(boston studies in the philosophy of science 177) hugues leblanc (auth.), mathieu marion, robert s....

QUEBEC STUDIES IN THE PHILOSOPHY OF SCIENCE

PART I

BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE

Editor

ROBERT S. COHEN, Boston University

Editorial Advisory Board

THOMAS F. GLICK, Boston University ADOLF GRUNBAUM, University of Pittsburgh

SAHOTRA SARKAR, McGill University SYLVAN S. SCHWEBER, Brandeis University

JOHN J. STACHEL, Boston University MARX W. WARTOFSKY, Baruch College of

the City University of New York

VOLUME 177

HUGUESLEBLANC Courtesy o/Virginia G. Leblanc

QUEBEC STUDIES IN THE PHILOSOPHY OF SCIENCE

Part I: Logic, Mathematics, Physics and History of Science

Essays in Honor of H ugues Leblanc

Edited by

MATHIEU MARION University of Ottawa

and

ROBERT S. COHEN Boston University

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging.inPublication Data

Owebec studies in the philosophy of science I edited by Mathieu Marion and Robert S. Cohen.

p. cm. Contents: pt. I. LogiC, mathematics, physics, and history of

science

alk. paper) 1. SCience--Phllosophy--Congresses. 2. Logic--Congresses.

I. Marion, Mathieu, 1962- II. Cohen, R. S. (Robert Sonne) 0174.043 1996 501--dc20 95-17467

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TABLE OF CONTENTS

EDITORIAL PREFACE ix

LOGIC

HUGUES LEBLANC / On Axiomatizing Free Logic - And Inclusive Logic in the Bargain 1 FRAN~OIS LEPAGE / Partial Propositional Logic 23 SERGE LAPIERRE / Generalized Quantifiers and Inferences 41 MARIE LA PALME REYES, JOHN MACNAMARA and GONZALO E.

REYES / A Category-Theoretic Approach to Aristotle's Term Logic, with Special Reference to Syllogisms 57

JOACHIM LAMBEK / On the Nominalistic Interpretation of Natural Languages 69

JEAN-PIERRE MARQUIS / If Not-True and Not Being True Are Not Identical, Which One Is False? 79

DANIEL V ANDER VEKEN / A New Formulation of the Logic of Propositions 95

YVON GAUTHIER / Internal Logic. A Radically Constructive Logic for Mathematics and Physics 107

JUDY PELHAM / A Reconstruction of Russell's Substitution Theory 123

PHILOSOPHY OF MATHEMATICS

MICHAEL HALLETT / Hilbert and Logic 135 MATHIEU MARION / Kronecker's 'Safe Haven of Real Mathematics' 189

PHILOSOPHY OF PHYSICS

MARIO BUNGE / Hidden Variables, Separability, and Realism 217 STORRS McCALL / A Branched Interpretation of Quantum Mechanics

Which Differs from Everett's 229 MICHEL J. BLAIS / ... and Chaos Shall Set You Free. . . 243

vii

viii T ABLE OF CONTENTS

HISTORY AND PHILOSOPHY OF SCIENCE

PAUL M. PIETROSKI / Other Things Equal, The Chances Improve 259 DA VID DAVIES / The Model-Theoretic Argument Unlocked 275 JEAN LEROUX / Helmholtz and Modern Empiricism 287 WILLIAM R. SHEA / Technology and the Rise of the Mechanical

Philosophy 297

NOTES ON THE AUTHORS 309

NAME INDEX 315

EDITORIAL PREFACE

By North-American standards, philosophy is not new in Quebec: the first men-tion of philosophy lectures given by a Jesuit in the College de Quebec (founded 1635) dates from 1665, and the oldest logic manuscript dates from 1679. In English-speaking universities such as McGill (founded 1829), philosophy began to be taught later, during the second half of the 19th century. The major influence on English-speaking philosophers was, at least initially, that of Scottish Empiricism. On the other hand, the strong influence of the Catholic Church on French-Canadian society meant that the staff of the facultes of the French-speaking universities consisted, until recently, almost entirely of Thomist philosophers. There was accordingly little or no work in modem Formal Logic and Philosophy of Science and precious few contacts between the philosophical communities. In the late forties, Hugues Leblanc was a young student wanting to learn Formal Logic. He could not find anyone in Quebec to teach him and he went to study at Harvard University under the supervision of W. V. Quine. His best friend Maurice L' Abbe had left, a year earlier, for Princeton to study with Alonzo Church.

After receiving his Ph.D from Harvard in 1948, Leblanc started his profes-sional career at Bryn Mawr College, where he stayed until 1967. He then went to Temple University, where he taught until his retirement in 1992, serving as Chair of the Department of Philosophy from 1973 until 1979. His achievements as a logician include seminal contributions to the development of Free Logic, in particular with the ground breaking paper, written jointly with Theodore Hailperin, 'Nondesignating Singular Terms' (Philosophical Review 68 (1959), pp. 239-43). After initial results by Bas van Fraassen, using supervaluation, Hugues Leblanc and Richmond Thomason obtained completeness results in 'Completeness Theorems for Some Presupposition-Free Logic' (Fundamenta Mathematicae 62 (1968), pp. 125-64). More recently, Leblanc also made seminal contributions to Truth-Value Semantics (cf. his Truth-Value Semantics, Amsterdam, North-Holland, 1976) and, inspired by appendices to Karl Popper's Logic of Scientific Discovery, to Probability Semantics and Probability Theory, in his paper 'Probabilistic Semantics for First-Order Logic' (ZeitschriJt for mathematische Logik und Grundlagen der Mathematik 25 (1979), pp. 498-509). In all, Leblanc has written more than one hundred scientific papers, the more recent of them in collaboration with Peter Roeper (Australian National University), and four books, he collaborated on two books and edited or co-edited four. Many logic students will remember learning the subject from his classic textbook, written with William A. Wisdom, Deductive Logic (3rd edn., Englewood Cliffs, Prentice Hall, 1993).

After a long and fruitful career in the United States, Hugues Leblanc is now

ix

x EDITORIAL PREFACE

back in Quebec, where the philosophical milieu has changed beyond recogni-tion since his student days. He came back to find studies in logic and in all aspects of philosophy of science in a flourishing state. As a result of the revolu-tion tranquille which took place among the French-speaking society in the sixties, philosophy in Quebec opened up to external influences such as, initially, phenomenology and Marxism and, increasingly in the past twenty years, Anglo-American analytic philosophy. As a result, there is now a growing number of French-speaking logicians and philosophers of science - although not all of them work from the point of view of analytical philosophy. Conditions were set for fruitful exchanges with the English-speaking philosophical community. (But we should add here that the essential role of immigrants in the evolution of the philosophical life in Quebec should not be overlooked. Contributors to the present volumes come not only from other parts of Canada, but also from Argentina, Australia, Belgium, Germany, Ireland, Switzerland, the United Kingdom and the United States).

Such exchanges have led recently to the creation of research groups across Quebec. These are now joined together under the name of Groupe de recherche sur la representation, I' action et Ie langage or GRRAL. Our two volumes of Quebec Studies in the Philosophy of Science comprise the first full-scale collection of studies in the philosophy and history of science from French- and English-speaking philosophers of Quebec to appear in English; they include in particular most members of the GRRAL. As editors, we are happy to join the contributors in dedicating these volumes to Hugues Leblanc, who is, among philosophers, the first logicien quebecois.

In our first volume, which opens with a new essay on Free Logic by Hugues Leblanc himself, we have collected together papers in logic, philosophy of mathematics, philosophy of physics and in general philosophy and history of science. This volume includes members of two of the research groups forming the GRRAL, the group on Fondements de la logique et fondement du raison-nement (M. Hallett, F. Lepage, S. Lapierre, J.-P. Marquis, and, at the time of writing, J. Pelham) and the group Actes du discours et grammaire universelle (D. Vanderveken & Y. Gauthier).

The papers in the section on logic show the great variety of logical investiga-tions work done across Quebec. Both Franyois Lepage and Serge Lapierre present their results, respectively, on partial functions in type theory and condi-tional quantifiers, within their more global context. The following three papers reflect the fundamental research in category theory which has been taking place in Montreal's various departments of mathematics. Marie La Palme Reyes, John Macnamara and Gonzalo Reyes argue in their paper for the replacement of the standard Boolean, class interpretation of syllogistic by a category-theoretical approach. Jim Lambek argues for an extension of his nominalistic interpretation of the language of mathematics, developed in collaboration with Jocelyne Couture and Phil Scott, to natural languages, and Jean-Pierre Marquis studies

EDITORIAL PREFACE xi

the distinction between true, not-true and not being true within the perspective of topos theory.

Further to his work on the logic of illocutionary forces, Daniel Vanderveken presents in his paper a framework for a new logic of propositions. Yvon Gauthier presents his own system of internal logic and system of finitist arith-metic, where the schema of complete induction is replacd by that of Fermat's infinite descent. Finally, building on joint results with Alasdair Urquhart on structured propositions, Judy Pelham presents in her paper a reconstruction of Russell's substitution theory (circa 1905-6) that provides a resolution of the paradoxes which originally caused Russell to abandon the theory in favour of the ramified theory of types.

In the section on the philosophy of mathematics, Michael Hallett studies the role of logic in Hilbert's approach to the foundations of mathematics, while Mathieu Marion examines the relations between Kronecker's philosophy of mathematics and the tradition of logical foundations. The section on philosophy of physics comprises a paper by the distinguished philosopher of science Mario Bunge, in which he argues that recent experiments which refuted hidden variable theories were not a refutation of realism, a paper by Storrs McCall presenting a new 'branched' interpretation of Quantum Mechanics and a paper by Michel Blais on Chaos Theory.

The last section of the volume contains studies in general philosopy of science and a study by the leading historian of science, William Shea. In his paper, Paul Pietro sky presents an original conception of ceteris paribus laws inspired by Ramsey's ideas about causation. Dave Davies argues that Putnam's recent emphasis on conceptual relativity against metaphysical realism does not vindicate critics of his model-theoretic argument, but actually clarifies Putnam's objections to these very criticisms. Jean Leroux presents elements of Helmholtz's epistemology which foreshadow modem forms of empiricism and he argues for an anti-realist reading of his theory of science. Finally, Shea studies the origin, in the development of new technologies from the medieval ages onwards, of the mechanical philosophy which underlay modem science, from Galileo to Newton.

We would like to thank Alain Voizard for his help in writing this preface, and also the editor of the Brazilian Journal of Physics, formerly Revista Brasileira de Fisica, for granting us permission to reprint Mario Bunge's essay 'Hidden Variables, Separability and Realism' (volume especial os 70 anos de Mario SchOnberg, 1984, pp. 150-168). We are especially grateful to Annie Kuipers for her professional assistance on behalf of Kluwer Academic Publishers and for her continued encouragement and patience.

Boston and Montreal April 1995

MATHIEU MARION & ROBERT S. COHEN

HUGUES LEBLANC

ON AXIOMATIZING FREE LOGIC -AND INCLUSIVE LOGIC IN THE BARGAIN*

Free Logic owes its name to its being free of two presuppositions of Standard Logic, one to the effect that something exists, a patent truth but surely a factual rather than a logical one, and the other to the effect that every singular term designates something, a patent falsehood. l In Standard Logic with Identity, it is this familiar law:

A. (3X) (X = T), T here any term you please, that most crisply encapsulates the two presup-positions.2 In Standard Logic without Identity, it may be this little known counterpart of it:

B. (3X) (A == A(T/X)), T as above and A(T/X) the result of replacing X everywhere in A by T, that does.

In such recent expositions of Free Logic as Bencivenga's [2] and Lambert's [8], Free Logic without Identity is axiomatized first; and the axiom schemata needed to convert it into a Free Logic with Identity, call them the axiom schemata for '=', are added next. The procedure is instructive, to be sure, but it is slightly inelegant. Indeed, one of the axiom schemata for Free Logic without Identity, to wit:

C. (\iX) (\iY)A ::> (\iY) (\iX)A, though independent of the other axiom schemata for that logic, is provable -it so happens - in the presence of those for '='.3 So, awkwardly, C must be dropped as one converts Free Logic without Identity into Free Logic with Identity.4 But what I regret more is that (i) in the process of ax iomati zing Free Logic with Identity, Bencivenga, Lambert, and others do not exploit Leonard's Specification Law for Free Logic with Identity in [19], to wit:

D. (3X) (X = T) ::> (\iX)A :::> A(T/X)),5 and (ii) in the process of ax iomati zing Free Logic without Identity, they do not exploit this Identityless counterpart of D in [10]:

E. (3X) (A == A(T/X)) :::> \iX)A :::> A(T/X)).

* Editorial note: a list of lettered fonnulas in Section I is given at the end of this essay.

M. Marion and R. S. Cohen (etis.), Quebec Studies in the Philosophy of Science I, 1-22. 1995 Kluwer Academic Publishers.

2 HUGUES LEBLANC

In contrast, I devise in Section II a partial logic with Identity whose axioms and one primitive rule of inference, Modus Ponens, (i) suit Standard as well as Free Logic and yet (ii) yield as theorems both D and this curious Identity Law, used by Tarski in [24]:

F. (3X)(X = T) ~ T = T. Since the consequent

G. (V'X)A ~ A(T/X) of D and that

H. T=T

of F are the only two of the customary axiom schemata for Standard Logic with Identity not among those for my partial logic, adding A to them obvi-ously extends it to a full-fledged Standard Logic with Identity.6 The resulting axiomatization of that logic may be new. In any event it is one in which, interestingly enough, the Specification Law for Free Logic with Identity begets G, the Specification Law for Standard Logic (with or without Identity).

In Section III I extend the partial logic of Section II to a full-fledged Free Logic with Identity. I do this in two different - and shown equivalent - ways, using in the first case H plus this generalization of A:

I. (V'Y) (3X) (X = Y), Y a variable distinct from X, and in the second case H plus Lambert's gen-eralization of G in [7]:

J. (VY) VX)A :J A(Y/X, Y this time a variable foreign to A and A(Y/X) the result of course of replacing X everywhere in A by Y.

In Section IV I drop the one axiom schema in Section II that exibits '=', and show the resulting partial logic to yield E as a theorem. So, enlisting B as a substitute for A is sure to extend that logic to a full-fledged Standard Logic without Identity. The resulting axiomatization may also be new, and this time around it is the Specification Law for Free Logic without Identity that begets G. I next extend the present partial logic to a full-fledged Free Logic without Identity. Following the precedent set in Section III, I do this in two different - and shown equivalent - ways, using in the first case this generalization of B:

K. (V'Y) (3X) (A == A(Y/X, Y a variable foreign to A, and in the second case Lambert's J. I then show B, which I called on page 1 a counterpart of A, to be provable from A for any statement A and A to be provable from (one special case of) B, this given the axioms of Section II and those of the sort H.7

In Section V, I sketch a truth-value semantics for Free Logic as well as a

ON AXIOMATIZING FREE LOGIC 3

Free Probability Theory (or, as more recent usage has it, a Free Probability Logic), and I remind the reader that Free Probability Theory does provide an alternative semantics for Free Logic, one in which probability functions relativized to possibly empty sets of terms substitute for truth-value functions relativized to such sets.

In Section VI, Lastly, I turn to what Quine called in [22] Inclusive Logic, provide both an axiomatization of and a semantics for it, and study its rela-tionship to Standard as well as to Free Logic. Used in the process will be these two axiom schemata:

L. (3X) (A v -A) ::J \tX)A ::J A(T/X and

M. (\tY) (3X) (A v -A). Note: For brevity's sake I shall refer to the partial logic with Identity of Section II as L., to the Standard Logic with Identity that L. extends to in that very section as SL., to the Free Logic with Identity that L. extends to in Section III as FL., to the partial logic without Identity of Section IV as L, and to the Standard Logic without Identity and the Free Logic without Identity that L extends to in that very Section as SL and FL, respectively. As for Inclusive Logic, I shall refer to the one with Identity as IL. and to the other as IL.

II

The primitive signs of L., SL., and FL. are the customary ones, to wit: (i) countably many predicates, '=' one of them, of course, (ii) a denumerable infinity of individual variables, say, 'X1" 'X2" . , 'xn', ... , (iii) a denu-merable infinity of singular or - for uniformity's sake - individual terms, say, 't1','t2 ', , 'tn', ... ,8 (iv) the three logical operators '-', '::J', and '\t', (v) the two parentheses '(' and ,)" and (vi) the comma','. As for the state-ments of L., SL., and FL., they are also the customary ones, except for (\tX)A counting as a statement only if A(T/X) - T once more any term you please - counts itself as a statement. Identical quantifiers, as a result, cannot overlap, hence the restriction placed three times in Section I on the variable y'9 Parentheses will be dropped unless ambiguity threatens; and the four logical operators '&', 'v', '=', and '3' - two of them already used in Section 1-will be presumed to be defined in the customary manner.

Lastly, extending my use so far of '/' and introducing its cognate 'II', suppose I and l'to be two individual variables, or two individual terms, or one an individual variable and the other an individual term. A(I'II) will then be the result of replacing I everywhere in A by I', and A(I'III) that of replacing I at zero or more places in A by 1/.10

Following in this Fitch's example in [5], I identify the axioms of L. (hence, those common to L., SL., FL., and ILJ recursively:

4 HUGUES LEBLANC

Basic Clause: Every statement of L_ of any of these six sorts:

AI. A ::J (B ::J A) A2. (A ::J (B ::J C)) ::J A ::J B) ::J (A ::J C)) A3. (-A ::J -B) ::J (B ::J A) A4. (VX)(A ::J B) ::J VX)A ::J (VX)B) AS. A ::J (VX)AII A6. T = T' ::J (A ::J A(T'//T))

counts as an axiom of L_,

Inductive Clause: If A counts as an axiom of L_, then so does (VX)A(XlT), so long as X is foreign to A. 12

Due to the second clause, which I shall occasionally refer to as Fitch's Clause, one rule of inference suffices, it turns out, Modus Ponens (MP, for short). So, a finite column of statements of L_ counts as a proof in L_ of a state-ment A of L_ if (i) every entry in the column is an axiom of L_ or follows by means of MP from two earlier entries in the column, and (ii) the last entry in the column is A. And A counts as a theorem of L_, or is said to be provable in L_, for short:

1-_ A,

if there exists a proof of A in L_. Preparatory to proving D and F, I put down eight lemmas, supplying proof

of just the two that involve quantifiers. Five of these lemmas are in effect derived rules of inference, and the most important one of the five is of course Lemma 6. It is a generalization of what I just called Fitch's Clause; and in axiomatizations of Standard or Free Logic that dispense with that clause, it must be adopted along with Modus Ponens as a primitive rule of inference. Called the Generalization Rule, it is refered to here as Gn.

LEMMA 1. If 1-_ A ::J B and 1-_ B ::J C, then 1-_ A ::J C, LEMMA 2. If 1-_ A ::J (B ::J C), then 1-_ B ::J (A ::J C), LEMMA 3. If 1-_ A ::J (A ::J B), then 1-_ A ::J B, LEMMA 4. 1-_ (A ::J B) ::J (-B ::J -A), LEMMA 5. If 1-_ -A ::J B, then 1-_ -B ::J A, LEMMA 6. If 1-_ A, then 1-_ (VX)A(XlT), so long as X is foreign to A.

Proof. Suppose the column made up of AI' A2, , and An constitutes a proof of A in L_; suppose X is foreign to A; and for each i from 1 through n let A; be Ai(X'/X), where X' is the alphabetically earliest individual variable of L_ that is foreign to all of AI' A2, , and An-I' It is easily verified that (i) if Ai (i = 1, 2, ... , or n) is an axiom of L_, then so is A;, and (ii) if for


some g from 1 through i-I Ai follows by means of MP from As and As ::J Ai' then A; does so from A~ and (As ::J A)" this because (As ::J AY is the same as A~ ::J A;. So the column made up of A~, A;, ... , and A~ consti-tutes a proof of A~ in L., and one to all of whose entries X is foreign. Consider then the column made up of ('v'X)A~ (X/T), ('v'X)A;(XlT), . . . , and

('v'X)A~(XlT). This third column constitutes a proof of ('v'X)A~(XlT) in L. if none of the entries in the second column was obtained by means of MP, and will do so in the contrary case upon the insertion in it of two entries per entry that was thus obtained. For suppose first that A; was an axiom of L . Then, X being foreign to A;, so by Fitch's Clause is ('v'X)A;(XlT), which justifies its presence in the third column. Suppose then that A; was obtained by means of MP from A~ and A~ ::J A; for some g from 1 through i-I. In this case inserting the following two lines:

('v'X)(A~(XlT) ::J A;(XlT)) ::J 'v'X)A~(XlT) ::J ('v'X)A;(XlT)) and

('v'X)A~(XlT) ::J ('v'X)A;(XlT) after whichever of ('v'X)A~(XlT) and ('v'X) (A~ ::J A;) (XlT) (=('v'X) (A~(XlT) ::J A;(XlT))) occurs second in the third column will justify the presence of ('v'X)A;(XlT) in that column. The first of these lines is indeed an axiom of L. of the sort A4, the second follows from the first and ('v'X) (A~ ::J A;) (XlT) by means of MP, and ('v'X)A;(XlT) follows from the second and ('v'X)A~(XlT) by means of MP also. But, X being by hypothesis foreign to A and hence to An' ('v'X)A~(XlT) is the same as ('v'X)A(XlT). So, if 1-_ A, then 1-. ('v'X)A(XlT), so long as X is foreign to A. 0

LEMMA 7. 1-. ('v'X)(A ::J B) ::J 3X)A ::J (3X)B). Proof Suppose T new. 13

(1) I-_ (A(T/X) ::J B(T/X)) ::J (-B(T/X) ::J -A(T/X))

(2) 1-. ('v'X) A ::J B) ::J (-B ::J -A)) (3) 1-. (2) ::J 'v'X) (A ::J B) ::J

('v'X) (-B ::J -A)) (4) 1-. ('v'X)(A ::J B) ::J ('v'X)(-B ::J -A) (5) 1-. ('v'X) (-B ::J -A) ::J 'v'X)-B ::J

('v'X)-A)

(Lemma 4) (Go, (1))

(A4) (MP, (2), (3))

(A4) (6) 1-. ('v'X)(A ::J B) ::J 'v'X)-B ::J ('v'X)-A) (Lemma 1, (4), (5)) (7) I-_ 'v'X)-B ::J ('v'X)-A) ::J

3X)A ::J(3X)B) (Lemma 4) (8) I-_ ('v'X)(A ::J B) ::J 3X)A ::J (3X)B) (Lemma 1, (6), (7)) 0

6 HUGUES LEBLANC

LEMMA 8. 1-. (V X) (A ~ B) ~ 3X)A ~ B), so long as X is foreign to B, Proof

(1) 1-. -B ~ VX)-B) (AS, hypo on X) (2) 1-. (3X)B ~ B (Lemma 5, (1 (3) 1-. (2) ::) 3X)A ::) 3X)B ::) B (AI) (4) 1-. (3X)A ::) 3X)B ::) B) (MP, (2), (3 (5) 1-. (4) ::) (3X)A ::) (3X)B) ::)

3X)A ::) B (A2) (6) 1-. 3X)A ::) (3X)B) ::) 3X)A ::) B) (MP, (4), (5 (7) 1-. (VX)(A ::) B) ::) 3X)A ::) (3X)B) (Lemma 7) (8) 1-. (VX)(A ::) B) ::) 3X)A ::) B) (Lemma 1, (7), (6 0

Proofs of D and F can now be had:

THEOREM 1. (3X)(X = T) ::) VX)A ::) A(T/X. Proof Suppose T' new.

(1) 1-. A(T'/X) ~ (T' = T ~ (A(T'/X(T/T' (Lemma 2, A6) i.e.

(1) 1-. A(T'/X) ~ (T' = T ~ A(T/X (Lemma 2, A6) (2) 1-. (VX)(A ::) (X = T ::) A(T/X) (Go, (1 (3) 1-. (2) ~ VX)A ~

(V X) (X = T ~ A(T/X) (A4) (4) 1-. (VX)A ::) (VX)(X = T ~ A(T/X (MP, (2), (3 (5) 1-. (VX)(X = T ::) A(T/X ::)

::IX) (X = T) ~ A(T/X (Lemma 8) (6) 1-. (VX)A ::) 3X)(X = T) ::) A(T/X (Lemma 1, (4), (5 (7) 1-. (3X)(X = T) ::) VX)A ::) A(T/X (Lemma 2, (6 0

THEOREM 2. 1-. (3X)(X = T) ::) T = T. Proof Suppose T' new.

(1) I-_ T' = T ::) (T' = T ::) T = T) (A6) (2) 1-. T' = T ::) T = T (Lemma 3, (1 (3) 1-. (2) ~ (-(T = T) ::) -(T' = T (Lemma 4) (4) I-_ -(T = T) ::) -(T' = T) (MP, (2), (3 (5) 1-. (VX)(-(T = T) ::) -(X = T (Go, (4

ON AXIOMA TIZING FREE LOGIC 7

(6) f-. (5) :J VX)-(T = T) :J (VX)-(X = T (A4)

(7) f-. (VX) -(T = T) :J (VX)-(X = T) (MP, (5), (6 (8) f-. -(T = T) :J (VX)-(T = T) (AS) (9) f-. -(T = T) :J (VX)-(X = T) (Lemma 1, (8), (7

(10) f-. (3X)(X = T) :J T = T (Lemma 5, (9 0

So, as claimed on page 2, enlisting as an extra axiom schema the antecedent A of Theorem 1 and Theorem 2 would extend the partial logic of this section to a full-fledged Standard Logic with Identity.

It could readily be seen, by the way, that the logic in question was a partial one with as well as without Identity. Assigning the truth-value 0 (for False) to all the atomic statements of L., evaluating its negations and conditionals in the customary manner, and assigning its quantifications the truth-value 1 (for True) ensure that all its axioms evaluate to 1 and that the consequent of a conditional of L. evaluates to 1 if the conditional in question and its antecedent themselves do. So, under this assignment, all statements of L. provable in L. evaluate to 1, but no statement of the sort H does and many a statement of the sort G, say, '(Vxj)(F(x j) & -F(xj :J (F(t j) & -F(tj', does not either.

III

To obtain the Free Logic with Identity promised on page 2, (i) substitute wherever appropriate 'FL.' for 'L.' in Section II, (ii) add to the six axiom schemata on page 4 either this axiom schema, labelled I on page 2:

FA7. (VY)(3X)(X = Y), or this one, labelled J on that page:

FA7'. (VY)VX)A:J A(Y/X, (iii) add also this axiom schema, familiar from page 2 as H:

FA8. T = T

and (iv) abridge A is provable in FL.

as

f-F_ A.

I now proceed to show that Theorem 1 on page 6, i.e. Leonard's Specification Law for Free Logic with Identity, and FA7 deliver FA7'.

8 HUGUES LEBLANC

THEOREM 3. I-F_ (V'Y) V'X)A ::J A(Y/X. Proof. Suppose T new. (1) I-F_ (V'Y)3X)(X = Y) ::J

V'X)A ::J A(Y/X) (Gn, Theorem 1) (2) I-F_ (1) ::J V'Y)(3X)(X = Y) ::J

(V'Y) V'X)A ::J A(Y/X) (A4) (3) I-F_ (V'Y) (3X) (X = Y) ::J

(V'Y) V'X)A ::J A(Y/X (MP, 0), (2 (4) I-F_ (V'Y) V'X)A ::J A(Y/X (MP, FA7, (3 0

This done, I go on to show that FA7' and FAS (not used in the foregoing proof) deliver FA7. Two new lemmas are used in the course of the proof.

LEMMA 9. I-F_ (A ::J -B) ::J (B ::J -A).

LEMMA 10. I-F_ (V'Y) (A(Y/X) ::J (3X)A). Proof. Suppose T new. (1) I-F_ V'X)-A ::J -A(T/X ::J (A(T/X) ::J

(3X)A) (Lemma 9) (2) I-F_ (V'Y) (V'X)-A ::J -A(Y/X ::J

(A(Y/X) ::J (3X)A (Gn, (1 (3) I-F- (2) ::J V'Y) V'X) -A ::J

-A(Y/X ::J (V'Y) (A(Y/X) ::J (3X)A (A4) (4) I-F_ (V'Y) V'X)-A ::J -A(Y/X ::J

(V'Y) (A(Y/X) ::J (3X)A) (MP, (2), (3 (5) I-F_ (V'Y) (A(Y/X) ::J (3X)A) (MP, FA7', (4 0

THEOREM 4. I-F_ (V'Y) (3X) (X = Y). Proof. (1) I-F_ (V'Y)(Y = Y ::J (3X) (X = Y (Lemma 10) (2) I-F_ (1) ::J V'Y) (Y = Y) ::J

(V'Y)(3X)(X = Y (A4) (3) I-F_ (V'Y) (Y = Y) ::J (V'Y) (3X) (X = Y) (MP, (1), (2 (4) I-F_ (V'Y) (Y = Y) (Gn, FAS) (5) I-F_ (V'Y)(3X)(X = Y) (MP, (3), (4 0

So, given FAS, FA7 and FA7' are provably equivalent means of extending the partial logic of Section II to a full-fledged Free Logic with Identity, a result mentioned by Bencivenga in [2] and known to Lambert.

Shown in effect at the close of Section II was that statements of L_, and hence of FL_, of the sort FAS are independent of the axioms of L_. Proof


that FA7 and FA7' are independent of the axioms of L_ and of the state-ments of FL_ of the sort FA8 calls for a bit more work.

:E1, :E2, ... , :En' ... being infinitely many non-empty sets of individual terms of FL_, take a statement A of FL_ to evaluate to I on a truth-value assignment a. (to the atomic statements of FL_) relative to the sequence (:E1, :E2, ... , :En' ... ) if the customary conditions are met when A is an atomic statement, a negation, or a conditional; but, in the case that A is a universal quantification (VX)B, X here the i-th individual variable of FL_ for some i or other from lon, take A to evaluate to 1 on a. relative to (:E1' :E2, ... , :En' ... ) if, and only if, B(T/X) evaluates to 1 on a. relative to that sequence for every term T in :Ej This done, consider the truth-value assign-ment a. that assigns 1 to every atomic statement of FL_ of the sort T = T but the truth-value 0 to every other one, and a sequence (:E1' :E2, ... , :En' ... ) that is arbitrary except for :E1 and :E2 being {'t1'} and {'t/}, respec-tively. Then all the axioms of L_ and statements of FL_ of the sort FA8 evaluate to 1 on a. relative to (:E1, :E2, ... , :En' ... ), as does the consequent of a conditional of FL_ if that conditional and its antecedent themselves do. Contrastingly, though, 't1 = t/ evaluates to 0 on a. relative to (:E1, ~, ... , :En' ... ), hence '(VX)-(x1 = t2), evaluates to I on a. relative to that sequence (this because 't1' is the only member of :E1), hence '(3x1) (X1 = t2), evaluates to 0 on a. relative to (:E1' ~, ... , :En' ... ), and hence '(Vx2)(3x1)(X1 = x2)' evaluates to 0 on a. relative to (:E1' :E2, ... , :En' ... ) (this because 't2' is the only member of :E2). So, at least one statement of FL_ of the sort FA7 evaluates to 0 on a. relative to that sequence. So, at least one statement of FL_ of the sort FA7 is independent of the axioms of L_ and of the statements of FL_ of the sort FA8. And at least one statement of FL_ of the sort FA7', to wit: 'CV'x2) Vx1)F(x1) ::J F(x2', will prove to be independent of the axioms of L_ and of the statements of FL_ of the sort FA8 if 'F(t1)' is assigned 1 rather than 0 by the truth-value assignment 0.. 14

IV

To obtain the partial logic L promised on page 2, (i) substitute wherever appro-priate 'L' for 'L_' in Section II, (ii) drop the predicate '=' on page 3 and, as already indicated, axiom schema A6, (iii) for the reason given on page 1, add this axiom schema, labelled C on that page:

FA6. (VX)(VY)A::J (VY)(VX)A,

and (iv) abridge

A is provable in FL

as

~F A.

10 HUGUES LEBLANC

And, to obtain the Free Logic FL also promised on page 2, (i) substitute wherever appropriate 'FL' for 'L' in Section II, (ii) add besides FA6 either this axiom schema, labelled K on page 2:

F7". (VY) (3X) (A == A(Y/X or the familiar FA", and (iii) abridge

A is provable in FL as

f-F A.

I first show that E is provable in L and hence that Free Logic without Identity boasts a Specification Law that exactly parallels

(3X)(X = T) ~ VX)A ~ A(T/X and thus outdoes the customary

(VY) VX)A ~ A(Y/X, to wit:

(3X) (A == A(T/X ~ VX)A ~ A(T/X. One additional lemma is needed in the process.

LEMMA 11. f- (A == B) ~ (A ~ B).

THEOREM 5. (3X)(A == A(T/X)) :J VX)A :J A(T/X)). Proof Suppose T' new. (1) f- (A(T'/X) == A(T/X ~

(A(T'/X) ~ A(T/X (Lemma 11) (2) f- A(T'/X) :J A(T'/X) == A(T/X ~

A(T/X (Lemma 2, (1 (3) f- (VX)(A ~ A == A(T/X :J ACT/X))) (Go, (2 (4) f- (3) ~ VX)A ~

(VX) A == A(T/X ~ ACT/X))) (A4) (5) f- (VX)A ~ (VX) A == A(T/X ~

A(T/X (MP, (3), (4 (6) f- (VX) A == ACT/X ~ A(T/X ~

3X) (A == A(T/X ~ A(T/X (Lemma 8) (7) f- (VX)A ~ 3X) (A == A(T/X ~

A(T/X (Lemma I, (5), (6 (8) f- (3X) (A == A(T/X ~

VX)A ~ A(T/X (Lemma 2, (7 0


So, as claimed on page 2, substituting the antecedent B of Theorem 5 for the axiom schema A6 of L. would extend the partial logic L to a full fledged Standard Logic without Identity. And proof that FA7" is independent of the axioms of L is easily retrieved from the proof in Section III that FA7' is independent of the axioms of L.: simply write 'FL' for 'FL.', 'L' for 'L.' and '(Vx2)(3x\)(F(x\) == F(x2))' for '(Vx2)(Vx\)F(x\) ::J F(x2))'. My main concern at this point, though, is to show that FA7" and FA7 are provably equiv-alent ways of extending L to a full-fledged Free Logic without Identity. Yet another lemma is needed in the process.

THEOREM 6. r-F (VY) (3X) (A == A(Y/X)) ::J (VY) VX)A ::J A(Y/X)). Proof Suppose T new. (1) r-F (VY) 3X) (A == A(Y/X)) ::J

VX)A ::J A(Y/X))) (Go, Theorem 5) (2) r-F (1) ::J VY)(3X)(A == A(Y/X)) ::J

(VY) VX)A ::J A(Y/X))) (A4) (3) r-F (VY) (3X) (A == A(Y/X)) ::J

(VY)VX)A ::J A(Y/X) (MP, (1), (2)) 0 So, by MP, FA7" yields FA7'.

LEMMA 12. r-F A == A.

THEOREM 7. r-F (VY)(3X)(A == A(Y/X). Proof Suppose T new. (1) r-F (VY) A == A(Y/X)) (Y/X) ::J

(3X)(A == A(Y/X))) (Lemma 10) i.e.

(1) r-F (VY) A(Y/X) == A(Y/X)) ::J (3X)(A == A(Y/X))) (Lemma 10)

(2) r-F (1) ::J (VY) (A(Y/X) == A(Y/X) ::J (VY) (3X) (A == A(Y/X))) (A4)

(3) r-F (VY) (A(Y/X) == A(Y/X) ::J (VY) (3X) (A == A(Y/X)) (MP, (1), (2))

(4) r-F A(T/X) == A(T/X) (Lemma 12) (5) r-F (VY) (A(Y/X) == A(Y/X)) (Go, (4)) (6) r-F (VY) (3X) (A == A(Y/X)) (MP, (5), (3)) 0

But FA7' is needed to obtain Lemma 10. So, FA7' yields FA7". On page 2 I talked of A and B being provably equivalent given H (=

FAS). Indeed, enlist H as an extra axiom schema of L . Then B is provable in L. from A, and A is provable in L. from (one special case of) B, as I proceed

12 HUGUES LEBLANC

to show. Four additional lemmas are needed to that effect, of which I prove only the two involving '='.

LEMMA 13. IJr_ A:::> (B :::> C) and r_ A:::> (C:::> B), then r_ A:::> (B == C).

LEMMA 14. T' = T :::> T = T'. Proof. (1) (2) (3) (4)

r _ T' = T :::> (T' = T' :::> T = T') r _ T' = T' :::> (T' = T :::> T = T') r_ T' = T' r _ T' = T :::> T = T'

LEMMA 15. r_ T' = T :::J (A == A(T/T'. Proof.

(A6) (Lemma 2, (1 (H) (MP, (3), (2

(1) r _ T' = T :::J (A :::J A(T/T' (A6) (2) r _ T = T' :::J (A(T/T') :::> (A(T/T' (T'/T (A6)

i.e.

(A6)

o

(2) (3) (4)

r _ T = T' :::J (A(T/T') :::J A) r _ T' = T :::J (A(T/T') :::J A) r _ T' = T :::> (A == A(T/T'

(Lemmas 1 and 14, (2 (Lemma 13, (1), (3 0

THEOREM 8. r _ (3X) (A == A(T/X. Proof Suppose T' new. (1) (2)

(3) (4) (5)

r _ (VX)(X = T :::J (A == A(T/X r _ (1) :::J 3X)(X = T) :::J (3X) (A == A(T/X))) r _ (3X) (X = T) :::> (3X) (A == A(T/X r _ (3X)(X = T) r _ (3X) (A == A(T/X)

(Gn, Lemma 15)

(Lemma 7, (1 (MP, (1), (2 (A) (MP, (4), (3

So, given H, B is provable in L_ from A, this for any statement A of L_.

LEMMA 16. IJr_ B, then r_ (A == B) :::J A.

THEOREM 9. r _ (3X) (X = T). Proof. Suppose T' new. (1) (2) (3)

r_ T = T r _ (T' = T == T = T) :::> T' = T r _ (VX)( (X = T == T = T) :::> X = T)

(H) (Lemma 16, (1 (Gn, (2

o

(4)

(5) (6) (7)

ON AXIOMA TIZING FREE LOGIC

f-. (3) ::J 3X)(X = T == T = T) ::J (3X) (X = T f-. (3X)(X = T == T = T) ::J (3X)(X = T) f-. (3X) (X = T == T = T) f-. (3X)(X = T)

(Lemma 7) (MP, (3), (4 (B) (MP, (6), (5

So, given H, A is provable in L. from this special case of B: (3X)(X = T == T = T).15

v

13

o

Of the various semantic accounts of Free Logic, the truth-value one in [11] is by far the simplest. Let ~ be a possibly empty set of terms of FL. and Ul; be a unary function from the statements of FL. to 0 and 1. Then Ul; is said to constitute an identity-normal truth-value function for FL= if it obeys the following six constraints:

BI. = 1 if ul;(A) = 0 = 0 otherwise

B2. ul;(A ::J B) = 1 if ul;(A) = 0 or ul;(B) = 1 = 0 otherwise

B3. ul;V'X)A) = 1 if~ = @ or ul;(A(T/X = 1 for every term T in ~ = 0 otherwise

B4. ul;(T = T) = 1 B5. If ul;(T = T') = 1, then ul;(A) = ul;(A(T'I/T, where A is atomic. B6. If one of T and T' belongs to ~ but the other one does not, then

ul;(T = T') = o. It follows from these constraints that

ul;3X) (X = T = 1 if, and only if, T E ~. Note indeed that ul;(T = T) = 1 by constraint B4. So, if T belongs to ~, then there exists a member T' of ~ such that ul;(T' = T) = 1. Hence ul;3X) (X = T = 1 by constraint B3, constraint Bl, and the definition of '3'. Suppose, on the other hand, that ul;3X)(X = T = 1. Then ul;(T' = T) "# 0 for at least one member T' of ~, and hence by constraint B6 either both T and T' belong to ~ or neither one does. So T as well as T' belongs to ~. The set ~ to which the truth-value function U is relativized thus consists of those, and those only, among the terms of FL. which - so far as a is con-cerned - designate something. With A required in constraint B5 to be atomic, the foregoing account of Ul; is of course a recursive one.

These matters attended to, declare a statement A of FL. logically true in the truth-value sense if ul;(A) = 1 for every identity-normal truth-value function

14 HUGUES LEBLANC

u}; for FL.. Proof can then be retrieved from [11] and Section 3 of [13] that

I-F A if, and only if, A is logically true in the truth-value sense. Free Probability Logic is the result of similarly relativizing the constraints

placed upon probability functions in Standard Probability Logic. With only absolute probability functions attended to here,16 let 1: again be a possibly empty set of terms of FL. and let p}; be a unary function from the statements of FL. to the reals. Then p}; is said to constitute an identity-normal proba-bility function for FL= if it obeys the following ten autonomous constraints, adaptations and simplifications of Popper's constraints in [21]:17

CI. 0 ~ P};(A) C2. P};(-(A & -A)) = 1 C3. P};(-A) = 1 - P};(A) C4. P};(A) = P};(A & B) + P};(A & -B) CS. P};(A & B) ~ P};(B & A) C6. P};(A & (B & C)) ~ P};A & B) & C)

P};(A) if 1: = 0, otherwise P};(A & ... (B(TI/X) & B(T2/X)) & ... ) & B(T jX))) or

C7. P};(A & (\fX)B) = limit P};(A & ... (B(T/X) & BCT/X))

cs. P};(T = T) = 1

n~co

& ... ) & B(TjX))), where T I , T 2, , and Tn in the first case and T I , T2, , and Tn' . . . in the second are in alphabetical order the various members of 1:

C9. If P};(T = T') = 1, then u};(A) = u};(A(T'IIT)), where A is atomic CIO. If one of T and T' belongs to 1: but the other one does not, then

P};(T = T') = O. Note as regards constraint C7 that Po(-(A & -A) & (\fX)B) = Po(-(A & -A)). But P};(-(A & -A) & (\fX)B) is easily shown to equal p};\fX)B). So. by C2, Po\fX)B) = 1, as expected. ls

These matters attended to, declare a statement A of FL. logically true in the probability sense if P};(A) = 1 for every identity normal probability function p}; for FL . Proof can then be retrieved from Section 4 of [13] that

I-F= A if, and only if, A is logically true in the probability sense.

Results analogous to the two just obtained hold of course for Free Logic without Identity: write 'FL' everywhere for 'FL.', delete all occurrences of the qualifier 'identity -normal', drop constraints B4-B6 on page 13 and con-

ON AXIOMA TlZING FREE LOGIC 15

straints C8-CIO above, write 'f-F' in place of 'f-F-' in the two results in question, and the trick is done. So, as axiomatized in this paper, Free Logic with and without Identity is sound and complete in the probability as well as the truth-value sense. 19

VI

Inclusive Logic is but a timid prefiguration of Free Logic, which it antedates by some eight years.20 Both logics acknowledge 0 as a domain, thus lifting the first of the two presuppositions mentioned on page 2. But, whereas Free Logic lifts the second as well, Inclusive Logic does not: given any domain D other than 0, it requires each of the terms 't1" 't2" ... , 't/, ... to designate a member of D. So Inclusive Logic is exactly like Standard Logic except for counting 0 a domain.21 Axiomatizing it, however, is a delicate affair, and one - we now know - that has not been properly attended to in the past.22 Helpful in the process will be Quine's phrase "holding for 0" in [22], and the test he proposed there for deciding whether a theorem of Standard Logic holds for 0. Adapting Quine's instructions to suit the present context, mark the atomic statements of the sort T = T as true, mark the universal quantifications as true and the existential ones as false, and apply truth-value considerations. If the theorem of Standard Logic you are testing turns out to be a tautology, then the theorem in question holds for 0; otherwise it does not.

The axioms of IL_ are to be: (i) all the statements of IL_ of the sorts AI-A6 and FA8, (ii) all the statements of IL_ of the sort

IA9. (3X)(A v -A) ~ VX)A ~ A(T/X)), an axiom schema I borrow from [15], (iii) all the statements of IL_ of the sort

IAtO. (VY) (3X) (A v -A), the axiom schema discussed in Note 22, plus of course (iv) all the axioms that can be gotten from the foregoing by means of Fitch's

Clause. IA9 obviously ensures that when one's domain is non-empty, Inclusive Logic is exactly like Standard Logic. As for IAIO, it ensures - it so turns out -that 0 qualifies in Inclusive Logic as a domain. Note indeed that IA9 yields by Gn

(VY) 3X) (A v -A) ~ VX)A ~ A(Y/X))), which by A4 and MP yields

(VY)(3X)(A v -A) ~ (VY)VX)A ~ A(Y/X)),

16 HUGUES LEBLANC

which by IAlO and MP again yields of course

(\ty) \tX)A :J A(Y/X, i.e. FA 7'. So all the theorems of Free Logic with Identity are provable in Inclusive Logic with Identity. But Free Logic with Identity was so axiomatized as to make room for 0 as a domain. Hence so is Inclusive Logic with Identity. Hence Inclusive Logic with Identity is so axiomatized here as to be exactly like Standard Logic with Identity except for owning 0 as a domain.

The truth-value semantics of Section V is easily adjusted to suit Inclusive Logic with Identity, as is the probability one: take 0 and the set {'tl', 't/, ... , 'tn', ... } of all the terms of IL_ to be the only :E's to which the truth-value functions and the probability functions there are relativized.23 Proof that, given the present axiomatization of and semantics for IL_, IL_ is sound and complete in both the truth-value and the probability sense can be retrieved from [10] and [13], but the retrieval is a bit laborious at places.

IL is readily gotten from IL_: (i) drop '=' of course, (ii) substitute Fine's axiom schema FA6 for A6 and drop axiom schema FA8, and (iii) drop con-straints B4-B6 on page 13 and constraints C8-ClO on page 14.

The relationship between Standard Logic, Inclusive Logic, and Free Logic can be depicted as follows, '=' ignored from now on (and without prejudice) to expedite matters:

SL

IL

Note for proof that (i) every theorem of FL is provable in IL, as we just saw, but some statements of the sort IA9 are not provable in FL, as I shall establish below, and (ii) every theorem of IL is obviously provable in SL, but some statements of the sort (\tX)A:J A(T/X) are not provable in FL, '(\txl)(F(xl) & -F(xl :J (F(tl) & -F(tl' being the most obvious case in point. So, claims to the contrary notwithstanding, Free Logic is but a sublogic of Inclusive Logic, and of course Inclusive Logic is but a sublogic of Standard Logic.24 25

So the only two items of business left concerning the present axiomatiza-tion of Inclusive Logic are showing that IA9 is independent of the axiom schemata of FL and that IAlO is independent of the other axiom schemata of IL.


As regards IA9, understand by the tl-rewrite of a statement A of IL the result of deleting all the quantifiers that occur in A and substituting the term 'tl' for every variable that occurs in the resulting quasi-statement; and let u r be any truth-value function for IL that assigns 1 to every atomic statement of IL except 'F(t2r. It is clear that every statement of IL of any of the sorts AI-AS, FA6, and FA7' evaluates to 1 on u r . It is also clear that the tl-rewrite of any statement of IL gotten from the preceding axioms by means of Fitch's Clause also evaluates to 1 on ur' and that the tl-rewrite of the con-sequent of a conditional of IL is sure to evaluate to 1 under ur if the tl-rewrite of the conditional in question and that of its antecedent evaluate themselves to 1 under ur. Yet the tl-rewrite

of this statement of IL of the sort IA9:

does evaluate to 0 on u r. As regards IAIO, consider the following 4-valued truth-value fuction for

IL due to Roeper:

u",(B) ~(A) ~(-A) u",(A => B) 2h 1/3 0

0 2h 1/3 0 2/3 Ih

u",(A) 2h 1 Ih Ih Ih 2/3 1/3 2/3 1 2h 0 1 0 1 1

~(A(T/X ~VX)A)

2/3 2/3 1/3 2/3 0 2/3

It is easily verified that all the axioms of IL not of the sort IAIO evaluate to 1 under u r. Yet this statement of IL of the sort IAIO:

evaluates to 2/3 whatever truth-value is assigned to 'F(t l),. Indeed,

hence

18 HUGUES LEBLANC

hence

hence

hence

LIST OF THE LETTERED FORMULAS IN SECTION I

A. (3X) (X = T) B. (3X)(A == A(T/X c. (V X) (VY)A ::) (VY) (VX)A D. (3X) (X = T) ::) VX)A ::) A(T/X E. (3X) (A == A(T/X ::) VX)A ::) A(T/X F. (3X)(X = T) ::) T = T G. (VX)A ::) A(T/X) H. T=T I. (VY) (3X) (X = Y) J. (VY) VX)A ::) A(Y /X K. (VY)(3X)(A == A(Y/X L. (3X) (A V -A) ::) VX)A ::) A(T/X M. (VY) (3X) (A V -A)

Universite du Quebec a Montreal

NOTES

(= FA6) (= Theorem 1) (= Theorem 5)

(= FAS) (= FA7) (= FA7') (= FA7") (= IA9) (= IAtO)

I Free Logic dates back to 1959, the year that saw the publication of [14], a paper by Leblanc and Hailperin, and the publication of [6], a paper by Hintikka. The Free Logic in both cases is one with Identity. Free Logic without Identity dates back to 1963, the year that saw the publi-cation of [7], a paper by Lambert. As regards the first of the two presuppositions, recall Russell's remark on p. 203 of [23]; "The primitive propositions in Principia Mathematica are such as to allow the inference that at least one individual exists. But I now regard this as a defect in logical purity." 2 A is an adaptation of an axiom of Tarski's in [24], where - free variables doing duty in effect for terms - a variable other than X occurs in place of T. 3 The result is Fine's in [4]. 4 This is explicitly done in [2], but should be done as well in [8]. s [19] predates [14] and [6], and influenced the writing of [14]. Leonard's Law appears as an


axiom schema in several axiomatizations of Free Logic with Identity and in all axiomatiza-tions of what is known as the Logic of Existence. On the latter occasions,

(3X)(X = T) is either abridged as or shown logically equivalent to

E!T,

'E!' a predicate familiar to readers of Principia Mathematica. 6 That D, generally and rightfully held characteristic of Free Logic with Identity, nonetheless follows from axioms suiting Standard as well as Free Logic was first shown in [14]. The proof in Section II is a simplification of that proof and a later one in [9]. 7 E and, on page 2, K first appeared in [10], p. 167. I also reported there that the upcoming formulas J and K are interprovable, a matter I did not pursue any further at the time. Incidentally, credit for E should be shared with Cocchiarella, who reported to me after the publication of [10] that E appeared in the original - though not in the published - version of [3]. 8 The order in which the individual variables and individual terms of L_, SL_, FL_, as listed here will be known as their alphabetical order. 9 The present treatment of universal quantifications allows one to dispense with the distinc-tion between bound variables and free ones, a distinction which has proved particularly irksome in writings on Free Logic and Inclusive Logic. See [15] on this matter. 10 When I does not occur in A, each of A(I'/I) and A(I'//I) is of course A. II Since A here is a statement and identical quantifiers cannot overlap, X is sure not to appear in A. A quantification ('ltX)A is said to be vacuous, and its quantifier ('ltX) to be a vacuous quantifier, when X does not occur in A. See Note 24 for more on the converse ('ltX)A :J A of AS. 12 The restriction on X guarantees of course that ('ltX)A(X/T) is a statement of L. A like remark applies on later occasions but will not be repeated. Note that when X is foreign to A, ('ltX)A(T/X) is but the vacuous quantification ('ltX)A. 13 The restriction placed either on T as here or on T' as in the proof of Theorem I and that of Theorem 2 on page 6 is crucial. One example will suffice. Though the conditional

is provable from

for every i larger than I, it is not from

14 Bencivenga had already shown in (1) that FA7 is independent of AI-A6, FA8, and Leonard's Specification Law for Free Logic with Identity (= Theorem I), hence in effect that the law in question could not substitute for either of FA7 and FA7' in the foregoing axiomatizations of FL_. He also showed there that this fascinating law:

('ltX)3Y)(Y = X) :J A) :J ('ltX)A,

is independent of AI-A6, FA8, and Leonard's Law. However, Bencivenga's Law - as I take leave to call it - readily follows from A4, Lemma 2, and FA7. Note indeed that

('ltX)3Y)(Y = X) :J A) :J 'ltX)(3Y)(Y = X) :J ('ltX)A)

by A4, hence

('ltX)(3Y)(Y = X) :J 'ltX)3Y)(Y = X) :J A) :J ('ltX)A)

20 HUGUES LEBLANC

by Lemma 2, and hence

(VX)3Y)(Y - X) :::> A) :::> (VX)A by FA7 and MP. 15 The relationship between

T=T'

and

A(T'/T) is even closer than the foregoing results suggest. Note indeed that

A:5 A(T/T), If A :5 A(TIT'), then A :5 A(T'IT),

and

If A :5 A(T/T') and A :5 A(T'IT"), then A - A(TIT"). So, like Identity, Substitutivity in the sense of '/' is an equivalence relation. Note also that as

(3X)(X - T) :::> T = T is provable in FL., so - trivially, to be sure - is its counterpart

(3X)(A:5 A(X/T :::> (A :5 A(T/T in FL. 16 For a similar treatement of the matter with conditional probability functions rather than absolute ones, see Section 5 of [13]. However, absolute probability functions particularly suit the present occasion: as shown in [12], truth-value functions are those (and those only) among absolute probability functions that are two-valued. 17 See [12] on this matter. The constraints in question are autonomous in that, far from presupposing this Interchange Law of other axiomatizations of absolute probability theory:

If 1-. A :5 B, then Pz;(A) = Pz;(B), they permit proof of it. 18 For more on constraint C7, particularly the presence in it of the conjunct A, see [17]. Unlike the account of (1z; that of Pz; is not a recursive one. In point of fact no recursive account of Pz; can be had since Pz;(A & B) for atomic A and B is not always a numerical function of Pz;(A) and Pz;(B). 19 Many a model-theoretic semantics for Free Logic will be found in the literature, the earliest being undoubtedly van Fraassen's in [25], which introduced the celebrated method of super-valuation. Two others, of quite a different character, will be found in [16] and [18]. The second of these introduced the two-domains method favored by many. 20 Two of the earliest papers on Inclusive Logic are [20] and [22], which appeared in 1951 and 1954, respectively. 21 My understanding of what counts as Free Logic and what as Inclusive Logic is, I believe, the more common one. In [2], however, Bencivenga considers it characteristic of an inclusive logic that it lifts the first presupposition on page I, and characteristic of a free logic that it lifts the second. Under this understanding of things, FL. and FL would be free logics that are inclusive, and IL. and IL would be inclusive logics that are not free. FL. and FL are readily made into free logics that are not inclusive: simply enlist (3X)(A v -A) as an extra axiom schema of both logics.


n Absent indeed from previous axiomatizations of IL_ and IL, the one in [15] among the latter, is IAI0 in (iii) on page 15, an axiom schema recently shown by Roeper to be indepen-dent of those in [15]. Yet, no matter the truth-value function 0.1: or the probability one P1: for IL_, statements of IL_ of the sort IAI0 all evaluate to 1 under 0.1: and P1: and hence are all logically true. 23 That the statements of IL_ of the sort IAI0 all evaluate to I under the resulting functions is obvious enough. 24 Studied in [20] and [15] are Inclusive Logics where (\iX)A, when vacuous, is provably equivalent to A. Quine's test will suit them if vacuous quantifiers are deleted before a state-ment is subjected to the test. The axiom schemata of IL in [15] are AI-A3, AS, the converse of AS, and this restricted version of A4:

(\iX)(A :::> B) :::> \iX)A :::> (\iX)B), so long as X is foreign to B.

Whether that logic is complete in the sense of this paper has yet to be ascertained. 25 The claim in question was made in [16] for instance. 26 Thanks are due to William A. Wisdom (Temple University) whose queries concerning the various axiomatizations of FL_ prompted the writing of this paper; to Ermano Bencivenga (University of California, Irvine) who brought to my attention his 1978 paper (hence, to what I call in Note 13 Bencivenga's Law) and to whose 1986 paper Section III owes much; to Willard V. Quine who noted that wanted in B, and hence in E and K, is the single slash rather than the double one which I had originally used; and to Peter Roeper (Australian National University) who suggested the present version of constraint B5 and contributed the closing independence proof of Section VI. Thanks are also due to Raymond Gumb (University of Lowell), Lisa Pastino (Temple University), Gilles St-Louis (I'Universite du Quebec i\ Montreal), and to William A. Wisdom (Temple University) who read earlier versions of the paper; and last, but not least, to Alain Voizard (Universite du Quebec i\ Montreal) who also read earlier versions of the paper and translated it into French. And thanks are due to the Social Sciences and Humanities Research Council of Canada which supported the research leading to it. A partial version of the text was read in December of 1992 at Concordia University (Montreal), the present version was read in October of 1993 at the Universitat Salzburg, and and a French translation of it was read in April of 1993 at l'Institut d'Histoire et de Philosophie des Sciences et des Techniques (Paris).

REFERENCES

1. Bencivenga, E., 1978, 'A Semantics for a Weak Free Logic', Notre Dame Journal of Formal Logic 19, 646-652.

2. Bencivenga, E., 1986, 'Free Logics', in Handbook of Philosophical Logic, vol. 3, D. Reidel Publishing Company, Dordrecht, pp. 373-426.

3. Cocchiarella, N. B., 1966, 'A Logic of Actual and Possible Objects', The Journal of Symbolic Logic 31, 689-690.

4. Fine, K., 1983, 'The Permutation Principle in Quantificational Logic', Journal of Philosophical Logic 12, 31-37.

5. Fitch, F. B., 1948, 'Intuitionistic Modal Logic with Quantifiers', Portugaliae Mathematica 7,113-118.

6. Hintikka, J., 1959, 'Existential Presuppositions and Existential Commitments', The Journal of Philosophy 56, 125-137.

7. Lambert, K., 1963, 'Existential Import Revisited', Notre Dame Journal of Formal Logic 4,288-292.

8. Lambert, K., 1991, 'The Nature of Free Logic', in Philosophical Applications of Free Logic, Oxford University Press, New York - Oxford.

9. Leblanc, H., 1968, 'On Meyer and Lambert's Quantificational Calculus FQ', The Journal of Symbolic Logic 33, 275-280.

22 HUGUES LEBLANC

10. Leblanc, H., 1971, 'Truth-Value Semantics for a Logic of Existence', Notre Dame Journal of Formal Logic 12, 153-168.

11. Leblanc, H., 1976, Truth-Value Semantics, North-Holland Publishing Company, Amsterdam New York Oxford.

12. Leblanc, H., 1982, 'Popper's 1955 Axiomatization of Absolute Probability', Pacific Philosophical Quarterly 63, 133-145.

13. Leblanc, H., 1983, 'Alternatives to Standard First-Order Semantics', in Handbook of Philosophical Logic, vol. 1, D. Reidel Publishing Company, Dordrecht, pp. 189-274.

14. Leblanc, H. and Hailperin, T., 1959, 'Nondesignating Singular Terms', The Philosophical Review 68, 239-243.

15. Leblanc, H. and Meyer, R. K., 1969, 'Open Formulas and the Empty Domain', Archiv fUr mathematische Logik und Grundlagensforschung 12, 78-84.

16. Leblanc, H. and Meyer, R. K., 1982, 'On Prefacing ('v'X)A ::J A(Y/X) with ('v'Y): A Free Quantification Theory without Identity', in Existence, Truth, and Provability, State University of New York Press, Albany, pp. 58-75. The paper there is an amended version of the original, which had appeared in ZeitschriJt fUr mathematische Logik und Grundlagen der Mathematik 12, 1971, pp. 153-168.

17. Leblanc, H. and Roeper, P. 1993, 'On Getting the Constraints on Popper's Probability Functions Right', Philosophy of Science 60, 151-157.

18. Leblanc, H. and Thomason, R. H., 1968, 'Completeness Theorems for Some Presupposition-Free Logics', Fundamenta Mathematicae 62,125-164.

19. Leonard, H. S., 1956, 'The Logic of Existence', Philosophical Studies 7,49-64. 20. Mostowski, A., 1951, 'On the Rules of Proof in the Pure Functional Calculus', The Journal

of Symbolic Logic 16, 107-111. 21. Popper, K. R., 1959, The Logic of Scientific Discovery, Basic Books, Inc., New York. 22. Quine, W. V., 1954, 'Quantification and the Empty Domain', The Journal of Symbolic Logic

19,177-179. 23. Russell, 8., 1919, Introduction to Mathematical Philosophy, George Allen and Unwin, Ltd.,

London. 24. Tarski, A., 1965, 'A Simplified Formulation of Predicate Logic with Identity', Archiv fUr

mathematische Logik und Grundlagen der Mathematik 7, 61-79. 25. van Fraassen, 8. C., 1%6, 'Singular Terms, Truth-value Gaps, and Free Logic', The Journal

of Philosophy 67, 481-495.

24 FRANyOIS LEPAGE

recent. Pavel Tichy [Tic82] not only formulated the project explicitly, but gave motivations for such an enterprise. His motivations remain, in my opinion, globally valid, and it is worth recalling them here.

The first motivation is that 'the logic underlying ordinary language - and hence our conceptual scheme - is that of the simple type theory'. What is meant here is without doubt, that type theory possesses sufficient resources for expressing the inferential structures one finds in natural languages; struc-tures whose complexity often exceeds the expressive capacity of first order calculi. The second motivation is that it is necessary to enrich the classical type theory from inside 'by dropping the totality assumption and treating partial functions on a par with total ones'. In fact, 'the partial theory seems to provide a medium which yields an analysis for any linguistic expression, and affords a universal explication of logical entailment'.

We see that for Tichy the idea of introducing partial functions in type theory is not presented as the adoption of a particular hypothesis but, on the contrary, as the lifting of an arbitrary restriction, i.e., that which consists of using only a subset of functions for semantic values - the subset of total functions, which are defined for all the arguments of the right category. It is difficult to determine whether TichY's motivation was primarily epistemic. What is certain, at any rate, is that his motives can be interpreted in an epistemic sense.

Tichy did not completely realize his project, in part because he did not know how to use the full power of the partial theory of types, lacking as he did a fundamental technical notion to which I will come back later. Now we will present the framework that will serve to introduce partial interpretations, i.e., the theory of propositional types.

THE THEORY OF PROPOSITIONAL TYPES

The partial theory of types results from generalizing the well-known simple theory of types by authorizing the presence of partially defined entities. Let us recall some elementary facts regarding type theory, in particular regarding the theory of propositional types.

Simple type theory is a simplified version of Russell's ramified theory of types. It was initially formulated by Chwistek and Ramsey, and the first full presentation is that of Church [Chu40].

In its contemporary versions, the theory of types consists first of the con-struction of a hierarchy of functions in the following way.2

DEFINITION 1. The set of types T is the smallest set such that

(i) e E T (e is the type of individuals) (ii) t E T (t is the type of individuals) (iii) if a, ~ E T, then (a~) E T.

The domains of each type are:

PARTIAL PROPOSITIONAL LOGIC 25

DEFINITION 2. For each a. E T, the set of entities of type a. is the set Do. such that

(i) De = E (where E is a non-empty set; E is the set of individuals); (ii) D, = {a, I} (the set of truth values); (iii) D(o.r.> = Dr,Do. (the set of functions of Do. in Dr.; we will simply

write Do.r.).

It is possible to provide a logical calculus comprising a denumerably infinite number of variables of each type and a very small number of logical constants, three in fact, these being the functional abstractor, its converse the functional application and identity. Henkin [Hen50] proposed an axiomatization of such a system and proved its completeness in a general sense. Gallin [GaI75] even proposed an intentional version and gave a complete axiomatization, as always, for general models.

For the time being we will concern ourselves with an essentially simple calculus, based on the hierarchy constructed exclusively from D, = {a, I} and dropping clause (i) of Definitions I and 2, which we will call the theory of propositional types. This calculus is interesting for two reasons. The first of these is purely theoretical: it is doing away with the hypothesis that objects exist. All valid statements of this calculus express properties that rest ultimately on propositional forms in general. In that sense, this calculus is a true logic (even though it contains variables for propositional functors of every order and allows quantification over these variables) and has a status vis-ii-vis the theory of types in general analogous to that which the propositional calculus has vis-ii-vis first order predicate calculus: its valid statements constitute a 'hard core' of the set of valid statements of the simple theory of types.

The second reason is more pragmatic: every domain Do. is finite, a fact which sometimes, simplifies things greatly, as we will see.

It would be a good idea to provide a date and place of birth to the theory of propositional types, both for internal reasons - its fundamental concepts are equally concepts of the simple theory of types and did not appear simul-taneously - and for external reasons - some concepts that were introduced in articles which went mostly unnoticed were simply reinvented later on. Here is an outline of the genesis of the theory in so far as I am capable of reconstituting it. To simplify things, let us start with a text which presents the theory in an complete way, a rather little-known text: 'A Theory of Propositional Types' by Leon Henkin [Hen63]. The calculus presented by Henkin uses only the abstractor A, its converse and identity as primitive symbols; it is complete and the proof of its completeness is constructive (in fact, for every element of every domain Do. there is an expression of the language of which that function is the value).

Here is a simplified presentation of that calculus. First of all, for every type a., there is a denumerable set Var a. = {xo.); EW' of variables of that type.

26 FRAN


When a formula A is closed ViA) is independent of J.l and V Il(A) is called the denotation of A and is written Ad.

The axiomatic system proposed by Henkin includes seven axioms and one replacement rule.

HI: [Aa == A] H2: [[At == 11 == A] H3: [[T /\ F] == F] H4: [lfrtT /\fF] == 'v'x,[.fxJ] H5: [xa == Ya] :J [[fa~ == ga~] :J [[fx] == [gym H6: ['v'xa[[fa~xaJ] == [ga~xa]] :J [j== gJ] H7: [[AxaABJ == A {Blx}] where B is free for x and A {Blx} is the formula

obtained from A by substituting B for each free occurrence of x.

RULE R. From (A == B) and C we can derive D, where D results from C by substituting an occurrence of B for an occurrence of A. If A is a theorem, we write f-A.5

To demonstrate the completeness of this system, Henkin proceeds in the following way:

(1) For any a E Da, there is a closed formula, call it an such that (an)d = a. This formula is a canonical name of a and Henkin gives an explicit algo-rithm for its construction. In particular r = T and Fn = F and Td = 1 and pi = O.

(2) Let All be the formula obtained from A by substituting the formula (J.l(xa)t. for every occurrence of the free variable Xa. Henkin shows that for any formula A we have f-[All == (J.l(A)tJ.

(3) In particular, if A is valid and closed, we will have f-[A == T] and from that we can easily show that f-A. (If A is valid and non-closed, we go by way of its universal closure.)

This article of Henkin's seems not to have received the attention it deserves. In fact I have found only two references to it, in Gallin's [Gal75] and Farmer's [Far90]. Nevertheless it seems to be the first presentation of a logic of superior order containing only three primitive symbols. Moreover, Henkin thought he was the first to suggest reducing the quantifiers to these operators. On page 324 of the first version of the text, after having paid homage to Tarski [Tar23t for having shown how to reduce the usual logical constants in terms of quantifiers and identity; he says that he seems to be the first, to propose an explicit reduction of the quantifiers in terms of the A operator and identity. In the published version, he adds a note thanking Peter Andrews for having pointed it out to him that this reduction had been done explicitly by Quine [Qui56].7

If Henkin's system is the first to use only the three primitive symbols in question, the first theory of propositional types is the protothetics of

28 FRANyOIS LEPAGE

Lesniewski. Unfortunately, we do not have Lesniewski's text, which may have been destroyed during the liberation of Warsaw in 1944. The most faithful version at our disposal is Jerzy Slupecki's [Slu53], published in the first issue of Studia Logica, entitled 'St. Lesniewski protothetics', which was drawn from notes taken by Lesniewski students. Curiously, Henkin did not know this text - indeed he thanks the referee for having pointed it out to him. This is all the more curious since Henkin himself refers to Lesniewski in the version of Andrzej Grzegorczyk [Grz55] 'The System of LeSniewski in relation to con-temporary Logical Research' which appeared in Studia Logica two years later. It must be said that Grzegorczyk does not note the existence of Slupecki's text. We will end this point by noting that Montague [Mon74] recapitulates the essence of the translations of Henkin in 'Universal Grammar' without pointing it out but refers explicitly to Tarski.

So much for the theory of propositional types. Let us now examine the matter of partiality.

PARTIAL LOGIC

As for the theory of propositional types, it would be difficult to give a birth date for partial logic. I noted earlier that the first explicit formulation of a system of partial logic was TichY's, but we can find many of the concepts he used in texts whose main purpose is not to make a contribution to partial logic.

The first serious attempt to make a systematic presentation of partial logic is that of Blarney [Bla86]. In fact, Blarney's text contains the essential ingre-dients for the elaboration of a general partial logic as well as the instructions for doing so, but he stops at the threshold of the enterprise and restrict himself to first-order logic.

First of all let us consider these essential elements. The first of these is the introduction of the undefined as a semantic object, that is the introduc-tion of the name of the undefined in the metalanguage. This may seem trivial but it is a crucial step: without this 'ontological commitment', it does not seem possible to describe the reiteration of functional application in a systematic and coherent way. This is the fundamental notion lacking in Tichy that we mentioned earlier, which led him to reject domains of functions in favour of domains of relations and thus essentially to be restricted to the first order.8

The argument is simple. Suppose - we reason on the basis of an example, the generalization of which goes without saying - that An' Bf'> and C"f are three expressions such that [BC] and [A[BC]] are well formed terms. The metalinguistic notions of 'being defined' and 'being undefined' apply to the values of the expressions (which are functions), and it is only be extension - by a trivial abuse of language - that the notion applies also to expressions depending on whether its value is or is not defined. It is necessary not to


confuse our use of the notion 'undefined' with another use, that which consists of calling an expression undefined if it is not well-formed, that is if it is not a meaningful term. In this case one should rather speak of nonsense.

In the classical context, the expressions A, B, C would take total func-tions as values, so the value of [BC] would be a total function of the type of the arguments of A and thus the value of [A[BC]] would be a total function as well. What would happen if the values of these expressions, still being of the right type, were partial? Specifically, what would happen if the value of B were a partial function undefined for the argument which is the value of C? The value of [BC] is not defined. In that case, [A[BC]] has no value, that is, it is not possible to give a sense to the expression 'value of [A[BC]]'.

A general solution to this problem consists of giving the undefined an intratheoretical status. The undefined becomes an object like the others and can thus be a value and an argument of a function. This simple addition enriches the metalanguage sufficiently for us to be able to describe the whole hierarchy of partial functions.

The idea of explicitly giving the undefined the status of 'object' goes back to Dana Scott [Sc073] who, curiously, introduced it in order to get rid of the hierarchy by constructing his reflexive domains which would serve to inter-pret the typeless A-calculus. We might equally say that the idea goes back to Kleene with his 'strong' connectors, where the undefined appeared explic-itly in the truth tables. My own idea [Lep84] was to use this trick to define the hierarchy of partial functions. Let us see how this is done.9

DEFINITION 7. For any a E T, the set PMa of partial functions of type a is

(i) PM,={O,l,

30 FRANC;OIS LEPAGE

and (iii) for x, Y E PM" x V y = x if y =


DEFINITION 9. For any a. E T, the set PTa s: PMa (the set of partial total functions) is the smallest set such that

(i) for a. = t, and x E PM" x E PT, iff x :t:-

32 FRANyOIS LEPAGE

such that ~(xa) E PMa. As before, we write ~(a/x) for the assignation that differs from ~ at the most by the fact that it assigns the value a to x

Finally, we define a partial valuation based on ~.

DEFINITION 14. A partial valuation based on ~ is a function

VIl: U Trma ~ U PMa

such that

(i) (ii)

(iii) (iv)

aeT aeT

v Il(xa) == ~(xa); V Il([Aa == Ba]) = 1 iff V Il(A) E PTa and V Il(A) "" V iB)

o iff VIl(A) ** VIl(B)


Let us return to Diagram 1. Clearly

o

represents propositional identity and that -,,/ = A.Xt[x == F]d will be the function represented by the following diagram

o 1

What about the binary connectors? An elementfof PMt(It) can be represented according to the following convention

f(1)(1) f(1)(O)

f(1)(q

f(O)( 1) f(O)(O)

f(O)(q

f(q(1) f(q(O)

f(q(q

Diagram 2.

If we execute the calculus by taking the denotation of A t(1I) for f, namely

At(tt)d = Ax,AYt[A.j,(tt)[[fx]y] == A.f[lfT]T]]d we get the function represented by Diagram 3

1 0 o 0

o

q> 0

Diagram 3.

Although there are no surprises in logic, it is remarkable to find Kleene's strong conjunction here. ll We clearly get Kleene's strong disjunction by the usual translation A v B =def JA A B).

34 FRAN


[Aa == Aa] is not verifiably valid but [AXaX == AXaX] is. The problem, one will have guessed, is to give a system (complete if possible) for the class of verifiably valid statements. The question is not yet resolved, and the major difficulty rests in the absolute impossibility of constructing an expression Elf of the object language, the interpretation of which would be [EA] is true if and only if A is not defined and false otherwise. It is easy to convince oneself that such a functor would not be monotone.

For the time being we must content ourselves with fragmentary results [Lrep92]. First of all, if it is not possible to introduce in the object language a functor [EA] the interpretation of which would be 'A is undefined', it is possible nonetheless to introduce a functor ~(A), the interpretation of which, roughly, is 'A is total'.

This is why we say 'roughly': ~(Aa) is true when A is total and undefined otherwise. ~(Aa) is never false and in this way the monotony is preserved. Such a functor is of great utility. In effect, it makes it possible to introduce axioms of the form 'if A, B, ... , are total, then ... ' and permits the elab-oration of a system of partial logic.

A DEDUCTIVE SYSTEM FOR PARTIAL PROPOSITIONAL LOGIC

The first stage consists in characterizing the subset of formulae of proposi-tional calculus as verifiably valid tautologies. We proceed in the following way.

DEFINITION 19. Let FP ~ Trm" be the set of propositional formulae, the smallest set such that

(i) Var, U {T, F} ~ FP (ii) if A, B E FP, then -,A, [A 1\ B], [A ::::) B), [A v B], [A == B) E FP.

Using an idea of Henkin's, we recursively define two subsets of FP:

DEFINITION 20. Let FP+ and FP- be two subsets of FP such that (i) T E FP+ and F E FP-; (ii) if A E FP+ and B E FP-, then -,B E FP+ and -,A E FP-; (iii) if A, B E FP+, then [A 1\ B] E FP+, and if A E FP- then for any

B E FP, [A 1\ B], [B 1\ A] E FP-; (iv) if A E FP+ and B E FP-, then [A ::::) B] E FP-, and for any C E

FP, [C ::::) A], [B ::::) C] E FP+; (v) if A E FP+ and B, C E FP-, then [B v C] E FP-, and for any D

E FP, [A v D], [D v A] E FP+;

36

(vi)

(vii)

FRAN


DEFINITION 21. There is a proof of A from a set H of hypotheses (in symbols H f- A) if and only if there is a series of n formulas AI' ... , An such that A = An and for all i :::; n:

(i) Ai E H, or (ii) Ai is such that => Ai is a rule or (iii) Ak, , Al => Ai (with k, I < i) is a rule, or (iv) Ai result from the application of one of the following improper rules: R1 * Ai = VX I . Vx"B, and there is a proof of B from S(x l ), ,

S(xn) R2* Ai is [B =::: C] and there is a proof of C from B and a proof of B

from C, and for a certain k < i, Ak = S(B) or Ak = S(C). R3* Ai = B,{Calxa} and there is a proof of B from the null set and C

is free for x in A.

If H is empty, we write I-A.

PROPOSITION 22. If A E Fr, then I-A.

One easily obtains the following generalization:

PROPOSITION 23. If A E FP is a classical tautological form whose free variables are XI' ... , Xn, then f-Vx I ... VxnA.

It does not seem possible, following Henkin, to generalize this result to the totality of valid formulae of the partial theory of propositional types. The unavoidable problem linked to this approach is the impossibility of having a canonical name in the object language for every partial function without modifying the theoretical framework in an essential way. Suppose, for example, that q> had a name, say q>n. One immediate consequence would be that, even for partial total valuations, some expressions would remain undefined. Even if such systems of logic can be of some interest, they are, from the point of view adopted here, unacceptable.

So far, we have tried to characterize classes of valid statements by departing from the very general context of the theory of propositional types. Finally, let us say a few words about another way of proceeding, more classical, which consists of defining more elementary partial logics and attempting to enrich them. It is possible in this way to define a first order functional calculus. The syntax of this calculus is completely classical, but the interpretation admits partials predicates and partial functions of arbitrary degrees of definition. Moreover, we can provide a complete system for this functors of superior order also raises problems of the same order as those encountered in the theory of propositional types.

Universite de Montreal

38 FRAN


[Ga175] Gallin, D., 1975, Intensional and Higher-Order Modal Logic, North-Holland Amsterdam.

[Grz55] Grzegorczyk, A., 1955, 'The Systems of LeSniewski in Relation to Contemporary Logical Research', Studia Logica 3, 77-95.

[HenS 0] Henkin, L., 1950, 'Completeness in the Theory of Types', The Journal of Symbolic Logic 15, 81-91.

[Hen63] Henkin, L., 1983, 'A Theory of Propositional Types', Fundamenta Mathematicce 52, 323-344.

[Lap92] Lapierre, S., 1992, 'A Partial Semantics for Intensional Logic', Notre Dame Journal of Formal Logic 33(4), 417-541.

[Lrep92] Lapierre, S. and Lepage F., 1992, 'Toward a Calculus of Partial Propositional Types', Cahier du departement de philosophie no 92-11, Universite de Montreal.

[Lrep93] Lapierre, S. and Lepage F., 1993, 'La completude du calcul propositionnel des prMi-cats du premier ordre avec identite pour les interpretations partielles', Cahier du departement de philosophie no 93-04, Universite de Montreal.

[Lep92] Lepage, F., 1992, 'Partial Functions in Type Theory', Notre Dame Journal of Formal Logic 33(4),493-516.

[Lep84] Lepage F., 1984, 'The Object of Belief', Logique et Analyse 36,106,193-210. [Mon74] Montague, R., 1974, 'Universal Grammar', in Formal Philosophy, Yale University

Press, New Haven, pp. 222-246. [Muss88] Musken, R., 1988, 'Going Partial in Montague Grammar', ITLI Prepublication Series,

LP-88-04, University of Amsterdam. [Qui56] Quine, W. V., 1956, 'Unification of Universes in Set Theory', The Journal of Symbolic

Logic 21, 267-279. [Sc073] Scott, D., 1973, 'Models for Various Type-free Calculi', in P. Suppes et al. (eds.),

Logic. Methodology and Philosophy of Science IV, North-Holland, Amsterdam, pp. 157-187.

[Siu53] Siupecki, J., 1953, 'St. Lesniewski protothetics', Studia Logica 1, 44-111. [Tar23] Tarski, A., 1923, 'Sur Ie terme primitif de la logistique', Fundamenta Mathematicce

IV, 59-74. [Thi87] Thijsse, G. C. E., 1987, 'Partial Logic and Modal Logic: A Systematic Survey',

manuscript. [Tic82] Tichy, P., 1982, 'Foundations of Partial Type Theory', Reports on Mathematical

Logic 14, 59-72. [Urq86) Urquhart, A., 1986, 'Many-Valued Logic', in Gabbay, D. and Guenthner, F. (eds.),

Handbook of Philosophical Logic Vol. 11/, Reidel, Dordrecht, pp. 71-116.

SERGE LAPIERRE

GENERALIZED QUANTIFIERS AND INFERENCES

INTRODUCTION

The semantic treatment of noun phrases and determiners in Montague (1973) presupposes the notion of "generalized quantifiers" proposed in Mostowski (1957). The importance of this notion for natural language has been brought out explicitly in Barwise and Cooper (1981) and Keenan and Stavi (1986). The basic idea is to let a noun phrase DA (all men, some women, most students, etc.) to denote a set of sets of individuals, that is to say, the set of the deno-tations of the verb phrases B for which (DA)B holds. For instance, given a fixed model with a non-empty universe E.

all A denotes {X I: E: [[AJ] I: X}, some A denotes {X I: E: [[AJ] n X"* 0}, most A denotes {X I: E: I[[AJ] n XI > I[[AJ] - XI},

where [[AJ] is the extension of the predicate A in the model. As Lindstrom (1966) pointed out, this analysis suggests that determiners denote binary relations between sets. For instance, given a non-empty universe E.

all denotes {(X, Y) E r:JP(E) x Q1>(E): X I: Y}; some denotes {(X, Y) E Q1>(E) x Q1>(E): X n Y"* 0}; most denotes {(X, Y) E Q1>(E) x Q1>(E): IX n YI > IX - YI}.

Such binary relations are called (local) binary generalized quantifiers. More generally, a (global) n-ary generalized quantifier (n ~ 1) is a function Q which assigns to every non-empty universe E an n-ary relation QE between subsets of E. From now on we will stick with global binary generalized quan-tifiers and we shall simply call them quantifiers. In order to define or exhibit a property of a quantifier Q operating on a universe E. we shall write "QEXY" instead of "X, Y I: E and (X, Y) E QE". Moreover, for denoting a familiar quan-tifier having a simple determiner, we shall write the determiner itself in italic. For instance:

allEXY iff X I: Y; someEXY iff X n Y "* 0; noEXY iff X n Y = 0; all and someEXY iff X I: Y and X n Y = 0; at least halfEXY iff IX n YI ~ IX - YI.

This abuse of notation is to be preferred to an abundance of brackets or quotes. Note that the specification of the parameter E is relevant only for context-

dependent quantifiers. For instance, according to a plausible intuition about

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M. Marion and R. S. Cohen (eds.), Quebec Studies in the Philosophy of Science I, 41-55. 1995 Kluwer Academic Publishers.

42 SERGE LAPIERRE

the meaning of the determiner "many", we may set manyEXY if and only if the proportion of Y-individuals in X is larger than the proportion of Y-indi-viduals in the whole universe E; formally:

manyEXY iff IX n YI/IXI > IYI/IEI Since the proportion IYIIIEI decreases when the size of E increases, many in this sense is not a context-free quantifier. However, most of the natural language quantifiers are context-free and even logical in the sense that only the cardinalities IX - YI and IX n YI are relevant for deciding whether QEXY or not. This is clearly the case for all, some, no, all and some, at least half, most and many others, From now on, we shall restrict our attention to logical quantifiers.

As relations, quantifiers have relational properties. For instance, all is reflexive and transitive, some is symmetric, not all is connected. These properties can be considered as inferential properties. For instance, symmetry allows us to infer someYX from someXY. If we consider other conditions, more inferential properties emerge. For instance, from allXY we may infer all(X n Z)Y, where Z is any set, because all is downward-left monotonic (i.e., if QXY and X' !;;;; X, then QX'Y).

The inferential properties of quantifiers can be studied from two opposite, but complementary perspectives. First, there is the "inverse logic" perspective, which consists of considering some specific inferential properties and finding which quantifiers have those properties. The opposite view - the "direct logic" perspective - consists of considering some specific quantifiers and studying their inferential behaviours. This paper belongs essentially to the inverse logic, since our purpose is to summarize some results in the analysis of the infer-ential properties of some quantifiers.

By its nature, our study depends on two things. First, it depends on a prior choice of formalism, expressing more or less the relevant inferential proper-ties. For instance, we may decide, as in Section 1, to study only simple relational properties; in that case a formalism consisting only of atoms of the form QXY is sufficient. But in order to express more specific properties of quantifiers, such as the various forms of monotonicity, we need a formalism allowing in addition Boolean set terms in the argument of the quantifier relation. This more expressive formalism will be used in Sections 2 and 3 when we study some "conditional quantifiers", that is to say quantifiers which behave as conditional relations.

Besides a prior choice of formalism, another important decision concerns the cardinality of the universes. Must we admit finite universes only, or infinite universes as well? Some theoreticians would prefer to stick with finite uni-verses. There is an empirical reason for this decision: natural language seems to require finite models only, infinite ones arising only through philosoph-ical or scientific considerations. But there is also a methodological reason for the finiteness restriction: it simplifies results and proofs and one notes that most of the results obtained under the finiteness assumption also hold

GENERALIZED QUANTIFIERS AND INFERENCES 43

on infinite universes of an arbitrary cardinality. For this, we will assume the finiteness restriction in Section I because all of the results obtained hold on finite universes as well as on infinite universes. The situation will be dif-ferent when we consider conditional quantifiers, since some of them do not have a stable behaviour when switching from finite universes to infinite ones, or from denumerable universes to infinite ones of higher cardinalities. Thus infinity appears relevant here and for this reason we will be obliged to drop the finiteness restriction.

1. PURE SYLLOGISTIC THEORIES OF SOME QUANTIFIERS

There are many different formalisms, having more or less expressing power, which can be used for expressing the inferential properties of a given quan-tifier. In this section we shall stick with the minimal formalism Lsyll, which consists only of elementary formulae of the form QXY, where Q is a quanti-fier symbol and X, Yare set variables. This formalism is rich enough for expressing "pure syllogistic" patterns of inference such as:

=>QXX => QXY QXY => QYX QXY => QXX QXY => QYY QXX => QXY QXY, QYZ => QXZ

reflexivity universality symmetry quasi-reflexivity weak-reflexivity quasi -universality transi

(boston studies in the philosophy of science 177) hugues leblanc (auth.), mathieu marion, robert s....

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