete existence, this principle is formulated asfollows:

(∀kA)(∀xA)�c [E!(x) → x ∈ A] . (K2)

In other words, where A is a natural kind, i.e., (∃kB)�c(A = B), and x is an A,i.e., x ∈ A, then x is an A whenever x exists in any causally possible world, i.e.,x is essentially an A. This is the most natural way to state this principle, butit can be assumed in this form only in the modal logic S5. In S4, the principleneeds to be formulated in the following somewhat stronger form, from which(K2) follows15:

(∀kA)(∀x)(♦c(x ∈ A) → �c [E!(x) → x ∈ A]).

15Because x ∈ A → ♦c(x ∈ A) is provable in S4, then (∀kA)(∀x)(x ∈ A →�c [E!(x) → x ∈ A]), is a consequence the S4 version of the principle, from which(∀kA)(∀xA)�c [E!(x) → x ∈ A], i.e., (K2), follows.

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12.4. ARISTOTELIAN ESSENTIALISM 285

Essential predication, which is represented here by the open formula,

�c [E!(x) → x ∈ A]

is of course one of Aristotle’s two types of predication. It can be formu-lated, of course, as a predicate, namely, as the λ-abstract [λx�c(E!(x) → x ∈A)](x). Accidental, or contingent, predication, is represented simply as either[λx(∃yA)(x = y)](x) or F (x). Thus, given that Socrates is a teacher, but onlycontingently so, then this “accidental” predication is represented as follows:

(∃xSocrates)[λx(∃yT eacher)(x = y)](x).

Similarly, the accidental, or contingent, predication that Socrates speaks Greekcan be symbolized as follows

(∃xSocrates)F (x),

where the predicate ‘speaks Greek’ is represented by the predicate constantF . Thus, as these examples illustrate, we have a natural and intuitive way torepresent Aristotle’s two types of predication in our formal ontology.

Another law of our logic is that the being of a natural kind, like that ofnatural properties, is its possibly being realized in nature.

(∀kA)♦c(∃ex)(x ∈ A). (K3)

The quantifier phrase ‘(∃ex)’ (‘there exists’) in (K3) can be replaced by themore general phrase ‘(∃x)’ (‘there is’), because we assume that only concreteexistents belong to natural kinds. The following principle, in other words, isalso an axiom of our logic of natural kinds.

(∀kA)(∀x)[x ∈ A→ E!(x)] (K4)

If we adopt the following abbreviatory notation for the subordination, andproper subordination, of one kind to another,

A ≤ B =df �c(∀x)[x ∈ A→ x ∈ B],A < B =df (A ≤ B) ∧ ¬(B ≤ A),

then the partition principle for natural kinds can be stated as follows:

(∀kA)(∀kB)[♦c(∃x)(x ∈ A ∧ x ∈ B) → A ≤ B ∨B ≤ A]. (K5)

In other words, if an object can belong to two natural kinds, then either theyare the same kind or one is subordinate to the other. Or, more simply, if twonatural kinds are not necessarily disjoint, then one must be subordinate tothe other. This means that the family of natural kinds to which any objectmay belong forms a chain of subordination of one natural kind to another—where each natural kind in the chain is, as it were, a template structure that is

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causally more determinate and finer-grained than the natural kinds to which itis subordinate.

An important consequence of (K2) and (K5) is the thesis that an objectcan belong to two natural kinds only if, as a matter of causal necessity, it belongsto the one kind when and only when it belongs to the other:

(∀kA)(∀kB)(∀x)[♦c(x ∈ A) ∧ ♦c(x ∈ B) → �c(x ∈ A↔ x ∈ B)].

Another version of the partition principle is a consequence of (K5), namelythat

natural kinds that are subordinate to the same immedi-ate genus are either identical or necessarily disjoint:

(∀kA)(∀kB)(∀kC)(A < C ∧B < C ∧ (∀kD)[A < D → C ≤ D] ∧(∀kD)[B < D → C ≤ D] → �c(A = B) ∨ ¬♦c(∃x)[x ∈ A ∧ x ∈ B])

Still yet another partition principle is the thesis that every genus is the sumof its species:

(∀kA)[(∃kB)(B < A) → �c(A = [x/(∃kB)(x ∈ B ∧B < A)])] (K6)

In terms of this view of natural kinds as template causal structures that canfit one within another, it is only natural to assume a summum genus principleto the effect that any chain of subordination between natural kinds must have asummum genus as an ultimate, superordinate template structure within whichall of the natural kinds of that chain must fit. It is only in this way that theindividuation of natural kinds of objects can even begin to take place in theuniverse as an ontological process.

Formally, the summum genus principle can be stated as follows:

(∀kA)(∀x)[x ∈ A→ (∃kB)(x ∈ B ∧ (∀kC)[x ∈ C → C ≤ B])]. (K7)

Thus, any object that belongs to a natural kind belongs, according to thisthesis, to a natural kind that is a summum genus—that is, a natural kind thathas subordinate to it every natural kind to which that object belongs. Giventhe partition principle, (K5), (K7) is equivalent to the following alternativeway of stating the summum genus principle, namely, that

every natural kind is subordinate to a natural kind that isproperly subordinate to no other natural kind:

(∀kA)(∃kB)[A ≤ B ∧ ¬(∃kC)(B < C)].

The opposite of a summum genus as the ultimate, superordinate causaltemplate structure of a natural kind of object is the infima species of thatobject. This is the finest grained template structure determining the causalnature of that object.

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12.5. GENERAL VERSUS INDIVIDUAL ESSENCES 287

The infima species principle stipulates that if an object belongs to a naturalkind, then it belongs to a natural kind that is subordinate to all of the naturalkinds to which that object belongs:

(∀kA)(∀x)(x ∈ A→ (∃kB)[x ∈ B ∧ (∀C)(x ∈ C → B ≤ C)]). (K8)

A consequence of (K8) is the following alternative version of the infima speciesprinciple, namely, that

every natural kind has subordinate to it a natural kind towhich no other natural kind is properly subordinate:

(∀kA)(∃kB)[B ≤ A ∧ ¬(∃kC)(C < B)].

12.5 General Versus Individual Essences

Aristotelian essentialism is an ontology of general essences and not of individualessences, such as are described by Alvin Plantinga in The Nature of Necessity.Unlike a general essence that can be common to many objects, an individualessence is unique to just one thing. Also, individual essences are properties,whereas general essences are natural kinds, and natural kinds are not properties,nor, as we have said, are they “conjunctions of properties”.

Thus, according to Plantinga, “to be an essence of Socrates, a property mustbe such that nothing else could have had it.16 That is,

“E is an essence of Socrates if and only if E is essential to Socratesand there is no possible world in which there is an object distinctfrom Socrates that has E.”17

Socrates, according to Plantinga, clearly has such an essence, namely “Socrate-ity, the property of being Socrates or being identical with Socrates is such aproperty.”18

Now although there is the predicable concept of being identical with Socrates,nevertheless, there is no natural property in Aristotelian essentialism that cor-responds to that concept. Nor is there a natural kind that corresponds to thename concept that ‘Socrates’ stands for. In other words, there is no such essenceas Socrateity, which is not to say that Socrates does not have an essential natureas a human being—assuming, of course, that being a man, i.e., a human being,is a natural kind that Socrates shares with all other humans.

As a form of conceptual natural realism, Aristotelian essentialism is an on-tology of general essences, not of individual essences. Of course, one might raisethe question of whether it is possible that each natural kind of object, i.e., eachobject that belongs to a natural kind, might itself constitute a natural kind that

16Plantinga 1974, p. 70.17Ibid.18Ibid., pp. 71–73.

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was both subordinate to the infima species to which that object belongs and yetso unique to that object as to distinguish it from all other objects belonging tothe same infima species. Could Socrates, for example, or perhaps his unique ge-netic structure, constitute such a natural kind, i.e., a natural kind that in effectamounted to an individual essence? Formally, this could be stated as follows:

(∃kA)[�c(E!(Socartes) → Socrates = A) ∧ �c(∀xA)(x = Socrates)].

That is, there is a natural kind A such that Socrates, and only Socrates, belongsto A (and hence is identical to A) in any world in which Socrates exists, and ifanything is A in a possible world, then it is Socrates, in which case, by (K4),it is a world in which Socrates exists. Could this be what Aristotle meansin Metaphysics Z6 (1032a4–6) when he says that each primary substance isidentical with its essence?

The notion of a natural kind that is also an individual essence seems du-bious in the case of natural kinds of objects in the microphysical world, suchas electrons, assuming, of course that there is a natural kind corresponding tothe common name Electron. How could a single electron have an individualessence that distinguished it from every other electron not only in this worldbut in every other possible world in which that electron exists? If that were so,would it not be like each point on a continuous line having its own individualessence, i.e., an essence that distinguished it from every other point on the line?Doesn’t the same doubt apply to all objects in the microphysical world? Andmight one not doubt that it applies to each grain of sand in the Sahara desertas well?

Could it be that only certain natural kinds of physical objects have an in-dividual essence, e.g., complex physical objects such as plants and animals?But then would a clone of Socrates, were such an individual actually broughtabout, have the same individual essence as Socrates? What scientific considera-tions would settle this question? That is, how could one determine by scientificinvestigations that Socrates had, or was, an individual essence?

These are matters we cannot settle here. In any case, the general frameworkof natural kinds as general essences is already a sufficiently strong and usefulformal ontology even without individual essences.

12.6 Summary and Concluding Remarks

• Conceptualism and natural realism presuppose each other as part of themore general ontology of conceptual natural realism.

• Conceptualism is based on a socio-biological theory of the human capacityfor language, culture and thought, and therefore presupposes some form of nat-ural realism as the causal ground of that capacity. As a formal ontology, naturalrealism in turn presupposes conceptualism as a framework in which we can ar-ticulate and form theories of the world and conjecture about natural propertiesand relations as part of the causal order.

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12.6. SUMMARY AND CONCLUDING REMARKS 289

• The connection between conceptualism and natural realism was implicitin Aristotle’s moderate realism as well as in Abelard’s more restricted view inhis Glosses on Porphyry. The difficulty for both was the so-called problem ofthe “double existence” of universals.

• It was Aquinas who resolved the problem of the “double existence” ofuniversals by noting that it is not an ontological problem but only a questionof explaining how the same predicate can stand for, or signify, a concept in themind on the one hand and a natural property in nature on the other. The twoare not really the same universal and do not even have the same mode of being.

• The sense in which a predicate stands for a concept is primary becauseit is the concept that determines the functional role of the predicate and theconditions under which it can be correctly used in a speech or mental act. Itis only by assuming as a result of empirical evidence that the truth conditionsdetermined by the concept have a causal ground based on a natural propertyor relation that we can then say that the predicate also stands for a naturalproperty or relation.

• The same predicate expression can stand in a double way for both a conceptand a natural property or relation. A predicate variable can be taken similarlyin a double way to have both concepts and natural properties and relations asits values.

• The difference in reference to concepts as opposed to natural propertiesand relations is represented not by a difference of types of predicate variables towhich predicate quantifiers can be affixed but by a difference between predicatequantifiers.

• Moderate realism, which maintains that natural properties and relations“exist” only in things as components of facts is too restrictive a view of naturalrealism. This is because natural properties and relations have a mode of beingwithin nature’s causal matrix and in that sense “exist” even when they arenot realized in things, e.g., at the moment of the Big Bang when the universeconsisted only of energy in the form of plasma.

• Instead of moderate realism, natural realism should be based on a modalmoderate realism. This is because the being of a natural property or relationis its possibly being in re, i.e., its being realizable in nature as a matter of anatural or causal possibility.

• Natural realism can be based on either a natural necessity and possibilityor a causal necessity and possibility. The modal logic S5 is the appropriatemodal logic for natural necessity, and the modal logic S4 is appropriate forcausal necessity.

• As part of nature’s causal matrix, natural properties and relations are“identical” when, as a matter of causal, or natural, necessity, they are co-extensive.

• Natural realism can be developed with or without being extended to Aris-totelian essentialism and an ontology of natural kinds.

• Aristotelian essentialism is part of cosmology as well as of ontology. It ispart of cosmology because it is based on natural kinds as causal structures, andit is part of ontology because it is characterized by two types of predication in

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reality, essential and accidental.• In addition to the natural properties and relations that may correspond

to some, but not all, of our predicable concepts, in Aristotelian essentialismthere are also natural kinds that may correspond to some, but not all, of ourcommon-name concepts.

• A natural kind is neither a natural property nor a conjunction of naturalproperties.

• A natural kind is a type of unsaturated causal structure that when sat-urated is the causal ground of the events and states of affairs containing thenatural properties that are said to “conjunctively define” that natural kind.

• A natural kind has a “substance-like” structure that is unsaturated in away that is complementary to the unsaturated predicative structures of naturalproperties and relations.

• It is because the being of a natural kind is its possibly being realized innature, that Aristotle’s problem of the fixity of species can be resolved in modalmoderate realism.

• A “real definition” is not a definition but an hypotheses that a certaincomplex common name names a natural kind, and in particular a species or agenus.

• An object belongs to a natural kind only if being of that kind is essential toit, i.e., only if it must belong to that kind whenever it exists as a real, concreteobject.

• Essential predication can be distinguished from accidental, or contingent,predication within the logic of natural kinds.

• Different principles for natural kinds, such as a partition principle, a sum-mum genus principle, an infima species principle, etc., are formulated in thelogic of natural kinds.

• The logic of natural kinds represents an ontology of general essences, notof individual essences.

• Unlike a general essence that can be common to many objects, an individ-ual essence is unique to just one thing.

• Whether it is possible that each object that belongs to a natural kind mightitself constitute a natural kind that was both subordinate to the infima speciesto which that object belongs and yet so unique to that object as to distinguishit from all other objects belonging to the same infima species is left open formacrophysical objects.

• But the notion of a natural kind that is also an individual essence is dubiousat best in the case of natural kinds of objects in the microphysical world.

12.7 Appendix 1

A strict actualist version of natural realism that is independent of conceptualrealism can be described axiomatically as follows:

(NR1a) (∀ex)(ϕ→ ψ) → [(∀ex)ϕ→ (∀ex)ψ],(NR1b) (∀nF )(ϕ→ ψ) → [(∀nF )ϕ→ (∀nF )ψ],

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12.8. APPENDIX 2 291

(NR2a) ϕ→ (∀ex)ϕ, if x is not free in ϕ,(NR2b) ϕ→ (∀nF )ϕ, if F is not free in ϕ,(NR3) (∀ex)(∃ey)(x = y),(NR4) ♦c(∃ex)(x = x),(NR5) �c(∃x)(x = x),(NR6) (∀n

F j)(∀x1)...(∀xj)[F (x1, ..., xj) → E!(x1) ∧ ... ∧ E!(xj)],(NR7) (∀nGj)�c(∃nF j)(G ≡c F ),(NR8) x = x,(NR9) (x = y) → (ϕ↔ ψ), where ψ is obtained from ϕ by replacing

one or more free occurrences of xby free occurrences of y,

(NR10) (∀nF j)♦c(∃x1)...(∃xj)F (x1, ..., xj),(NR11) �cϕ→ ϕ,(NR12) �cϕ→ �c�cϕ,(NR13) �c(ϕ→ ψ) → (�cϕ→ �cψ).

As inference rules we have modus ponens (MP) and universal generalization(UG) with respect to both object and predicate variables. We also need thefollowing rule of modal generalization:

(MG) If � ϕ, then � �cϕ,

and for actualist quantifiers a rule regarded embedded universal generalizationwithin the scope of iterated occurrences of �c:

(�UG) If � ϕ1 → �c(ϕ2 → ... → �c(ϕn → �cψ)...) and x is not freein ϕ1, ..., ϕn, then

� ϕ1 → �c(ϕ2 → ...→ �c(ϕn → �c(∀x)ψ)...).

For a strict actualist natural realism based on the modal logic S5 we needonly replace axiom (NR12) with (♦ϕ→ �♦ϕ). Axioms (NR3) and (NR4) repre-sent the strict actualist view that only concrete objects are values of the objectvariables, i.e., that there can be only concrete objects, and that there is a con-crete object in every possible world, i.e., that there can be no empty possibleworlds.

A set-theoretic semantics with respect to which the above axiom set for thisstrict actualist version of natural realism can be shown to be complete can befound in Cocchiarella 1986, Chapter 3, §§5–6.

12.8 Appendix 2

We describe in this appendix an axiomatization of the logic of natural kindsbased on the simple logic of names as described in chapter ten, §2 (includingthe λ-operator complex predicates), but extended to include the logic of actualobjects as well, and with possibilist quantifiers ranging over objects in general,whether causally possible or not. Without quantification over predicable con-cepts, the predicate E! is understood as defined in terms of the actualist quan-tifier ∃e, and ≡c is assumed to be defined as in this chapter. The grammar is

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extended to also include predicate variables, the quantifier ∀n for quantificationover natural properties and relations, and the quantifier ∀k for quantificationover natural kinds. The quantifiers ∃n and ∃k are assumed to be defined asabbreviatory devices of the metalanguage. The specific axioms for possibilistnatural realism to be added to the simple logic of names are as follows:

(NR1) (∀nF )[ϕ→ ψ[→ [(∀nF )ϕ→ (∀nF )ψ],(NR2) ϕ→ (∀nF ), if F is not free in ϕ,(NR3) (∀nF k)[F (x1, ..., xk) → E!(x1) ∧ ... ∧ E!(xn)],(NR4) (∀nGk)�c(∃nF k)�c(∀x1)�c...�c(∀xk)�c(F ≡c G),(NR5) (∀nF k)♦c(∃x1)...(∃xk)F (x1, ..., xk),(NR6) ♦c(x = y) → �c(x = y),(S41) �ϕ→ ϕ,(S42) �(ϕ→ ψ) → (�ϕ→ �ψ),(S43) �ϕ→ ��ϕ.The inference rules are modus ponens, universal generalization with respect

to ∀ and ∀n, and the rule of necessitation.The possibilist logic of natural kinds is obtained by extending the above logic

of natural realism by adding the quantifier ∀k as a logical constant (with (∃kA)defined as ¬(∀kA)¬). We add to the above axioms the following for naturalkinds19:

(K1) (∀kA)�c(∃kB)([λx(∃yA)(x = y)] ≡c [λx(∃yB)(x = y)]),(K2) (∀kA)(∀x)(♦c(∃yA)(x = y) → �c[E!(x) → (∃yA)(x = y)]),(K3) (∀kA)♦c(∃x)(∃yA)(x = y),(K4) (∀kA)�c(∀xA)(∃eyA)(x = y),(K5) (∀kA)(∀kB)[♦c(∃x)((∃yA)(x = y)∧(∃yB)(x = y)) → A ≤ B∨B ≤

A], partition principle,(K6) (∀kA)[(∃kB)(B < A) → (A ≡c [λx(∃kB)((∃yB)(x = y) ∧ B <

A)])],(K7) (∀kA)(∀x)[(∃yA)(x = y) → (∃kB)((∃yB)(x = y)∧(∀kC)[(∃yC)(x =

y) → C ≤ B])],(K8) (∀kA)(∀x)((∃yA)(x = y) → (∃kB)[(∃yB)(x = y)∧(∀C)((∃yC)(x =

y) → B ≤ C)]).

The only new rule needed is universal generalization with respect to ∀k. Ifthe modal logic is extended to S5, then (K2) can be formulated as

(∀kA)(∀xA)�c [E!(x) → x ∈ A] .

If the logic is extended to include the logic of classes as many, then the formulas(∃yA)(x = y), (∃yB)(x = y), (∃yC)(x = y) in the above axioms can be replacedx ∈ A, x ∈ B, x ∈ C, respectively.

19If the logic is further extended to include the logic of classes as many, then axiom (K1)can be formulated as in §4 above, i.e., as

(∀kA)�c(∃kB)(A = B).

In axioms (K3)–(K8) the formulas of the form (∃yA)(x = y), (∃yB)(x = y), (∃yC)(x = y)can be replaced by x ∈ A, x ∈ B, x ∈ C, respectively.

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Afterword on Truth-Makers

We have made much of the distinction between two levels of analysis in ourdevelopment of conceptual realism, one level having to do with the cognitivestructure of our speech and mental acts and the other having to do with alogically perspicuous representation of the truth conditions of those acts, andtherefore of their logical consequences as well. This distinction is relevant to theissue of a logic of events and states of affairs as truthmakers, which, because ofthe added complexity involved, we have not gone into in this book. However,because this is an area of research that is important to further development andapplication of conceptual realism as a formal ontology, we will say a little aboutthis subject here in this afterword.

The role of events in truth conditions has been an important part of logicalanalysis ever since Donald Davidson first argued that action verbs carry a hiddenargument place for events.20 For example, according to Davidson, the logicalform of an assertion of ‘Shem kicked Shawn’ is not as the traditional analysiswould have it—namely, Kick(a, b), where the names ‘Shem’ and ‘Shawn’ arerepresented by individual parameters a and b, and the verb ‘kick’ by a two-place predicate Kick.21 Nor is it as our conceptualist view of activated anddeactivated referential concepts would have it, namely,

(∃xShem)[λxKick(x, [∃yShawn])](x),

where the grammatical subject, ‘Shem’ (a proper name sortal), stands for thereferential concept that is activated (exercised) in the assertion—and which,presumedly, is used with existential presupposition, and therefore is representedby (∃xShem)—and ‘Shawn’, the direct object of the sentence, stands for a deac-tivated, and hence nominalized occurrence of (∃yShawn), which is symbolizedhere by [∃yShawn].22

Note that on both accounts the transitive verb ‘kick’ is interpreted as atwo-place predicate but that the traditional account fails to account for thecognitive structure of the assertion. For Davidson, the verb ‘kick’ contains ahidden argument place for events, which means that it should be represented by

20See Davidson 1967.21For convenience we ignore the representation of tense by means of tense operators through-

out this afterword.22See chapter seven, §§6-7, for an account of deactivated (nominalized) referential expres-

sions.

293

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294 AFTERWORD ON TRUTH-MAKERS

a three-place predicate, say, KickD, and that the logical form of the assertionin question is given by,

(∃e)KickD(e, a, b),

which is read as, ‘There is an event e such that e is a kicking of Shawn byShem’.23 This is not just an equivalent way of saying that Shem kicked Shawn,according to Davidson, but in fact it is the only logical form of the assertion inquestion. Logical forms are not designed to give perspicuous representations ofthe cognitive structures of our speech acts, in other words, but only of the truthconditions of those acts.

The truth conditions determined by the relational predicable concept thatthe verb ‘kick’ stands for, we agree, do involve events of a certain sort—namely,kicking events, with, in this particular case, Shem as agent and Shawn as object,or what linguists call the theme of the event. Indeed, incorporating a logic ofevents and adding thematic role concepts (such as agent, theme, experiencer,source, goal, etc.) for characterizing the different participants of events indicatesan important way in which our conceptualist theory of logical form might bedeveloped.24 Thus, for example, assuming (as Davidson does), that action verbsstand for sorts of events that can be represented by a common name, e.g.,Kicking, we can describe the truth conditions of an assertion of ‘Shem kickedShawn’ as follows:

(∃eKicking)(∃xShem)(∃yShawn)[Agent(x, e) ∧ Theme(y, e)],

where the assumption that a kicking is a sort of event is made explicit by thefollowing meaning postulate:

(∀z) [(∃eKicking)(z = e) → (∃eEvent)(z = e)] .

What is important for us to keep in mind here is that a logically perspicuousrepresentation of truth conditions is not necessarily the same as a logicallyperspicuous representation of the cognitive structure of our speech and mentalacts, and that logical forms for the latter purpose are no less essential to atheory of logical form than are those that represent only the truth conditionsof those acts. They are two different kinds of representations, serving differentpurposes, and a theory of logical form should include both kinds along with therelevant transformational rules for showing the connection between them. Thiscan be done to some extent by means of meaning postulates. Thus, for example,given the following meaning postulate for the two-place predicate Kick (as thesymbol for the transitive verb ‘kick’),

Kick(x, y) ↔ (∃eKicking)[Agent(x, e)∧ Theme(y, e)],23Davidson 1967, p. 92.24See Parsons 1990 and Dowty 1989 for a description and explanation of various thematic

roles. Dowty, incidentally, thinks of the thematic role analysis as an alternative and competitorto the traditional theory of logical forms, and not, as we suggest, as a supplement to thelatter—i.e. as a supplement when the traditional theory is extended to include an account ofthe cognitive structure of our speech and mental acts and not just of the truth conditions ofthose acts.

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and a meaning postulate stipulating that Kick is extensional in its second ar-gument place, e.g.,

[λxKick(x, [∃y S])] = [λx(∃y S)Kick(x, y)],

it then follows (by λ-conversion and elementary transformations) that the abovelogical form representing the cognitive structure of an assertion of ‘Shem kickedShawn’ is equivalent to the logical form representing the truth conditions of thatassertion, i.e.,

(∃xShem)[λxKick(x, [∃yShawn])](x)] ↔(∃eKicking)(∃xShem)(∃yShawn)[Agent(x, e) ∧ Theme(y, e)],

is then provable.A more complex example given by Davidson involves adverbs and preposi-

tions, which on his analysis become predicates describing some aspect of theevent in question. Davidson gives the following sentence as an example:

Jones butters the toast with a knife in the bathroom.

Where a is a term for ‘the toast’ as theme, b is a term for ‘the bathroom’ (forthe thematic role of location), and With for the thematic role of instrument,the analysis of the statement would be as follows:

(∃xJones)(∃eButters)(∃yKnife)[Agent(x, e) ∧ Theme(a, x) ∧ Location(e, b) ∧With(e, y)].

This analysis, however, while it specifies the event and truth conditions forthe speech act in question, it does not in any sense represent the cognitivestructure of that speech act. What the latter requires is an extension of ourframework to include a logic of predicate modifiers, i.e., operators that apply topredicates to generate another predicate, which, as in the case of the prepositions‘in’ and ‘with’ might increase the degree or adicity of the original predicate, i.e.,the number of argument positions of the predicate.25 Thus, beginning withthe two-place predicate Butters(x, y), we apply the predicate modifier In toget the three-place predicate In(Butters)(x, y, z), and then apply the predicatemodifier With to get the four place predicate With(In(Butters))(x, y, z, w),which is read as ‘x butters y in z with w’. The logical form that respects thecognitive structure of the assertion in question then becomes:

(∃xJones)[λxWith(In(Butters))(x, [∃1Toast], [∃1Bathroom], [∃wKnife])](x).

Meaning postulates stipulating that buttering is an event and that the pred-icates generated by the predicate modifiers In and With are extensional in thenew argument positions, etc., will then result in an equivalence between thislogical form and the above logical form referring to a buttering event. We willnot go into those details here, however.

25See, e.g., Montague 1970a, Clark 1970, 1986, Parsons 1970, and Bennett 1988 for anaccount of predicate modifiers.

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296 AFTERWORD ON TRUTH-MAKERS

It is clear that a fuller account of the connection between the logical formsthat represent the cognitive structures of our speech and mental acts and thelogical forms that represent the truth conditions of those acts in terms of eventsand states of affairs requires adding a category of predicate modifiers for therepresentation of prepositional phrases as well as the development of a logic ofevents and states of affairs.26 We leave these topics to further research andperhaps another book in the future.

26States of affairs, unlike events, do not occur. Yet they are needed for certain kindsstatements, such as ‘Socrates is a Greek philosopher’, whose truth conditions do not involveevents. Nevertheless, the logic of states of affairs is similar to that for events.

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Bibliography

[1] Aczel, Peter, 1988, Non-Well-Founded Sets, CSLI, Stanford.

[2] Adams, Marilyn McCord, 1977 “Ockham’s Nominalism and Unreal Enti-ties,” Philosophical Review 86 (1977): 144-76.

[3] Adams, Robert M., 1974, “Theories of Actuality,” Nous 8: 211–231;reprinted in Loux 1979.

[4] Armstrong, David M., 1978, A Theory of Universals, Cambridge Univer-sity Press, Cambridge.

[5] Beth, E.W., 1960, “Extension and Intension,” Synthese 12: 375–379.

[6] Basti, Gianfranco, 2004, “Analogia, Ontologia Formale e Problema deiFondamenti,” in Analogia e Autorefrenza, eds., G. Basti and C. Testi,Mariaetta 1820, Milan-Genoa, 2004, pp. 159–236.

[7] van Benthem, J.F.A.K., 1983, The Logic of Time, D. Reidel PublishingCo., Dordrecht.

[8] Bealer, G., 1982, Quality and Concept, Oxford: Oxford University Press.

[9] Bennett, Jonathan, 1988, Events and their Names, Indianapolis, HackettPublishing Co.

[10] Bochenski, I.M., 1974, “Logic and Ontology,” Philosophy East and West24 (1974): 275–292.

[11] Bozon, Serge, 2004, “Why λHST∗ and HST∗λ do not solve the Russell-

Myhill paradox after all,” Proceedings of the Ninth ESSLLI Student Ses-sion, 2004.

[12] Brentano, Franz, 1933, The Theory of Categories, translated by R.Chisholm and N. Guterman, Nijhoff, the Hague, 1981.

[13] Burkhardt, Hans, and Barry Smith, 1991, eds., Handbook of Metaphysicsand Ontology, Philosophia Verlag, Munich.

297

Page 314: Formal Ontology and Conceptual Realism

298 BIBLIOGRAPHY

[14] Carnap, Rudolf, The Logical Structure of the World, (1928) translatedby R. George, Berkeley and Los Angeles: University of California Press,1967.

[15] Carnap, Rudolf, “Foundations of Logic and Mathematics,” in Interna-tional Encyclopedia of Unified Science, Vol. 1, University Chicago Press,1938.

[16] Carnap, Rudolf, 1946, “Modalities and Quantification”, Journal of Sym-bolic Logic 11: 33-64.

[17] Carnap, Rudolf, 1958, Introduction to Symbolic Logic and Its Applications,Dover Publications, NYC, 1958.

[18] Chisholm, Roderick M., 1960, Realism and the Background of Phe-nomenology, Glenco, Ill.: The Free Press.

[19] Church, Alonzo, 1956, Introduction to Mathematical Logic, Princeton Uni-versity Press, Princeton.

[20] Clark, Romane, 1970, “Concerning the Logic of Predicate Modifiers,”Nous 41: 311–335.

[21] Clark, Romane, 1986, “Predication and Paronymous Modifiers,” NotreDame Journal of Formal Logic 27: 376–392.

[22] Cocchiarella, Nino B., 1966, Tense Logic: A Study of Temporal Reference,Ph.D. dissertation, UCLA.

[23] Cocchiarella, Nino B., 1969, “A Completeness Theorem in Second OrderModal Logic,” Theoria, vol. 35: 81-103.

[24] Cocchiarella, Nino B., 1974, “Formal Ontology and the Foundations ofMathematics,” in Nakhnikian 1974, pp. 29–46.

[25] Cocchiarella, Nino B., 1975, “On the Primary and Secondary Semanticsof Logical Necessity,” Journal of Philosophical Logic 4.1:13–27.

[26] Cocchiarella, Nino B., 1976, “On the Logic of Natural Kinds,” Philosophyof Science 43: 202–222.

[27] Cocchiarella, Nino B., 1980, “Nominalism and Conceptualism as Predica-tive Second-order Theories of Predication,” Notre Dame Journal of FormalLogic, vol. 21: 481–500.

[28] Cocchiarella, Nino B., 1982, “Meinong Reconstructed versus Early RussellReconstructed,” Journal of Philosophical Logic, vol. 11 (1982), pp. 183-214.

[29] Cocchiarella, Nino B., 1984, “Philosophical Perspectives on Quantificationin Tense and Modal Logic,” in Handbook of Philosophical Logic, vol. 11,eds. Gabbay, D. and F. Guenthner, Dordrecht: Reidel, (1984), pp. 309-53.

Page 315: Formal Ontology and Conceptual Realism

BIBLIOGRAPHY 299

[30] Cocchiarella, Nino B., 1986, “Conceptualism, Ramified Logic, and Nomi-nalized Predicates,” Topoi, vol. 5, no. 1: 75-87.

[31] Cocchiarella, Nino B., 1986, Logical Investigations of Predication Theoryand the Problem of Universals, Naples: Bibliopolis.

[32] Cocchiarella, Nino B, 1987, Logical Studies in Early Analytic Philosophy,Columbus: Ohio State University Press.

[33] Cocchiarella, Nino B., 1988, “Predication Versus Membership in the Dis-tinction between Logic as Language and Logic as Calculus,” Synthese 75:37-72.

[34] Cocchiarella, Nino B., 1989, “Conceptualism, Realism, and IntensionalLogic”, Topoi 8: 15-34.

[35] Cocchiarella, Nino B, 1991, “Higher-Order Logics” in Handbook of Meta-physics and Ontology, H. Burkhardt and B. Smith, eds., Philosophia Ver-lag, Munich: 466–470.

[36] Cocchiarella, Nino B., 1992, “Cantor’s Power-Set Theorem Versus Frege’sDouble-Correlation Thesis,” History and Philosophy of Logic 13: 179-201.

[37] Cocchiarella, Nino B., 1996, “Conceptual Realism as a Formal Ontology,”in Formal Ontology, Roberto Poli and Peter Simons, eds., Kluwer Aca-demic Pub., Dordrecht, pp. 27–60.

[38] Cocchiarella, Nino B., 1998, “Reference in Conceptual Realism,” Synthese114, no. 2: 169–202.

[39] Cocchiarella, Nino B., 2001a, “A Conceptualist Interpretation ofLesniewski’s Ontology,” History and Philosophy of Logic 22: 29–43.

[40] Cocchiarella, Nino B., 2001b, “A Logical Reconstruction of Medieval Ter-minist Logic in Conceptual Realism,” Logical Analysis and History of Phi-losophy, vol. 4: 35–72.

[41] Cocchiarella, Nino B., 2002, “On the Logic of Classes as Many,” StudiaLogica 70, no. 3: 303–338.

[42] Cocchiarella, Nino B., 2005, “Denoting Concepts, Reference, and theLogic of Names, Classes as Many, Groups, and Plurals,” in Linguisticsand Philosophy, vol. .

[43] Cohen, Jonathan, 1954, “On the Project of a Universal Character,” Mind63: 49-63.

[44] Davidson, Donald, 1967, “The Logical Form of Action Sentences,” in TheLogic of Decision and Action, N. Rescher, ed., University of PittsburghPress, Pittsburgh, pp. 81-98.

Page 316: Formal Ontology and Conceptual Realism

300 BIBLIOGRAPHY

[45] De Witt, B. and N. Graham, eds., 1973, The Many-Worlds Interpretationof Quantum Mechanics, Princeton University Press, Princeton.

[46] Dowty, D.R., 1989, “On the Semantic Content of the Notion of ‘ThematicRole’,” in Properties, Types and Meaning, II, G. Chierchia, B.H. Partee,and R. Turner, eds., Kluwer Academic Publishers, Dordrecht, pp. 69-129.

[47] Dummett, Michael, 1976, “Is Logic Empirical?”, in Truth and Other Enig-mas, by M. Dummett, Harvard University Press, Cambridge, MA, 1978.

[48] Fodor, Jerry, 1993, Psychosemantics, MIT Press, Cambridge, MA.

[49] Frege, Gottlob 1893, Der Grundgesetze der Arithmetik, Pagination is tothe partial translation by M. Furth as The Basic Laws of Arithmetic,University of California Press, 1964.

[50] Frege, Gottlob 1979, Posthumous Writings, edited by H. Hermes, F. Kam-bartel, and F. Kaulbach, University of Chicago Press, Chicago.

[51] Frege, Gottlob, 1879, “Begriffsschrift, A Formula Language, ModeledUpon that of Arithmetic, for Pure Thought,” in From Frege to Goedel,J. van Heijenoort, ed., Cambridge: Harvard University Press, 1967.

[52] Frege, Gottlob, 1952, Translations From the Philosophical Writings ofGottlob Frege, P. Geach and M. Black, eds., Oxford, Blackwell.

[53] Frege, Gottlob, 1972, Conceptual Notation and Related Articles, edited byT.W. Bynum, Oxford University Press, Oxford, 1972.

[54] Gabbay, Dov M., 1976, Investigations in Modal and Tense Logics withApplications to Problems in Philosophy and Linguistics, D. Reidel, Dor-drecht.

[55] Gabbay, Dov, and Franz Guenthner, eds., Handbook of PhilosophicalLogic, 2nd. edition, vol. 7, Kluwere Academic Publishers, Dordrecht, 2002.

[56] Gallin, Daniel, 1975, Intensional and Higher-Order Logic, North HollandPress, Amsterdam.

[57] Geach, Peter, Mental Acts, 1957, The Humanities Press, N.Y.

[58] Geach, Peter, 1967, “Intentional Identity”, Journal of Philosophy 74(1967); reprinted in Logic Matters, University of California Press, 1980.

[59] Geach, Peter, 1980, Reference and Generality, 3rd. edition, Cornell Uni-versity Press, Ithaca.

[60] Goodman, Nelson, The Structure of Appearance, Cambridge: HarvardUniversity Press, 1951.

Page 317: Formal Ontology and Conceptual Realism

BIBLIOGRAPHY 301

[61] Goodman, Nelson, 1956, “A World of Individuals,” in J. M. Bochenski, A.Church and N. Goodman, eds., The Problem of Universals. A Symposium,Notre Dame, Ind., University of Notre Dame Press.

[62] Goodman, Nelson and W.V. Quine, “Steps Toward a Constructive Nom-inalism,” Journal of Symbolic Logic, vol. 12 (1947): 105–122.

[63] Guarino, Nicola, 1998, ed., Formal Ontology in Information Systems, Pro-ceedings of the First International Conference, IOS Press.

[64] van Heijenoort, J., 1967, “Logic as Language and Logic as Calculus,”Synthese 17 (1967): 324-30.

[65] Henry, D.P., Medieval Logic and Metaphysics, Hutchison University Li-brary, London, 1972.

[66] Hintikka, Jaakko, 1956, “Identity, Variables and Impredicative Defini-tions,” Journal of Symbolic Logic 21: 225–245.

[67] Hintikka, Jaakko, 1973, Time and Necessity, Studies in Aristotle’s Theoryof Modality, Oxford University Press, London.

[68] Holmes, R., 1999, Elementary Set Theory with a Universal Set, Cahiersdu Centre de Logique, Bruylant-Academia, Louvain-la-Neuve, Belgium.

[69] Humphreys, Paul W. and James H. Fetzer, 1998, The New Theory ofReference, Kluwer Academic Publishers, Dordrecht, 1998.

[70] Husserl, Edmund, 1900, Logical Investigations, Volumes I and II, trans-lated by J.N. Findlay, Routledge & Kegan Paul, N.Y. 1970.

[71] Husserl, Edmund, 1913, Ideas, General Introduction to Pure Phenomenol-ogy, translated by W.R. Boyce Gibson, Collier-Macmilan Ltd., London,1962.

[72] Husserl, Edmund, 1929, Formal and Transcendental Logic, translated byD. Cairns, Martinus Nijhoff, The Hague, 1969.

[73] Iwanus, B., 1973, “On Lesniewski’s Elementary Ontology,” Studia Logicaxxxi: 73-119. (Reprinted in Srzednicki et al 1984.)

[74] Kaku, Michio, 2005, Parallel Worlds, Doubleday, New York, 2005.

[75] Kalish, D. and R.M. Montague, 1965, “On Tarski’s Formalization of Predi-cate Logic with Identity,” Arch. fur Math. Logik und Grundl., vol. 7 (1965):61–79.

[76] Kanger, Stig, 1957, Provability in Logic, Univ. of Stockholm, 1957.

[77] Kenny, Anthony 1969, “Intellect and Imagination in Aquinas”, in Aquinas,A Collection of Critical Essays, ed., A. Kenny, Anchor Books, GardenCity, N.Y.

Page 318: Formal Ontology and Conceptual Realism

302 BIBLIOGRAPHY

[78] Knowlson, J., 1975 Universal Language Schemes in England and France,1600 -1800, Toronto: University of Toronto Press.

[79] Kripke, Saul, 1959, “A Completeness Theorem in Modal Logic,” Journalof Symbolic Logic, 24: 1–14.

[80] Kripke, Saul, 1962, “The Undecidability of Monadic Modal QuantificationTheory,” Zeitschrift fur Math. Logik und Grundlagen der Math, 8: 113–116.

[81] Kripke, Saul, 1971, “Identity and Necessity,” in M. Munitz, ed., Identityand Individuation, New York University Press, 1971.

[82] Kung G., and T. Canty, 1979, “Substitutional Quantification andLesniewskian Quantifiers,” Theoria, 36: 165–182.

[83] Lackey, Douglas, 1973, Essays in Analysis by Bertrand Russell, BrazillerPress, N.Y.

[84] Lambert, Karel, 1991,ed., Philosophical Applications of Free Logic, OxfordUniversity Press, New York and Oxford.

[85] Lewjewski, Czes�law, 1958, “On Lesniewski’s Ontology”, Ratio 1: 150–176.

[86] Lewis, David, 1973, Counterfactuals, Harvard University Press, Cam-bridge.

[87] Lorenz, Konrad, 1962, “Kant’s Doctrine of the A Priori in the Light ofContemporary Biology,” in L. van Bertanlanffy and A. Rapaport, eds.,General Systems, Yearbook for the Society for General Systems Research,vol. VII, and reprinted in Konrad Lorenz: the Man and his Ideas, by R.I.Evans, Harcourt Brace Jovanovich, N.Y., 1975, pp. 181–217.

[88] Loux, Michael J., 1979, The Possible and the Actual, Reading in the Meta-physics of Modality, Cornell University Press, Ithaca.

[89] Loux, Michael J., 1974, Ockham’s Theory of Terms, Part 1 of the SummaLogicae, University of Notre Dame Press.

[90] Nakhnikian, George, 1974, Bertrand Russell’s Philosophy, edited byGeorge Nakhnikian, Duckworth, London, 1974.

[91] McKay, T., 1975, “Essentialism in Quantified Modal Logic,” Journal ofPhilosophical Logic, 4: 423–438.

[92] Meinong, Alexius, 1904, “The Theory of Objects,” translated by I. Levi,D.B. Terrell, and R. Chisholm, in Realism and the Background of Phe-nomenology, R. Chisholm, editor, The Free Press of Glencoe, Ill., pp.76–117.

Page 319: Formal Ontology and Conceptual Realism

BIBLIOGRAPHY 303

[93] Montague, Richard M., 1970a. “English as a Formal Language,” reprintedin Formal Philsophy.

[94] Montague, Richard M., 1970b, “The Proper Treatment of Quantificationin Ordinary English,” reprinted in Formal Philosophy.

[95] Montague, Richard M., 1974, Formal Philosophy, edited by R. Thoma-son,Yale University Press, New Haven.

[96] Montague, Richard M., 1960, “Logical Necessity, Physical Necessity,Ethics and Quantifiers,” Inquiry, 4 (1960): 259–269’; reprinted in R.Thomason, ed., Formal Philosophy, Yale University Press, 1974.

[97] Nakhnikian, George, ed., 1974, Bertrand Russell’s Philosophy, Duck-worth, London.

[98] Normore, Calvin, 1985, “Buridan’s Ontology,” in How Things Are, eds.,J. Bogen and J.E. McGuire, D. Reidel Pub. Co., Dordrecht, pp. 189-203.

[99] Normore, Calvin, 1990, “Ockham on Mental,” in Historical Foundationsof Cognitive Science, J.C. Smith, ed., Kluwer Academic Pub., Dordrecht,1990, pp. 189-203.

[100] Parsons, Terence, 1969, “Essentialism and Quantified Modal Logic,”Philosophical Review 78: 35–52.

[101] Parsons, Terence, 1970, “Some Problems Concerning the Logic of Gram-matical Modifiers,” Synthese 21: 320–333.

[102] Parsons, Terence, 1980, Nonexistent Objects, Yale University Press, NewHaven.

[103] Parsons, Terence, 1990, Events in the Semantics of English, A Study ofSubatomic Semantics, MIT Press.

[104] Penrose, Roger, 2004, The Road to Reality, A Complete Guide to the Lawsof the Universe, Alfred A. Knopf, N.Y.

[105] Piaget, Jean, 1972, The Principles of Genetic Epistemology, Routledgeand Kegan Paul, N.Y.

[106] Piaget, Jean, 1977, The Development of Thought: Equilibration of Cogni-tive Structures, Viking Press, N.Y.

[107] Plantinga, Alvin, 1974, The Nature of Necessity, Oxford University Press,Oxford.

[108] Poincare, Henri, 1906, “Les Mathematiques et la Logique,” Revue deMetaphysique et de Moral 14: 17–34.

[109] Poli, Roberto, Ontologia Formale, Marietti, Genova, 1992.

Page 320: Formal Ontology and Conceptual Realism

304 BIBLIOGRAPHY

[110] Poli, Roberto, and Peter Simons, 1996, eds., Formal Ontology, KluwerAcademic Publishers, Dordrfecht.

[111] Popper, Karl, 1967, “Epistemology Without a Knowing Subject,” Proc. ofthe Third International Congress for Logic, Methodology and Philosophyof Science, reprinted in Objective Knowledge, Oxford at the ClarendonPress, London, 1975.

[112] Popper, Karl and J. Eccles: 1977, The Self and Its Brain, Routledge andKegan Paul.

[113] Prior, Arthur N. and Kit Fine, 1977, Worlds, Selves and Times, Duck-worth Press, London.

[114] Prior, Arthur N., 1967, Past, Present and Future, Oxford University Press,Oxford, 1967.

[115] Putnam, Hilary,1967, “Time and Physical Geometry,” The Journal of Phi-losophy, vol. LXIV, 8 (1967); reprinted in Punam’s Mathematics, Matterand Method, Philosophical Papers, vol. 1, Cambridge University Press,1975.

[116] Putnam, Hilary, 1969, “Is Logic Empirical?” in Boston Studies for thePhilosophy of Science, vol. V, R.S. Cohen and M. Wartovsky, pp. 216–41.

[117] Quine, W.V., 1943, “Notes on Existence and Necessity,” The Journal ofPhilosophy, vol. 40. : 113–127.

[118] Quine, W. V., 1947, “The Problem of Interpreting Modal Logic,” TheJournal of Symbolic Logic, vol. 12, no. 2: 43–48.

[119] Quine, W.V., 1953, “Three Grades of Modal Involvement,” Proc. of the11th International Congress of Philosophy, vol. 14: 65–81. Reprinted inQuine 1966.

[120] Quine, Willard V.O., 1960, Word and Object, MIT Press, Cambridge.

[121] Quine, Willard V.O., 1966, The Ways of Paradox, Random House, N.Y.,1966.

[122] Ramsey, F.P., 1960, The Foundation of Mathematics, edited by R.B.Braithwaite, Littlefield, Adams, Paterson, 1960.

[123] Rickey, V.F., 1977, “A Survey of Lesniewski’s Logic,” Studia Logica, 36:407–426).

[124] Russell, Bertrand, 1903, The Principles of Mathematics, second edition,Norton & Co., N.Y., 1938.

[125] Russell, Bertrand, 1906, “Les Paradoxes de la Logique,” Revue de Meta-physique et de Moral 14: 627–50.

Page 321: Formal Ontology and Conceptual Realism

BIBLIOGRAPHY 305

[126] Russell, Bertrand, 1907, “The Regressive Method of Discovering thePremises of Mathematics,” reprinted in Lackey 1973.

[127] Russell, Bertrand and Alfred N. Whitehead, 1910, Principia Mathematica,vol. I, Cambridge University Press, London.

[128] Russell, Bertrand, 1912, The Problems of Philosophy, Oxford UniversityPress, N.Y. and Toronto, 1912.

[129] Russell, Bertrand, 1914 Our Knowledge of the External World, London:George Allen & Unwin Ltd., reprinted 1952.

[130] Russell, Bertrand, Logic and Knowledge, 1956, R.C. Marsh, ed., London,George Allen & Unwin Ltd., 1956.

[131] Russell, Bertrand, 1919, Introduction to Mathematical Philosophy, GeorgeAllen & Unwin, LTD., London.

[132] Russell, Bertrand, 1959, My Philosophical Development, George Allen &Unwin, LTD., London.

[133] Saarinen, E., 1976, “Backwards-looking Operators in Tense Logic andNatural Language,” in J. Hintikka et al (eds), Essays on MathematicalLogic, Reidel, Dordrecht, 1976.

[134] Schein, Barry, 1993, Plurals and Events, MIT Press, Cambridge.

[135] Scott, Theodore K., 1966a, John Buridan: Sophisms on Meaning andTruth, Appleton-Century-Crofts, N.Y., 1966.

[136] Scott, Theodore K., 1966b, “Geach on Supposition Theory,” Mind 75:586-88, 1966.

[137] Sellars, Wilfrid F., 1963, “Grammar and Existence: A Preface to On-tology,” in Science, Perception and Reality, London, Routledge & KeganPaul.

[138] Sellars, Wilfrid, 1981, “Mental Events,” Philosophical Studies 39:325-45,1981.

[139] Sellars, Wilfrid, 1963, Science, Perception and Reality, Routledge & KeganPaul, London.

[140] Slupecki, J., 1955, “S. Lesniewski’s Calculus of Names,” Studia Logica,vol III: 7-72. (Reprinted in Srzednicki et al 1984.): 150-176.

[141] Smith Barry, ed., 1982. Parts and Moments: Studies in Logic and FormalOntology, Munich: Philosophia.

[142] Spade, Paul V., 1996, Thoughts, Words and Things: An Introduction toLate Medieval Logic and Semantic Theory, electronic version 1.0, availableat http://pvspade.com/Logic.

Page 322: Formal Ontology and Conceptual Realism

306 BIBLIOGRAPHY

[143] J. Srzednicki, V.F. Rickey, and J. Czelakowski, eds., 1984, Lesniewski’sSystems, Ontology and Mereology, Martinus Nijhoff Pub., The Hague.

[144] Stalnaker, Robert C., 1976, “Possible Worlds,” in Nous 10: 65–75;reprinted in Loux 1979.

[145] Tegmark, Max, 2003, “Parallel Universes,” in Science and Ultimate Real-ity: From Quantum to Cosmos, honoring John Wheeler’s 90th birthday,J.D. Barrow, P.C.W. Davies, & C.L. Harper, eds., Cambridge UniversityPress, Cambridge.

[146] Trentman, John, 1970, “Ockham on Mental,” Mind 79: 586-90, 1970.

[147] Van Themaat, P., 1962 “Formalized and Artificial Languages”, Synthese14: 320-326.

[148] Welty, Christopher and Barry Smith, 2001, eds., Formal Ontology andInformation Systems, Proceedings of the Second International Conference,ACM Press.

[149] Wittgenstein, Ludwig, 1961, Tractatus Logico-Philosophicus, Routledge &Kegan Paul, N.Y., 1961.

[150] Varzi, Achille and Laure Vieu, 2004, eds., Formal Ontology and Informa-tion Systems, Third International Conference, IOS Press, 2004.

[151] Vlach, F., 1973, ‘Now’ and ‘Then’: A Formal Study in the Logic of TenseAnaphora, Ph.D. dissertation, UCLA.

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Index

“All” as a plural quantifier, 257“That”-operator, 159

co-reference, 210“all” possible worlds, 61, 122“in the story”-operator, 163“standard” second-order predicate logic,

83“standard” set-theoretic semantics for

second-order logic, 83λ-conversion, 96�λHST∗ and HST∗

λ�, 123

a prori knowledge, 17logic of events and states of

affairs, 293

Abelard, xxii, 171, 274, 275, 289abstract facts

with universals as constituents, 115abstract nouns, 84abstract objects, 13, 110, 163

as evolutionarfy products of cul-ture and language, 110

as object-ifications of concepts, 110as products of cultural evolution,

161in conceptual realism, 153

abstract singular term, 152abstract singular terms

nominalized predicates, 22accessibility relation between possible

worlds, 61, 75accidental (contingent) predication

vs essential predication, 290accidental predication, 281

accidental/contingent predication, 285active vs deactivated referential con-

cepts, 201Actualism

two main theses, 125actualism, 124

possibilism, 26, 121Adams, M., 175, 177ampliation, 171Anscombe, E., 204Aquinas, xxii, 66, 274, 281, 289Aristotelian essentialism, xix, xxiii, 11,

22, 24, 60, 61, 281, 289essentialism, 66

Aristotle, xix, xxii, 3, 10, 14, 17, 23, 53,66, 121, 274, 281, 288, 289

natural kinds, 273temporal possibilism, 41, 45

Armstrong, D., 281ars combinatoria, 5, 23artificial intelligence

cognitive science, 18artificial vs natural languages, 172atomic states of affairs, 121

logical atomism, 61, 141atoms

Nelson Goodman’s notion of an atom,239

Barcan, R., 63Basti, G., xi, 275being

univocal vs multiple forms, 12being vs existence, xv, 26, 52

tense logic, 26

307

Page 324: Formal Ontology and Conceptual Realism

308 INDEX

Bell, J., 270Bennett, J., 295Bochenski, I.M., 3Boolean operations for classes as many

intersection, union, and comple-ments, 246

bound to objects, 107Bozon, S., 111Bradley, F.H.

infinite regress argument, 179Buridan, 170Burkhardt, H., 297Burley, 175Burley, Walter, 170

calculus ratiocinator, 5, 23Cantor’s power-set theorem, 248, 265Cantor, G., 19, 231, 248Canty, T, 218Carnap, R., 7, 50, 63, 67, 196Carnap-Barcan formula

Barcan, R.Carnap. R., 63

categorematic concepts, 170categorematic expressions

vs syncategorematic expressionsmedieval logic, 170

categorial analysis, 14Aristotle’s, 14Kant’s categorial analysis,

16, 23categories

ontologicalcategories of being, 3

categorycosmological

time, 39category of names, 235causal future, 49causal or natural modalities, 121causal past, 49causal possibility, 279causal signal relation, 39characteristica realis, 5, 23characteristica uiversalis, 23characteristica universalis, xiv, 4

Church, A., 222lambda-operator, 95

Clark, R., 295class as many, 235class as the extension of a property or

conceptvs sets, 104

classes as many, xxi, 215as the extensions of names, 231vs classes as ones, 167

co-reference, xxicognitive capacities, 87

two major types, 142cognitive schemata

characterizing our orientation intime, 48

represented by standard tense op-erators

represented by causal tense op-erators, 51

temporalmodal, 35

cognitive theory of predication, xviiicogntive equilibrium, 88common and proper names

the category of names, 145common name, 185, 188common names, 171, 255common supposition, 183complementarity of referential and pred-

icable concepts, 178complementary cognitive structures, 143completeness problems

completeness of cartegoriesdeductive completeness, 17

complex names, 146, 207, 219complex referential expressions, 146comprehension principle

for actualist quantifiers, 126computional theory of mind, 18concept of (concrete) existence

definedimpredicative, 125

concept-formation, 87idealized closure of logical opera-

tions, 87

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INDEX 309

conceptsas intersubjectively realizable cog-

nitive capacities, 108, 143conceptual intensional realism, xix, 22,

99conceptual natural realism, xix, 22, 175conceptual necessity and possibility

in conceptual realism, 133vs metaphysical necessity and pos-

sibility, 122conceptual Platonism

vs conceptual intensional realism,109

vs conceptual realism, 153conceptual Platonism

indirect Platonism vs direct Pla-tonism, 109

conceptual priority, 36conceptual realism, 116, 121, 177

realism, 75conceptual realism’s theory of reference,

117conceptualism, 11, 142, 172

and natural realism, 273intensional realism, 108of terminist logicians, 170

conceptualist theory of reference, 195concordance model of cosmology, 76confused and distributive supposition,

189, 190confused supposition, 183constructive conceptualism, 88counterpart theory, 165culture, 163cut-down on all possible worlds, 70, 75

secondarfy meaning of necessity“all” possible worlds, 61

Davidson, D., 293de dicto modality, 60de re modality, 59, 60, 66De Witt and Graham, 77deactivation, 151deactivation of concepts, xxidefinite descriptions

with and without existential pre-suppositions, 148

definition by abstraction, 7demonstrative “that”-opedrators, 185demonstrative phrases

existential presuppositions, 185denials

and negative judgmentsdeactivation of what is denied,

151descent to, or ascent from, singular propo-

sitionsin terminist logic, 184

determinate supposition, 183, 187Diodorus, 44, 53, 121discrete supposition, 183doctrine of supposition proper

identity theory of copula, 180doctrine of the modes of supposition,

183double existence of universals

Abelard, 289double reflexive abstraction, 166double reflexive abstraction , 154Doyle, Conan, 164

Einstein, A., 76theory of special relativity, 39

empty class, 265epistemic accessibility of natural num-

bers, 162epistemology

naturalized, 18Eriugena, John Scottus, 13Esperanto, 4essence-accident disrtinction

Aristotelian essentialism, 14essential predication, 281, 285

vs accidental predication, 290essentialism, 59events

in Ruissell’s 1910-13 ontology, 115Everett, H., 77exemplification, 13existence

actual existence vs being, 123

Page 326: Formal Ontology and Conceptual Realism

310 INDEX

as an impredicative concept, 133existence-entailing concepts and prop-

erties, 124existential presupposition, 147

of demonstrativesof definite description, 186

extensional logicFrege’s commitment, 104vs intensional logic, 109

extensionality axiom, 103, 242

fabels and myths, 163facts

abstract or physical are extensionalentities, 116

in Russell’s 1910-13 ontology, 115ficta as intensional objects

in medieval logic, 175fictional characters as intensional ob-

jects, 164fictional objects, xx

vs actual objects, 125fixity of species, 15formal ontology, xiii, 3

comparative, 12, 19, 21formal theories predication, 99free logic, 148free logic for predicate quantifiers, 89Frege’s basic law V, 105Frege’s Basic Laws Va and Vb, 118Frege’s double-correlation thesis, 163,

262Frege, G., xiv, xvii, 8, 84, 98, 109, 139,

161, 231, 262, 273Basic Law V, 105characteristica universalis, 6Russell’s paradox, 104

functionsin Frege’s ontology, 108

Gabbay, D., 300Gallin, D., 132Geach, P., xxi, 62, 147, 169, 196, 200,

213, 230on the grammar of Mental, 172

Geach-Kaplan sentence, 251

general essencesindividual essences, 66vs individual essences, 287, 290

general reference, xx, xxi, 147, 195, 213general vs individual essences, xxiiiGodel, K.

incompleteness theorem, 17incompletgeness theorem, 19second incompleteness theorem, 21

Goodman, N., 8, 12, 13, 239, 264nominalist dictum, 241

groupsand the semantics of plurals, 251

groups and cardinal numbers, 259Guenthner, F., 300

Henry, D.P., 215Heraclitus, 191Hintikka, J., 45, 67holistic conceptualism, 88, 94Holmes, R., 103homogeneous stratification, 102λHST∗, 103HST∗

λ, 107Hubble volume, 76Husserl, E., xiii, 16, 23hyperintensional logic, 111, 119hypertensionality paradox, 111, 119

idealized transition to a limit, 93, 99identity

indiscernibility, 126identity criteria, 171identity in constructive conceptualism,

90identity in Lesniewski’s logic, 217identity in nominalism, 90identity theory of the copula, 194

for categorical propositions, 169,180

imagesas concepts, 171

impossible objects, 165, 175impredicative

concept of existence, 125

Page 327: Formal Ontology and Conceptual Realism

INDEX 311

impredicative comprehension principle,82

impredicative definitions, 87, 99in nominalism vs constructive con-

cepotualism, 90incompleteness of fictional objects, 165incompleteness theorem

quantified modal logic, 69indiscernibility

in possibilism and actualism, 126individual concepts, 59, 60individual essences, 66

vs general essences, 287, 290individuals

vs plural objects, 239infima species priinciple, 287initial level of analysis

of speech and mental acts, 146vs deductive level of analysis, 157

intensional content, 163of a predicable concept, 109of a referential concept, 202

intensional content of a predicable con-cept, 152

intensional content of a referential con-cept, 153

intensional objects, 22, 111, 170, 175intensional possible worlds

as properties of propositions, 131as propositions, 129

intensional posssible worlds, xxintensional verbs, 193intensionality

Platonic vs conceptual, 35intentional content, 160intentional contexts

belief, desire, etc., 110, 116, 119intentional objects

ficta, 194intentionality

aboutness of a speech or mentalact, 201

the directedness of a speech or men-tal act, 142

internal relations, 128Iwanus, B., 218

judgment, 143

Kung, G., 218Kaku, M., 75, 279Kant, I., 16, 23

transcendental logic, 17Kenny, A., 274Kripke, S., 60, 62, 73, 74

lambda-conversion in free logic, 106Lamberrt of Auxerre, 170Lambert, K., 27Landini, G., xilaws of concept-formation, 93laws of nature, 75, 279Leibniz’s law

applies only to atoms (single ob-jects), 265

Leibniz, G., xiv, 4Lejewski, C., 216Lejewski, C. , 218Lesniewski’s logic of names, 215, 224,

248conceptualist logic of names, 233

Lesniewski’s ontology, 235Lesniewski

ontology, 22Lesniewski’s ontology, xxilevels of analysis

cognitive level vs deductive level,117

Lewis, D., 130counterpart theory, 164

lingua philosophica, 23local time, 48

Eigenzeit, 36, 38logic of predicate modifiers, 295logical atomism, xvi, 60, 113, 121, 273

as a form of natural realism, 141Russell’s version

reducing general reference to sin-gular reference, 196

logical categories, 23logical essentialism, 60logical forms that only represent truth

conditions, 151

Page 328: Formal Ontology and Conceptual Realism

312 INDEX

logical forms that represent cognitivestructure of speech or mentalact, 150

logical grammar, 96, 107logical modalities, xvilogical notion of class, 20logical objects

numbers, 104logical realism, xvii, 82, 99, 101, 109,

121, 139, 141conceptual intensional realism

hyperintensionality, 113Frege’s and Russell’s ontology, 116Frege’s variant, 113holistic conceptualism, 97modern form of Platonism, 273Russell’s 1910-13 ontology

not a simples+complexes ontol-ogy, 116

Russell’s early 1903 form, 102, 113Russell’s later 1q910-13 form, 114vs intensional realism, 95, 99

logical space, 64Lorenz, K.

biological Kantianiism, 18Loux, M., 175lower level of analysis

of deductive transformations, 146�Lukasiewicz, 216

Malatesta, M., ximany-worlds interpretation of quantum

mechanics, 76, 127McKay, T., 69meaning postulate

for extensional relation, 203meaning postulate for “That”-operator,

211meaning postulate for the “that”-operator,

159meaning postulates, 90

and Leibniz’s lawfor reactivating quantifier phrases,

160for proper names, 179

medieval logic, 169

medieval suppositio theory, 195medieval supposition theory, xximedieval terminist logic of 14th cen-

tury, 193Meinong, A., 165Mental

a tensed and modal languagemedieval logic, 170

mental acts, 87mental images

as particular mental occurrences,108

mental language, 170mental propositions

in medieval logicjudgments or thoughts, 170

merely confused personal supposition,178

merely confused supposition, 183medieval logic, 191

mereology, 13, 215, 216part-to-whole relation, 13

metaphysical essentialism, 60metaphysical modality, 122metaphysical necessity, 60

logical necessity, 74metaphysical necessity and possibility

in logical realism, 133metaphysical possible worlds

vs conceptually possible worlds, 133metaphysical vs physical possible world,

122metaphysics, 23methodological solipsism, 8mind-body problem, 10Minkowski, H., 49modal logical realism, xxmodal moderate realism, xxiii, 14, 23,

279, 289modal thesis of anti-essentialism, 60,

66modal-moderate realist theory of pred-

ication, 175moderate realism, xix, 23, 277, 289

moderate realismmodal moderate realism, 14

Page 329: Formal Ontology and Conceptual Realism

INDEX 313

Montague, R.M., 64, 132, 154, 156, 295multiverse, 75, 279mutual saturation

of referential and predicable con-cepts, 108

mutual saturation of complementary cog-nitive structures, 179

namesthe simple logic of names, xxi

names as a logical category, 145names, proper and common, 235naming vs referring, 179natural kinds, xviii, xxiii, 60, 66, 175,

281species and genera, 14

natural likeness between concepts andthings

in medieval logic, 171natural numbers, xxnatural numbers

as object-correlates of numericalquantifiers, 161

natural numbers and their ontology, 160natural possibility, 279natural properties and relations

as modes of configuration in statesof affairs

nexus of predication, 141natural property or relation, 84natural realism, 14, 66, 121, 141, 273

and conceptualism, xxiinaturalistic epistemology, 24necessity

logical, 15metaphysical

logical, 62natural, 15

necessity and possibilitylogical, xvi

metaphysical, 59negative judgment

internal negation, 145new theory of reference, 196nexus of predication, xx, 23, 141, 273

nexus of predication in conceptualism,142

NFU, 231NFU set theory, 103, 105, 244nominal definition

vs real definition, 284nominalism, xvii, 10, 84, 99, 139nominalistic dictum

Nelson Goodman, 241, 243, 265nominalization, 95, 110nominalized predicates

as abstract objectual terms, 95, 99Normore, C., 170, 174, 182now-operator, 43numbers

as cultural products, 110numerical quantifiers, 161

object-fication of conceptsintensional content, 109

object-ification of truth conditions, 153objectivity of concepts, 143, 177objects of fiction

as intensional objects, 167Ockham, xx, 170, 172, 173, 175, 179,

180, 188, 192, 193theory of ficta as intensional ob-

jects, 174two kinds of mental language, 176

Ockham’s intellectio, or mental-act, the-ory, 176

omnium, 76ontological categories, 23ontological logicism, 15, 95

Frege’s, 106ontological projection

based on nominalization, 110ontological soplipsism, 8ontology

a simples+complex ontology, 113Frege’s vs Russell’s, 107logical atomism, 61of fictional or mythological obe-

jcts, 125of natural numbers as logical ob-

jects, 109

Page 330: Formal Ontology and Conceptual Realism

314 INDEX

Plato’s ontology, 15Russell’s early 1903 form, 107

ontology of fiction, 163

Parsons, T., 295particle’s wave-function

collapse of a wave-function, 76partition principle for natural kinds, 285Pegasus, 124Penrose, R., 77personal supposition

in terminist logic, 178Peter of Spain, 170Piaget, J., 48, 88, 93

genetic epistemology, 18Plantinga, A., 66, 128, 287Plato, xix, 15, 95Platonic forms, 153Platonism, xixplural objects, 231

vs individual as single objects, 235plural operator, 265plural predication, 254, 255, 265plural reference, 251, 255, 265Poincare, 87Poincare-Russell vicious-circle principle,

87positive and negative facts

logical atomism, 61possibilia, 60possibilism, 121, 124

actualism, 26in medieval logic, 171

possible worldsaccessibility relation, 53, 59

possible worlds as propertiesthe ways things might have been,

130pragmatics, 199

vs semantics, 195predicable concepts

as unsaturated cognitive capacities,109

predicationin language and thought

concepts as cognitive capacities,108

preeminent being, 14vs Frege’s functionality, 108

predication in language, 142predication in thought, 142predication of cardinal numbers, 265predication theory, 81predication vs membership, 20principle of descent

medieval logic, 188principle of rigidity, 119, 132, 134Prior, A., 27, 38, 44, 52proper and common names

as parts of quantifier phrases, 169proper and common, xxi

proper names, 196as singular terms, 240proper, 62Russell’s view, 195

propositionconceptual, 35

propositionsas objective truth or falsehoods,

116in Russell’s early form of logical

realism, 139propositions as intensional objects, 163protothetic, 215Putnam, H., 48, 50

quality theory of conceptsmental acts as qualities of the mind

Ockham’s logic, 177quantities

numerical quantifier phrases, 161quantum mechanics, 18, 76Quine’s set theory NF, 244Quine, W., 12

the system NF, 231Quine, W.V.O., 59, 66, 229

ramified second-order predicate logicin nominalism vs in constructive

conceptualism, 92ramified type theory, 101

Page 331: Formal Ontology and Conceptual Realism

INDEX 315

ramified types, 85Ramsey, F.P., 63real definitions, 284

vs nominal definitions, 290real vs nominal definitions, 169realia, 51, 59realism, 11, 170Rees, M., 75reference

in conceptual realism, 169reference and predication, 21reference to past and future objects, 42referential concepts

vs predicable concepts, 108referential concepts and expressions

as quantifier phrases, 144reflective abstraction

projection of concepts, 93reflexive abstraction

abstract objects, 154and nominalization, 153representing what is not an object

as an object, 110reflexive pronouns

vs pronouns of laziness, 205relative clauses, 146relative pronouns, 209Remigius, 13Rickey, C.F., 216rigid designators, 59, 60

proper names, 62Rimini, Gregory, 170Roger Bacon, 170Romeo and Juliet in Flatland, 165Russell’s paradox, xiv, 89, 103, 105,

118of predication, 98

Russell’s paradox for classes, 238, 264Russell’s paradox of predication, 101Russell, B ., 113Russell, B., xiv, xvii, xxi, 61, 84,

86, 95, 98, 101, 104, 115,151, 195, 231, 232, 235, 239,258, 273

characteristica universalis, 6no-classes doctrine, 162

on denoting, 149Russell’s early form of logical re-

alism, 139theory of logical types, 15

Schein, B., 238Scott, T.K., 169, 175, 180, 183second-order predicate logic, 81Sellars, W., 172, 230semantics

internal vs external, 20sense-data, 7set theory, 81, 115, 244

external semantics, 23set-theoretical semantics, 20the iterative concept of set, 19theory of membership, 13vs predication, 81

Sherlock Holmesas a fictional character, 165

signal relation, 49signification, 174

semantic relation of medieval Men-tal language

narrow and wide, 170simple acts of naming

vs reference, 230simple logic of names, 219simple properties and relations, 113simple supposition

in terminist logic, 178simple theory of types

vs ramified theory of types, 102simple type theory

vs ramified type theory, 98simple+complexes ontology, 113singular reference, xx, xxi, 147

vs general reference, 195, 213singular terms, 169, 196Smith, B., 297sortal concepts, 22, 171space-time, 51

world-lines, 36Spade, P., 170, 173, 175, 177, 180, 183,

190, 192special relativity

Page 332: Formal Ontology and Conceptual Realism

316 INDEX

tense operators, 53special relativity theory, 39, 48special theory of relativity, xvispecies and genera, 66speech and mental acts, 108, 118

in conceptualism, 116stages of cognitive development, 88Stalnaker, R., 130stands for a concept, 172state of affairs, 48state-description semantics, 196subordination

Ockham, 172substitutional interpretation of predi-

cate quantifiers, 85, 90summum genus principle, 286suppositio theory

medieval, 169

Tegmark, M., 76temporal possibilism, 40tense logic, xvi, 25, 121terminist logic, xx, xxi, 174

medieval logic, 169the problem of the “double existence”

of universals, 274thematic roles, 294theories of universals, 23theory of types, 6, 215Transcendental logic, 17transcendental logic, 16, 23transcendental phenomenology, 17transcendental subjectivity, 108Trentman, J., 170, 173truthmakers, 293two levels of analysis, 156, 293two-name (identity) theory of the copula

in terminist logic, 180type theory, 118

unified account of both general and sin-gular reference, 166

unified account of general and singularreference, 214

unified encyclopedia of sciencecharacteristica realis, 8

unity of a proposition in Frege’s ontol-ogy, 140

unity of a proposition in Russell’s earlylogical realism, 139

unity of a speech or mental act, 166unity of speech and mental acts, 108unity of the nexus of predication, 139unity of thought, 87universal class, 19universal class as many, 244universal grammar, 18universal set, 231, 244universals, 82, 84, 139, 175

in Frege’s ontology, 104in Frege’s ontology vs in concep-

tualism, 108in natural realism, 141in predication, 115Nominalism

conceptualism and realism, 10theories

predication, 81universals, theory of, 83unsaturated function

Frege, G., 140unsaturatedness

Frege vs conceptualism, 107

value-ranges, 262vicious-circle principle, 93, 99vicious-circle principle , 93

wh-questions, 205William of Sherwood, 170Wittgenstein, L., 61, 62, 67, 141world properties, xxworld propositions, xxworld-line, 38, 49

Zermelo set theory, 103, 105Zermelo-Frankel set theory

ZF, 244

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16. G. Patzig, Aristotle’s Theory of the Syllogism. A Logical-philosophical Study of Book A of thePrior Analytics. Translated from German by J. Barnes. 1968 ISBN 90-277-0030-3

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62. K. Ajdukiewicz, Pragmatic Logic. Translated from Polish by O. Wojtasiewicz. 1974ISBN 90-277-0326-4

63. S. Stenlund (ed.), Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger onHis 50th Birthday. 1974 ISBN 90-277-0438-4

64. K. F. Schaffner and R. S. Cohen (eds.), PSA 1972. Proceedings of the Third Biennial Meeting ofthe Philosophy of Science Association. [Boston Studies in the Philosophy of Science, Vol. XX]1974 ISBN 90-277-0408-2; Pb 90-277-0409-0

65. H. E. Kyburg, Jr., The Logical Foundations of Statistical Inference. 1974ISBN 90-277-0330-2; Pb 90-277-0430-9

66. M. Grene, The Understanding of Nature. Essays in the Philosophy of Biology. [Boston Studiesin the Philosophy of Science, Vol. XXIII] 1974 ISBN 90-277-0462-7; Pb 90-277-0463-5

67. J. M. Broekman, Structuralism: Moscow, Prague, Paris. Translated from German. 1974ISBN 90-277-0478-3

68. N. Geschwind, Selected Papers on Language and the Brain. [Boston Studies in the Philosophyof Science, Vol. XVI] 1974 ISBN 90-277-0262-4; Pb 90-277-0263-2

69. R. Fraisse, Course of Mathematical Logic – Volume 2: Model Theory. Translated from French.1974 ISBN 90-277-0269-1; Pb 90-277-0510-0(For Volume 1 see under No. 54)

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70. A. Grzegorczyk, An Outline of Mathematical Logic. Fundamental Results and Notions explainedwith all Details. Translated from Polish. 1974 ISBN 90-277-0359-0; Pb 90-277-0447-3

71. F. von Kutschera, Philosophy of Language. 1975 ISBN 90-277-0591-772. J. Manninen and R. Tuomela (eds.), Essays on Explanation and Understanding. Studies in the

Foundations of Humanities and Social Sciences. 1976 ISBN 90-277-0592-573. J. Hintikka (ed.), Rudolf Carnap, Logical Empiricist. Materials and Perspectives. 1975

ISBN 90-277-0583-674. M. Capek (ed.), The Concepts of Space and Time. Their Structure and Their Development.

[Boston Studies in the Philosophy of Science, Vol. XXII] 1976ISBN 90-277-0355-8; Pb 90-277-0375-2

75. J. Hintikka and U. Remes, The Method of Analysis. Its Geometrical Origin and Its GeneralSignificance. [Boston Studies in the Philosophy of Science, Vol. XXV] 1974

ISBN 90-277-0532-1; Pb 90-277-0543-776. J. E. Murdoch and E. D. Sylla (eds.), The Cultural Context of Medieval Learning. [Boston

Studies in the Philosophy of Science, Vol. XXVI] 1975ISBN 90-277-0560-7; Pb 90-277-0587-9

77. S. Amsterdamski, Between Experience and Metaphysics. Philosophical Problems of the Evo-lution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975

ISBN 90-277-0568-2; Pb 90-277-0580-178. P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976

ISBN 90-277-0570-4; Pb 90-277-1200-X79. H. von Helmholtz: Epistemological Writings. The Paul Hertz / Moritz Schlick Centenary

Edition of 1921 with Notes and Commentary by the Editors. Newly translated from Germanby M. F. Lowe. Edited, with an Introduction and Bibliography, by R. S. Cohen and Y. Elkana.[Boston Studies in the Philosophy of Science, Vol. XXXVII] 1975

ISBN 90-277-0290-X; Pb 90-277-0582-880. J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975

ISBN 90-277-0584-4; Pb 90-277-0612-281. S. G. Harding (ed.), Can Theories Be Refuted? Essays on the Duhem-Quine Thesis. 1976

ISBN 90-277-0629-8; Pb 90-277-0630-182. S. Nowak, Methodology of Sociological Research. General Problems. 1977

ISBN 90-277-0486-483. J. Piaget, J.-B. Grize, A. Szeminsska and V. Bang, Epistemology and Psychology of Functions.

Translated from French. 1977 ISBN 90-277-0804-584. M. Grene and E. Mendelsohn (eds.), Topics in the Philosophy of Biology. [Boston Studies in

the Philosophy of Science, Vol. XXVII] 1976 ISBN 90-277-0595-X; Pb 90-277-0596-885. E. Fischbein, The Intuitive Sources of Probabilistic Thinking in Children. 1975

ISBN 90-277-0626-3; Pb 90-277-1190-986. E. W. Adams, The Logic of Conditionals. An Application of Probability to Deductive Logic.

1975 ISBN 90-277-0631-X87. M. Przełecki and R. Wojcicki (eds.), Twenty-Five Years of Logical Methodology in Poland.

Translated from Polish. 1976 ISBN 90-277-0601-888. J. Topolski, The Methodology of History. Translated from Polish by O. Wojtasiewicz. 1976

ISBN 90-277-0550-X89. A. Kasher (ed.), Language in Focus: Foundations, Methods and Systems. Essays dedicated to

Yehoshua Bar-Hillel. [Boston Studies in the Philosophy of Science, Vol. XLIII] 1976ISBN 90-277-0644-1; Pb 90-277-0645-X

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90. J. Hintikka, The Intentions of Intentionality and Other New Models for Modalities. 1975ISBN 90-277-0633-6; Pb 90-277-0634-4

91. W. Stegmuller, Collected Papers on Epistemology, Philosophy of Science and History ofPhilosophy. 2 Volumes. 1977 Set ISBN 90-277-0767-7

92. D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems inPhilosophy and Linguistics. 1976 ISBN 90-277-0656-5

93. R. J. Bogdan, Local Induction. 1976 ISBN 90-277-0649-294. S. Nowak, Understanding and Prediction. Essays in the Methodology of Social and Behavioral

Theories. 1976 ISBN 90-277-0558-5; Pb 90-277-1199-295. P. Mittelstaedt, Philosophical Problems of Modern Physics. [Boston Studies in the Philosophy

of Science, Vol. XVIII] 1976 ISBN 90-277-0285-3; Pb 90-277-0506-296. G. Holton and W. A. Blanpied (eds.), Science and Its Public: The Changing Relationship.

[Boston Studies in the Philosophy of Science, Vol. XXXIII] 1976ISBN 90-277-0657-3; Pb 90-277-0658-1

97. M. Brand and D. Walton (eds.), Action Theory. 1976 ISBN 90-277-0671-998. P. Gochet, Outline of a Nominalist Theory of Propositions. An Essay in the Theory of Meaning

and in the Philosophy of Logic. 1980 ISBN 90-277-1031-799. R. S. Cohen, P. K. Feyerabend, and M. W. Wartofsky (eds.), Essays in Memory of Imre Lakatos.

[Boston Studies in the Philosophy of Science, Vol. XXXIX] 1976ISBN 90-277-0654-9; Pb 90-277-0655-7

100. R. S. Cohen and J. J. Stachel (eds.), Selected Papers of Leon Rosenfield. [Boston Studies inthe Philosophy of Science, Vol. XXI] 1979 ISBN 90-277-0651-4; Pb 90-277-0652-2

101. R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds.), PSA 1974. Proceedingsof the 1974 Biennial Meeting of the Philosophy of Science Association. [Boston Studies in thePhilosophy of Science, Vol. XXXII] 1976 ISBN 90-277-0647-6; Pb 90-277-0648-4

102. Y. Fried and J. Agassi, Paranoia. A Study in Diagnosis. [Boston Studies in the Philosophy ofScience, Vol. L] 1976 ISBN 90-277-0704-9; Pb 90-277-0705-7

103. M. Przełecki, K. Szaniawski and R. Wojcicki (eds.), Formal Methods in the Methodology ofEmpirical Sciences. 1976 ISBN 90-277-0698-0

104. J. M. Vickers, Belief and Probability. 1976 ISBN 90-277-0744-8105. K. H. Wolff, Surrender and Catch. Experience and Inquiry Today. [Boston Studies in the

Philosophy of Science, Vol. LI] 1976 ISBN 90-277-0758-8; Pb 90-277-0765-0106. K. Kosık, Dialectics of the Concrete. A Study on Problems of Man and World. [Boston Studies

in the Philosophy of Science, Vol. LII] 1976 ISBN 90-277-0761-8; Pb 90-277-0764-2107. N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G. Hellman.

[Boston Studies in the Philosophy of Science, Vol. LIII] 1977ISBN 90-277-0773-1; Pb 90-277-0774-X

108. K. Ajdukiewicz, The Scientific World-Perspective and Other Essays, 1931-1963. Translatedfrom Polish. Edited and with an Introduction by J. Giedymin. 1978 ISBN 90-277-0527-5

109. R. L. Causey, Unity of Science. 1977 ISBN 90-277-0779-0110. R. E. Grandy, Advanced Logic for Applications. 1977 ISBN 90-277-0781-2111. R. P. McArthur, Tense Logic. 1976 ISBN 90-277-0697-2112. L. Lindahl, Position and Change. A Study in Law and Logic. Translated from Swedish by P.

Needham. 1977 ISBN 90-277-0787-1113. R. Tuomela, Dispositions. 1978 ISBN 90-277-0810-X114. H. A. Simon, Models of Discovery and Other Topics in the Methods of Science. [Boston Studies

in the Philosophy of Science, Vol. LIV] 1977 ISBN 90-277-0812-6; Pb 90-277-0858-4

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115. R. D. Rosenkrantz, Inference, Method and Decision. Towards a Bayesian Philosophy of Sci-ence. 1977 ISBN 90-277-0817-7; Pb90-277-0818-5

116. R. Tuomela, Human Action and Its Explanation. A Study on the Philosophical Foundations ofPsychology. 1977 ISBN 90-277-0824-X

117. M. Lazerowitz, The Language of Philosophy. Freud and Wittgenstein. [Boston Studies in thePhilosophy of Science, Vol. LV] 1977 ISBN 90-277-0826-6; Pb 90-277-0862-2

118. Not published 119. J. Pelc (ed.), Semiotics in Poland, 1894–1969. Translated from Polish.1979 ISBN 90-277-0811-8

120. I. Porn, Action Theory and Social Science. Some Formal Models. 1977 ISBN 90-277-0846-0121. J. Margolis, Persons and Mind. The Prospects of Nonreductive Materialism. [Boston Studies

in the Philosophy of Science, Vol. LVII] 1977 ISBN 90-277-0854-1; Pb 90-277-0863-0122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical

Logic. 1979 ISBN 90-277-0879-7123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation. 1978

ISBN 90-277-0882-7124. E. Saarinen, R. Hilpinen, I. Niiniluoto and M. P. Hintikka (eds.), Essays in Honour of Jaakko

Hintikka on the Occasion of His 50th Birthday. 1979 ISBN 90-277-0916-5125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston Studies

in the Philosophy of Science, Vol. LVIII] 1978 ISBN 90-277-0921-1; Pb 90-277-0922-X126. P. Mittelstaedt, Quantum Logic. 1978 ISBN 90-277-0925-4127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi.

1979 ISBN 90-277-0929-7128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of Machine

Theories of Memory. 1978 ISBN 90-277-0933-5129. M. W. Wartofsky, Models. Representation and the Scientific Understanding. [Boston Studies

in the Philosophy of Science, Vol. XLVIII] 1979 ISBN 90-277-0736-7; Pb 90-277-0947-5130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the Philosophy

of Science, Vol. XXIV] 1979 ISBN 90-277-0953-X; Pb 90-277-0954-8131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston Studies in

the Philosophy of Science, Vol. XXIX] 1979 ISBN 90-277-0723-5; Pb 90-277-0956-4132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist. 1979 ISBN 90-277-0958-0133. P. Bieri, R.-P. Horstmann and L. Kruger (eds.), Transcendental Arguments in Science. Essays

in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5134. M. Markovic and G. Petrovic (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodol-

ogy of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXXVI] 1979ISBN 90-277-0727-8; Pb 90-277-0968-8

135. R. Wojcicki, Topics in the Formal Methodology of Empirical Sciences. Translated from Polish.1979 ISBN 90-277-1004-X

136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science. [BostonStudies in the Philosophy of Science, Vol. LIX] 1979

ISBN 90-277-0994-7; Pb 90-277-0995-5137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism. 1980

ISBN 90-277-1046-5138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor of Virgil

C. Aldrich. 1979 ISBN 90-277-1013-9139. L. Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the Marxian

Idea of Science. 1980 ISBN 90-277-1014-7

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140. C. Perelman, The New Rhetoric and the Humanities. Essays on Rhetoric and Its Applications.Translated from French and German. With an Introduction by H. Zyskind. 1979

ISBN 90-277-1018-X; Pb 90-277-1019-8141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979

ISBN 90-277-1020-2142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning. Translated

from French and German. With an Introduction by H.J. Berman. 1980ISBN 90-277-1089-9; Pb 90-277-1090-2

143. S. Kanger and S. Ohman (eds.), Philosophy and Grammar. Papers on the Occasion of theQuincentennial of Uppsala University. 1981 ISBN 90-277-1091-0

144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. 1980ISBN 90-277-1096-1

145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics andGalileo’s Methodology. Proceedings of the 1978 Pisa Conference on the History and Philosophyof Science, Volume I. 1981 ISBN 90-277-1126-7

146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Thermodynamics,and the Interaction of the History and Philosophy of Science. Proceedings of the 1978 PisaConference on the History and Philosophy of Science, Volume II. 1981 ISBN 90-277-1127-5

147. U. Monnich (ed.), Aspects of Philosophical Logic. Some Logical Forays into Central Notionsof Linguistics and Philosophy. 1981 ISBN 90-277-1201-8

148. D. M. Gabbay, Semantical Investigations in Heyting’s Intuitionistic Logic. 1981ISBN 90-277-1202-6

149. E. Agazzi (ed.), Modern Logic – A Survey. Historical, Philosophical, and Mathematical Aspectsof Modern Logic and Its Applications. 1981 ISBN 90-277-1137-2

150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for Explanatory Principlesbelow the Level of Physics. 1981 ISBN 90-277-1214-X

151. J. C. Pitt, Pictures, Images, and Conceptual Change. An Analysis of Wilfrid Sellars’ Philosophyof Science. 1981 ISBN 90-277-1276-X; Pb 90-277-1277-8

152. R. Hilpinen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Foundations ofEthics. 1981 ISBN 90-277-1278-6; Pb 90-277-1346-4

153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the Relation betweenSuccessive Scientific Theories. 3rd rev. ed., 1994 ISBN 0-7923-2487-0; Pb 0-7923-2488-9

154. D. Woodruff Smith and R. McIntyre, Husserl and Intentionality. A Study of Mind, Meaning,and Language. 1982 ISBN 90-277-1392-8; Pb 90-277-1730-3

155. R. J. Nelson, The Logic of Mind. 2nd. ed., 1989 ISBN 90-277-2819-4; Pb 90-277-2822-4156. J. F. A. K. van Benthem, The Logic of Time. A Model-Theoretic Investigation into the Varieties

of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 1991 ISBN 0-7923-1081-0157. R. Swinburne (ed.), Space, Time and Causality. 1983 ISBN 90-277-1437-1158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz.

1983 ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3159. T. Chapman, Time: A Philosophical Analysis. 1982 ISBN 90-277-1465-7160. E. N. Zalta, Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983

ISBN 90-277-1474-6161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on Episte-

mology, Metaphysics, Methodology, and Philosophy of Science. 1983ISBN 90-277-1496-7; Pb 90-277-1538-6

162. M. A. Stewart (ed.), Law, Morality and Rights. 1983 ISBN 90-277-1519-X

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163. D. Mayr and G. Sussmann (eds.), Space, Time, and Mechanics. Basic Structures of a PhysicalTheory. 1983 ISBN 90-277-1525-4

164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I: Elements ofClassical Logic. 1983 ISBN 90-277-1542-4

165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II: Extensions ofClassical Logic. 1984 ISBN 90-277-1604-8

166. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. III: Alternative toClassical Logic. 1986 ISBN 90-277-1605-6

167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV: Topics in thePhilosophy of Language. 1989 ISBN 90-277-1606-4

168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic. 1983ISBN 90-277-1543-2

169. M. Fitting, Proof Methods for Modal and Intuitionistic Logics. 1983 ISBN 90-277-1573-4170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984

ISBN 90-277-1574-2171. R. Tuomela, A Theory of Social Action. 1984 ISBN 90-277-1703-6172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical Analysis in Latin

America. 1984 ISBN 90-277-1749-4173. P. Ziff, Epistemic Analysis. A Coherence Theory of Knowledge. 1984

ISBN 90-277-1751-7174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984

ISBN 90-277-1773-7175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure, Examples,

Philosophical Problems. 1984 ISBN 90-277-1811-3176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory of Legal Science. Proceedings of

the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983).1984 ISBN 90-277-1834-2

177. I. Niiniluoto, Is Science Progressive? 1984 ISBN 90-277-1835-0178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative Perspective.

Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Refer-ence. 1985 ISBN 90-277-1870-9

179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985 ISBN 90-277-1894-6180. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and

Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2182. M. Detlefsen, Hilbert’s Program. An Essay on Mathematical Instrumentalism. 1986

ISBN 90-277-2151-3183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human Affairs. Studies in Honor

of Chaim Perelman. 1986 ISBN 90-277-2255-2184. H. Zandvoort, Models of Scientific Development and the Case of Nuclear Magnetic Resonance.

1986 ISBN 90-277-2351-6185. I. Niiniluoto, Truthlikeness. 1987 ISBN 90-277-2354-0186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The Structuralist

Program. 1987 ISBN 90-277-2403-2187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8188. L. M. Vaina (ed.), Matters of Intelligence. Conceptual Structures in Cognitive Neuroscience.

1987 ISBN 90-277-2460-1

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189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological Relativism. 1987ISBN 90-277-2469-5

190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm Program, with aComplete Evolutionary Epistemology Bibliograph. 1987 ISBN 90-277-2582-9

191. J. Kmita, Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon, with an

Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical Studies of

Scientific Change. 1988 ISBN 90-277-2608-6194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988 ISBN 90-277-2640-X195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights and New

Perspectives on Their Relation. 1988 ISBN 90-277-2645-0196. J. Osterberg, Self and Others. A Study of Ethical Egoism. 1988 ISBN 90-277-2648-5197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence, Cognitive

Science, and Philosophy. 1988 ISBN 90-277-2711-2198. J. Wolenski, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X199. R. Wojcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988

ISBN 90-277-2785-6200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic.

Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2808-9202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 ISBN 90-277-2814-3203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical Knowledge. 1989

ISBN 0-7923-0131-5204. A. Melnick, Space, Time, and Thought in Kant. 1989 ISBN 0-7923-0135-8205. D.W. Smith, The Circle of Acquaintance. Perception, Consciousness, and Empathy. 1989

ISBN 0-7923-0252-4206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of Arthur W.

Burks. With his Responses, and with a Bibliography of Burk’s Work. 1990ISBN 0-7923-0325-3

207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husserl,Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4

208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory ofPerception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0

209. P. Kosso, Observability and Observation in Physical Science. 1989 ISBN 0-7923-0389-X210. J. Kmita, Essays on the Theory of Scientific Cognition. 1990 ISBN 0-7923-0441-1211. W. Sieg (ed.), Acting and Reflecting. The Interdisciplinary Turn in Philosophy. 1990

ISBN 0-7923-0512-4212. J. Karpinski, Causality in Sociological Research. 1990 ISBN 0-7923-0546-9213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991 ISBN 0-7923-0823-9214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein’s Philosophy of Psychology. 1990

ISBN 0-7923-0850-6215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and Epistemological

Implications of the Work of W.V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical

Perspectives. 1991 ISBN 0-7923-1046-2217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe.

1991 ISBN 0-7923-1322-4

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218. M. Kusch, Foucault’s Strata and Fields. An Investigation into Archaeological and GenealogicalScience Studies. 1991 ISBN 0-7923-1462-X

219. C.J. Posy, Kant’s Philosophy of Mathematics. Modern Essays. 1992 ISBN 0-7923-1495-6220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Con-

nectionism. 1992 ISBN 0-7923-1519-7221. J.C. Nyıri, Tradition and Individuality. Essays. 1992 ISBN 0-7923-1566-9222. R. Howell, Kant’s Transcendental Deduction. An Analysis of Main Themes in His Critical

Philosophy. 1992 ISBN 0-7923-1571-5223. A. Garcıa de la Sienra, The Logical Foundations of the Marxian Theory of Value. 1992

ISBN 0-7923-1778-5224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual

Order. 1992 ISBN 0-7923-1803-X225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human

Reality. 1993 ISBN 0-7923-2047-6226. P. Suppes, Models and Methods in the Philosophy of Science: Selected Essays. 1993

ISBN 0-7923-2211-8227. R. M. Dancy (ed.), Kant and Critique: New Essays in Honor of W. H. Werkmeister. 1993

ISBN 0-7923-2244-4228. J. Wolenski (ed.), Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional Logic.

1993 ISBN 0-7923-2342-4230. B.K. Matilal and A. Chakrabarti (eds.), Knowing from Words. Western and Indian Philosophical

Analysis of Understanding and Testimony. 1994 ISBN 0-7923-2345-9231. S.A. Kleiner, The Logic of Discovery. A Theory of the Rationality of Scientific Research. 1993

ISBN 0-7923-2371-8232. R. Festa, Optimum Inductive Methods. A Study in Inductive Probability, Bayesian Statistics,

and Verisimilitude. 1993 ISBN 0-7923-2460-9233. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 1: Probability and Probabilistic

Causality. 1994 ISBN 0-7923-2552-4234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of Physics,

Theory Structure, and Measurement Theory. 1994 ISBN 0-7923-2553-2235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and

Psychology. 1994 ISBN 0-7923-2862-0Set ISBN (Vols 233–235) 0-7923-2554-0

236. D. Prawitz and D. Westerstahl (eds.), Logic and Philosophy of Science in Uppsala. Papersfrom the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994

ISBN 0-7923-2702-0237. L. Haaparanta (ed.), Mind, Meaning and Mathematics. Essays on the Philosophical Views of

Husserl and Frege. 1994 ISBN 0-7923-2703-9238. J. Hintikka (ed.), Aspects of Metaphor. 1994 ISBN 0-7923-2786-1239. B. McGuinness and G. Oliveri (eds.), The Philosophy of Michael Dummett. With Replies from

Michael Dummett. 1994 ISBN 0-7923-2804-3240. D. Jamieson (ed.), Language, Mind, and Art. Essays in Appreciation and Analysis, In Honor

of Paul Ziff. 1994 ISBN 0-7923-2810-8241. G. Preyer, F. Siebelt and A. Ulfig (eds.), Language, Mind and Epistemology. On Donald

Davidson’s Philosophy. 1994 ISBN 0-7923-2811-6242. P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of Continua. 1994

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243. G. Debrock and M. Hulswit (eds.), Living Doubt. Essays concerning the epistemology ofCharles Sanders Peirce. 1994 ISBN 0-7923-2898-1

244. J. Srzednicki, To Know or Not to Know. Beyond Realism and Anti-Realism. 1994ISBN 0-7923-2909-0

245. R. Egidi (ed.), Wittgenstein: Mind and Language. 1995 ISBN 0-7923-3171-0246. A. Hyslop, Other Minds. 1995 ISBN 0-7923-3245-8247. L. Polos and M. Masuch (eds.), Applied Logic: How, What and Why. Logical Approaches to

Natural Language. 1995 ISBN 0-7923-3432-9248. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Com-

putation. Volume One: Surveys. 1995 ISBN 0-7923-3448-5249. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Com-

putation. Volume Two: Contributions. 1995 ISBN 0-7923-3449-3Set ISBN (Vols 248 + 249) 0-7923-3450-7

250. R.A. Watson, Representational Ideas from Plato to Patricia Churchland. 1995ISBN 0-7923-3453-1

251. J. Hintikka (ed.), From Dedekind to Godel. Essays on the Development of the Foundations ofMathematics. 1995 ISBN 0-7923-3484-1

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255. P. Hugly and C. Sayward: Intensionality and Truth. An Essay on the Philosophy of A.N. Prior.1996 ISBN 0-7923-4119-8

256. L. Hankinson Nelson and J. Nelson (eds.): Feminism, Science, and the Philosophy of Science.1997 ISBN 0-7923-4162-7

257. P.I. Bystrov and V.N. Sadovsky (eds.): Philosophical Logic and Logical Philosophy. Essays inHonour of Vladimir A. Smirnov. 1996 ISBN 0-7923-4270-4

258. A.E. Andersson and N-E. Sahlin (eds.): The Complexity of Creativity. 1996ISBN 0-7923-4346-8

259. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Logic and Scientific Meth-ods. Volume One of the Tenth International Congress of Logic, Methodology and Philosophyof Science, Florence, August 1995. 1997 ISBN 0-7923-4383-2

260. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Structures and Normsin Science. Volume Two of the Tenth International Congress of Logic, Methodology andPhilosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4384-0

Set ISBN (Vols 259 + 260) 0-7923-4385-9261. A. Chakrabarti: Denying Existence. The Logic, Epistemology and Pragmatics of Negative

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Language. 1997 ISBN 0-7923-4425-1263. D. Nute (ed.): Defeasible Deontic Logic. 1997 ISBN 0-7923-4630-0264. U. Meixner: Axiomatic Formal Ontology. 1997 ISBN 0-7923-4747-X265. I. Brinck: The Indexical ‘I’. The First Person in Thought and Language. 1997

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Individual Action. 1997 ISBN 0-7923-4753-6; Set: 0-7923-4754-4

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267. G. Holmstrom-Hintikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 2:Social Action. 1997 ISBN 0-7923-4752-8; Set: 0-7923-4754-4

268. B.-C. Park: Phenomenological Aspects of Wittgenstein’s Philosophy. 1998ISBN 0-7923-4813-3

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271. K. Szaniawski, A. Chmielewski and J. Wolenski (eds.): On Science, Inference, Informationand Decision Making. Selected Essays in the Philosophy of Science. 1998

ISBN 0-7923-4922-9272. G.H. von Wright: In the Shadow of Descartes. Essays in the Philosophy of Mind. 1998

ISBN 0-7923-4992-X273. K. Kijania-Placek and J. Wolenski (eds.): The Lvov–Warsaw School and Contemporary Phi-

losophy. 1998 ISBN 0-7923-5105-3274. D. Dedrick: Naming the Rainbow. Colour Language, Colour Science, and Culture. 1998

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ogy and Mathematics. 1999 ISBN 0-7923-5246-7276. P. Fletcher: Truth, Proof and Infinity. A Theory of Constructions and Constructive Reasoning.

1998 ISBN 0-7923-5262-9277. M. Fitting and R.L. Mendelsohn (eds.): First-Order Modal Logic. 1998

Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335-8278. J.N. Mohanty: Logic, Truth and the Modalities from a Phenomenological Perspective. 1999

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for Intuitionism. 1999 ISBN 0-7923-5630-6280. A. Cantini, E. Casari and P. Minari (eds.): Logic and Foundations of Mathematics. 1999

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1999 ISBN 0-7923-5810-4283. F. Vollmer: Agent Causality. 1999 ISBN 0-7923-5848-1284. J. Peregrin (ed.): Truth and Its Nature (if Any). 1999 ISBN 0-7923-5865-1285. M. De Caro (ed.): Interpretations and Causes. New Perspectives on Donald Davidson’s Phi-

losophy. 1999 ISBN 0-7923-5869-4286. R. Murawski: Recursive Functions and Metamathematics. Problems of Completeness and

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290. G. Sommaruga: History and Philosophy of Constructive Type Theory. 2000ISBN 0-7923-6180-6

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ISBN 0-7923-7004-X300. P. Gardenfors: The Dynamics of Thought. 2005 ISBN 1-4020-3398-2301. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An

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Nonstandard Views of the Continuum. 2001 ISBN 1-4020-0152-5307. S.D. Zwart: Refined Verisimilitude. 2001 ISBN 1-4020-0268-8308. A.-S. Maurin: If Tropes. 2002 ISBN 1-4020-0656-X309. H. Eilstein (ed.): A Collection of Polish Works on Philosophical Problems of Time and Space-

time. 2002 ISBN 1-4020-0670-5310. Y. Gauthier: Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. 2002

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315. P. Gardenfors, J. Wolenski and K. Kijania-Placek: In the Scope of Logic, Methodology andPhilosophy of Science. Volume One of the 11th International Congress of Logic, Methodologyand Philosophy of Science, Cracow, August 1999. 2002

ISBN 1-4020-0929-1; Pb 1-4020-0931-3316. P. Gardenfors, J. Wolenski and K. Kijania-Placek: In the Scope of Logic, Methodology and

Philosophy of Science. Volume Two of the 11th International Congress of Logic, Methodologyand Philosophy of Science, Cracow, August 1999. 2002

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Perception, Induction and Vagueness. 2003 ISBN 1-4020-1176-8318. D.O. Dahlstrom (ed.): Husserl’s Logical Investigations. 2003 ISBN 1-4020-1325-6319. A. Biletzki: (Over)Interpreting Wittgenstein. 2003

ISBN Hb 1-4020-1326-4; Pb 1-4020-1327-2320. A. Rojszczak, J. Cachro and G. Kurczewski (eds.): Philosophical Dimensions of Logic and

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321. M. Sintonen, P. Ylikoski and K. Miller (eds.): Realism in Action. Essays in the Philosophy ofthe Social Sciences. 2003 ISBN 1-4020-1667-0

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323. J. Hintikka, T. Czarnecki, K. Kijania-Placek, T. Placek and A. Rojszczak † (eds.): Philosophyand Logic In Search of the Polish Tradition. Essays in Honour of Jan Wolenski on the Occasionof his 60th Birthday. 2003 ISBN 1-4020-1721-9

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ISBN 1-4020-3906-9331. B. Feltz, M. Crommelinck and P. Goujon (eds.): Self-organization and Emergence in Life

Sciences. 2005 ISBN 1-4020-3916-6332.333. L. Albertazzi: Immanent Realism. An Introduction to Brentano. 2006 ISBN 1-4020-4201-9334. A. Keupink and S. Shieh (eds.): The Limits of Logical Empiricism. Selected Papers of Arthur

Pap. 2006 ISBN 1-4020-4298-1335. M. van Atten: Brouwer meets Husserl. On the Phenomenology of Choice Sequences. 2006

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ISBN 978-1-4020-5651-2Ontology, and Semiotics. 2007

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