# etchemendy, the concept of logical consequence

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THE CONCEPT OF LOGICAL CONSEQUENCE

JOHN ETCHEMENDY

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THE DAVID HUME SERIES PHILOSOPHY AND COGNITIVE SCIENCE REISSUES

CSLI PUBLICATIONS

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Copyright © 1999

CSU Publications

Center for the Study of Language and Information

Leland Stanford Junior University

Printed in the United States

03 02 0100 99 1 2 3 4 5

Library of Congress Cataloging-in-Publication Data

Etchemendy, John, 1952-

The concept of logical consequence / John Etchemendy.

p. cm.

Originally published: Cambridge, Mass.: Harvard University Press, 1990.

Includes bibliographical references and index.

I S B N 1-57586-194-1 (pbk.: alk. paper)

1. Logic, Symbolic and mathematical 1. Title. [B C 135.E 83 1999]

i6o-dc2i 99-12538

CIP

00 The acid-free paper used in this book meets the minimum requirements of the

American National Standard for Information Sciences - Permanence of Paper for

Printed Library Materials, a n s i Z39. 48- 1984.

The David Hume Series of Philosophy and Cognitive Science Reissues consists of

previously published works that are important and useful to scholars and students

working in the area of cognitive science. The aim of the series is to keep these

indispensable works in print in affordable paperback editions.

In addition to this series, CSLI Publications also publishes lecture notes, monographs,

working papers, and conference proceedings. Our aim is to make new results, ideas,

and approaches available as quickly as possible. Please visit our web site at http://csli-publications.stanford.edu/

for comments on this and other titles, as well as for changes and corrections by the author and publisher.

http://csli-publications.stanford.edu/ http://csli-publications.stanford.edu/

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For Nancy and Max

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Acknowledgments

I owe many thanks to many people. For their help and encourage- ment, without which I may never have finished the book, and their criticism, without which I would certainly have finished too soon, I

would like to thank Ian Hacking, Calvin Normore, Ned Block, GregO’Hair, Richard Cartwright, Leora Weitzman, and, in particular, John Perry, Genoveva Marti, and Paddy Blanchette. For their pa- tience, I thank my family, and especially my wife, Nancy. And for all of the above and more, I thank my friend and colleague Jon Barwise. Finally, I am indebted to the Mrs. Giles Whiting Foundation and to the Center for the Study of Language and Information for support while working on various stages of this book.

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Contents

1 Introduction 1

2 Representational Semantics 12

3 Tarski on Logical T ru th 27

4 Inte rpre tational Semantics 51

5 Interpre ting Quantifiers 65

6 Modality and Consequence 80

7 The Reduction Principle 95

8 Substantive Generalizations 107

9 The Myth of the Logical Constant 125

10 Logic from the Metatheory 136 11 Completeness and Soundness 144

12 Conclusion 156

Notes 161

Bibliography 171

Index 173

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Introduction

1

The highest compliment that can be paid the author of a piece of conceptual analysis comes not when his suggested definition survives whatever criticism may be leveled against it, or when the analysis is acclaimed unassailable. The highest compliment comes when the sug gested definition is no longer seen as the result of conceptual analy sis—when the need for analysis is forgotten, and the definition is treated as common knowledge. Tarski’s account of the concepts of logical truth and logical consequence has earned him this compliment.

Anyone whose study of logic has gone beyond the most rudimentary stages is familiar with the standard, model-theoretic definitions of the logical properties. According to these definitions, a sentence is logically true if it is true in all models; an argument is logically valid, its conclusion a consequence of its premises, if the conclusion is true in every model in which all the premises are true. These definitions, along with the additional machinery needed to understand them, are set forth in every introductory textbook in mathematical logic.1 In these texts we are taught how to delineate a class of models for a simple language and how to provide a recursive definition of truth in a model—in short, how to construct a simple model-theoretic semantics. Once this semantic theory is in place, the model-theoretic definitions of the logical properties can be applied.

This method of defining logical truth and logical validity is gener ally traced to Tarski’s 1936 article, “On the Concept of Logical Conse quence.”2 In this article Tarski sets out to give a precise and general account of what he calls the “intuitive” consequence relation and the

corresponding property of logical truth. The definitions that result aremeant to be applicable to any language whose truth predicate can be

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2 Introduction

defined, and to remain, as Tarski puts it, “close in essentials” to the common, everyday concepts.

Tarski devotes most of his attention in this brief, twelve-page article to shortcomings of other attempts to define the consequence relation, in particular attempts to characterize it syntactically, by means of formal systems of deduction. His own, semantic account, sketched in a mere four pages, is devoted in part to the exposition of some ancillary notions treated at length in his earlier monograph on truth. The main thrust of the article is not to discuss details of the semantic account of consequence, or even to give a simple example of its application, but rather to urge that “in considerations of a general theoretical nature the proper concept of consequence must be placed in the foreground” (1956, p. 413).

Tarski begins his article by emphasizing the importance of the intu

itive notion of consequence to the discipline of logic. He dryly notes that the introduction of this concept into the field “was not a matter of arbitrary decision on the part of this or that investigator” (1956, p. 409). The point is that when we give a precise account of this notion, we are not arbitrarily defining a new concept whose properties we then set out to study—as we are when we introduce, say, the concept of a group, or that of a real closed field. It is for this reason that Tarski takes as his goal an account of consequence that remains faithful to the ordinary, intuitive concept from which we borrow the name. It is for

this reason that the task becomes, in large part, one of conceptual analysis.

Tarski’s account of the logical properties is widely regarded as suc cessful in this respect, as capturing, in mathematically tractable form, the “proper” concepts of logical truth and logical consequence. We can see this not only from explicit acknowledgments of its success by many philosophers and logicians, but also from the treatment given it by those not interested in conceptual analysis as such. Perhaps the most striking indication is the different status afforded syntactic characteri zations of consequence, formal systems of deduction.

It has long been acknowledged that the purely syntactic approach does not yield a general analysis of the ordinary notion of conse quence, and in principle cannot. The reason for this is simple. It is obvious, for starters, that the intuitive notion of consequence cannot

be captured by any single deductive system. For one thing, such a system will be tied to a specific set of rules and a specific language, while the ordinary notion is not so restricted. Thus, by “consequence”

we clearly do not mean derivability in this or that deductive scheme.But neither do we mean derivability in some deductive system or other, for any sentence is derivable from any other in some such system.

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Introduction 3

So at best we might mean by “consequence” derivability in some sound deductive system. But the notion of soundness brings us straight back to the intuitive notion of consequence: a deductive system is sound if it allows us to prove only genuinely valid arguments, those whose con clusions follow logically from their premises.

We recognize that a syntactic definition does not capture the ordinary notion of consequence, and we recognize this even though we may be convinced, for one reason or another, that a given deductive system is adequate for a given language—that is, even if we believe that all valid arguments, and o