Entailment, Vol. II: The Logic of Relevance and Necessity 9781400887071

Table of contents :
CONTENTS
ANALYTICAL TABLE OF CONTENTS
PREFACE
ACKNOWLEDGMENTS
SUMMARY REVIEW OF VOLUME I
CHAPTER VI. THE THEORY OF ENTAILMENT
CHAPTER VII. INDIVIDUAL QUANTIFICATION
CHAPTER VIII. ACKERMANN'S STRENGE IMPLIKATION
CHAPTER IX. SEMANTICS
CHAPTER X. PROOF THEORY AND DECIDABILITY
CHAPTER XI. FUNCTIONS, ARITHMETIC, AND OTHER SPECIAL TOPICS
CHAPTER XII. APPLICATIONS AND DISCUSSION
BIBLIOGRAPHY
INDEX OF NAMES
INDEX OF SUBJECTS
SPECIAL SYMBOLS

Citation preview

ENTAILMENT

ENTAILMENT THE LOGIC OF RELEVANCE AND NECESSITY by

ALAN ROSS ANDERSON and

NUEL D. BELNAP, JR. and

J. MICHAEL DUNN with contributions by K I T FINE ALASDAIR URQUHART

and further contributions by DANIEL COHEN

G L E N HELMAN

STEVE GIAMBRONE DOROTHY L. GROVER ANIL G U P T A

ERROL P. M A R T I N MICHAEL A, MCROBBIE STUART SHAPIRO

and including a Bibliography of Entailment by ROBERT G. W O L F

VOLUME II

PRINCETON UNIVERSITY PRESS

COPYRIGHT ©

1992 BY PRINCETON UNIVERSITY PRESS

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford All Rights Reserved Library of Congress Cataloging-in-Publication Data (Revised for vol. 2) Anderson, Alan Ross. Entailment: the logic of relevance and necessity. Vol. 2. written by Alan Ross Anderson, Nuel D. Belnap and J. Michael Dunn with contribution by Kit Fine . . . et al. Includes bibliographical references. 1. Entailment (Logic). I. Belnap, Nuel D., 1930—joint author. II. Dunn, J. Michael, 1941- III. Fine, Kit. IV. Title. BC199.E58A53 1975 160 72-14016 ISBN 0-691-07192-6 (v. 1) ISBN 0-691-07339-2 (v. 2) This book has been composed in Monotype Times Roman by Syntax International Pte Ltd Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton University Press Princeton, New Jersey 10

9 8 7 6 5 4 3 2 1

Princeton Legacy Library edition 2017 Paperback ISBN: 978-0-691-60042-0 Hardcover ISBN: 978-0-691-65464-5

Dedicated to six generous parents SELMA WETTELAND ANDERSON ELIZABETH DAFTER BELNAP PHILOMENA LAUER DUNN

ROSS ELMER ANDERSON ·

NUEL DINSMORE BELNAP ·

JON HARDIN DUNN

CONTENTS

VOLUME I Analytical Table of Contents Preface Acknowledgments I. II. III. IV. V.

ix xxi xxix

THE PURE CALCULUS OF ENTAILMENT ENTAILMENT AND NEGATION ENTAILMENT BETWEEN TRUTH FUNCTIONS THE CALCULUS E OF ENTAILMENT NEIGHBORS OF E

Appendix: Grammatical propaedeutic Bibliography for Volume I Indices to Volume I VOLUME II Analytical Table of Contents Preface Acknowledgments Summary Review of Volume I

3 107 150 231 339

473 493 517

ix xvii xix xxiii

VI. THE THEORY OF ENTAILMENT VII. INDIVIDUAL QUANTIFICATION

3 70

VIII.

ACKERMANN'S strenge Implikation IX. SEMANTICS X. PROOF THEORY AND DECIDABILITY

129 142 267

XI. FUNCTIONS, ARITHMETIC, AND OTHER SPECIAL TOPICS XII. APPLICATIONS AND DISCUSSION

392 488

Bibliography of Entailment (by Robert G. Wolf) Indices to Volume II

565 711

vn

ANALYTICAL TABLE OF CONTENTS

VOLUME II

SUMMARY REVIEW OF VOLUME I

§R1. Grammatical review xxiii §R2. Axiomatic review xxiv §R3. Natural deduction review xxv VI.

THE THEORY OF ENTAILMENT

3

§30. Propositional quantifiers 3 §30.1. Motivation 3 §30.2. Notation 7 §31. Natural deduction: F E V 3 p 9 §31.1. Universal quantification 10 §31.2. Existential quantification 14 §31.3. Distribution of universality over disjunction 16 §31.4. Necessity 16 §31.5. F E V 3 p and its neighbors: Summary 18 §32. E V 3 p and its neighbors: Summary and equivalence 19 §33. Truth values 25 §33.1. TVVp 26 §33.2. For every individual χ, χ is president of the United States between 1850 and 1857 28 §33.3. E f d e and truth values 29 §33.4. Truth-value quantifiers 31 §33.5. R v 3 p a n d T V 32 §34. First degree entailments in E V 3 p (by Dorothy L. Grover) 33 §34.1. The algebra of first degree entailments of E v 3 p 33 §34.2. A consistency theorem 36 §34.3. Provability theorems 37 §34.4. Completeness and decidability 45 §35. Enthymemes 45 §35.1. Intuitionistic enthymemes 46 §35.2. Strict enthymemes 47 §35.3. Enthymematic implication in E V 3 p 50 §35.4. Summary 53 §36. Enthymematic implications: Embedding H and S4 in E V 3 p 55 §36.1. H i n E v + 3 p 55 §36.1.1. Under translation, E+ 3p contains at least H 57 IX

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Analytical table of contents

§36.1.2. Under translation, contains no more than H 60 36.2. H and S 4 + in 62 §37. Miscellany 64 37.1. Prenex normal forms (in 64 37.2. The weak falsehood of 66 37.3. is not a conservative extension of 67 37.4. Definitions of connectives in R with propositional quantifiers 68 V I I . INDIVIDUAL QUANTIFICATION

38.

39.

40.

41. 42.

43.

70

and 70 38.1. Natural deduction formulations 71 38.2. Axiomatic formulations and equivalence 72 Classical results in first-order quantification theory 73 39.1. Godel completeness theorem 73 39.2. Lowenheim-Skolem theorem 81 39.3. Gentzen's cut elimination theorem 84 Algebra and semantics for first degree formulas with quantifiers 87 40.1. Complete intensional lattices 88 40.2. Some special facts about complete intensional lattices 40.3. The theory of propositions 99 40.4. Intensional models 103 40.5. Branches and trees 107 40.6. Critical models 111 40.7. Main theorems 114 40.7.1. Quantificational sequences 114 40.7.2. Quantifier-free sequences 115 Undecidability of monadic first degree formulas 117 Extension of to et al. 119 42.1. Terminology for logics and theories 120 42.2. The Way Up 123 42.3. The Way Down 126 42.4. Admissibility of in et al. 127 Miscellany 128

V I I I . ACKERMANN'S strenge

Implikation

44. Ackermann's -systems 129 44.1. Motivation 129 44.2. E 131 44.3. E contains E 132 44.4. E contains E 134 §45. , and E (historical) 134 45.1. / goes 137 45.2. (.5) goes 138

129

96

Analytical table of contents §45.3. (γ) goes 138 §46. Miscellany 139 §46.1. Ackermann on strict "implication" §46.2. An interesting matrix 141 IX.

SEMANTICS

139

142

§47. Semilattice semantics for relevance logics (by Alasdair Urquhart) 142 §47.1. Semantics for R^ 142 §47.2. Semantics for E^ 146 §47.3. Semantics for T ^ 147 §47.4. Variations on a theme 149 §48. Relational semantics for relevance logics 155 §48.1. Algebraic vs. set-theoretical semantics 155 §48.2. Set-theoretical semantics for first degree relevant implications 158 §48.3. Three-termed relational (Routley-Meyer) semantics for R + 161 §48.4. Strong completeness for R + 169 §48.5. Relational semantics for all of R 170 §48.6. Relational semantics for E 171 §48.7. Relational semantics for T, RM, etc. 172 §48.8. Spinoffs from relational semantics 173 §48.9. Relational semantics for quantifiers 175 §49. Binary relational semantics for the mingle systems RM and RM V 3 * 176 §49.1. Binary relational semantics for RM 176 §49.1.1. The binary semantics 177 §49.1.2. Informal interpretation 178 §49.1.3. Semantical soundness 179 §49.1.4. Semantical completeness 181 §49.1.5. Decidability by filtration 184 §49.1.6. RM models and Sugihara matrices 184 §49.1.7. The binary semantics with "star operation" §49.1.8. Limitations of the binary semantics 187 §49.2. Quantification and RM 188 §49.2.1. Grammar and proof theory of RM V * 188 §49.2.2. Semantics 189 §49.2.3. Soundness 190 §49.2.4. Completeness of R M V x 190 §50. Intuitive semantics for first degree entailments and "coupled trees" 193 §50.1. Introduction 194 §50.2. Relevantly coupled trees 195 §50.3. Intuitive semantics 197

Analytical table of contents

XU

§51.

§52. §53.

§54.

X.

§50.4. Coupled trees and the semantics 203 §50.5. Tautological entailments and the semantics 203 §50.6. An earlier semantical gloss of essentially the same mathematics 205 §50.7. Ruminations 208 Models for entailment: Relational-operational semantics for relevance logics (by Kit Fine) 208 §51.1. Models 209 §51.2. Logics 212 §51.3. The minimal logic 213 §51.4. The systems E and R 217 §51.5. Alternative models 222 §51.6. Finite models 226 §51.7. Admissibility of (γ) 229 No fit between constant-domain semantics and R v 3 x 231 Semantics for quantified relevance logic (by Kit Fine) 235 §53.1. Models 239 §53.2. Truth 245 §53.3. The logics 253 §53.4. Soundness 254 §53.5. Completeness 255 KR_ & : A conjunction-arrow fragment corrupted by Boolean structure 262 §54.1. Axioms for KR _ & 0 , and their consistency 263 §54.2. Completeness 264

PROOF THEORY AND DECIDABILITY

267

§60. Relevant analytic tableaux (with Michael A. McRobbie) 267 §60.1. The tableau systems 267 §60.2. Equivalence via left-handed consecution calculuses 274 §60.3. Problems 278 §61. A consecution calculus for positive relevant implication with necessity (with Anil Gupta) 279 §61.1. History 279 §61.2. Postulates for L ( = LR°°') 281 §61.3. Translation and equivalence 282 §61.4. Some definitions and the normality property 284 §61.5. Elimination theorem: Outline of proof 287 §61.6. Closure under substitution and case 1.2 288 §62. Display logic 294 §62.1. Introduction 294 §62.2. Grammar 296 §62.2.1. Indices and families 296 §62.2.2. Formula-connectives and structure-connectives 297

Analytical table of contents

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§62.2.3. Formulas, structures, and consecutions 299 §62.2.4. Interpretation 299 §62.3. Postulates for DL 300 §62.3.1. Identity axioms 300 §62.3.2. Display-equivalence 300 §62.3.3. Connective postulates 302 §62.3.4. Reduction 303 §62.4. Subformula and elimination theorems 305 §62.4.1. Analysis, parameter, congruence 306 §62.4.2. Conditions on an analysis 307 §62.4.3. Proofs of subformula and elimination theorems 310 §62.5. Some families and logics 313 §62.5.1. Boolean family and two-valued logic 314 §62.5.2. Relevant implication 314 §62.5.3. Entailment 316 §62.5.4. Ticket entailment 319 §62.5.5. Semantics of relevance logics 320 §62.5.6. Modal logics 320 §62.5.7. Intuitionist logic 324 §62.5.8. Interfamilial relations 326 §62.6. Further developments 327 §62.6.1. Demarcation 327 §62.6.2. Quantifiers 328 §62.6.3. Interpolation 328 §62.6.4. Algebra 328 §62.6.5. Other connectives 328 §62.6.6. Restricted rules 329 §62.6.7. Incompatibility 330 §62.6.8. Binary structuring and infinite premiss sets 331 §62.6.9. Priority of the right? 332 §63. Decidability: Survey 332 §63.1. Decidability of fragments limited by degrees 333 §63.2. Decidability of fragments limited by connectives 334 §63.3. Decidability of neighbors 335 §64. Which entailments entail which entailments? 336 §64.1. Reducibility of the decision question to the second degree 337 §64.2. The positive case 337 §64.3. The case with negation 344 §65. The undecidabihty of all principal relevance logics (by Alasdair Urquhart) 348 §65.1. Relevant implication and projective geometry 348 §65.1.1. Models for relevance logics 349

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§65.1.2. The logic KR 349 §65.1.3. Projective spaces 350 §65.1.4. Model structures constructed from projective spaces 353 §65.1.5. Undecidability 354 §65.1.6. More geometrical ruminations 357 §65.2. The undecidability of entailment and relevant implication 358 §65.2.1. Introduction 358 §65.2.2. Coordinate frames in ordered monoids §65.2.3. The algebra of relevance logics 364 §65.2.4. De Morgan monoids and vector spaces §65.2.5. Undecidability 371 §65.2.6. Further undecidability results 374 §66. Minimal logic again (by Errol P. Martin) 375 §66.1. Three-valued metalogic 376 §66.2. S-models 377 §66.3. Reduced valuations 379 §66.4. The guarded merge theorem 380 §66.5. Powers's conjecture 382 §66.6. Significance of all this 384 §67. Decision procedures for contractionless relevance logics (by Steve Giambrone) 384 §67.1. Introduction 385 §67.2. LTW": and LRW* 385 §67.3. Vanishing t 386 §67.4. Denesting 388 §67.5. Reduction 388 §67.6. Degree and decidability 389 §67.7. E W ; 391 XI. FUNCTIONS, ARITHMETIC, AND OTHER SPECIAL TOPICS

359 369

392

§70. Functions that really depend on their arguments 392 §70.1. Mathematical concept of dependence 393 §70.2. Semantic and syntactic concepts of dependence 397 §70.3. Church's !-/-calculus and Scott's strictness 399 §71. Relevant implication and relevant functions (by Glen Helman) 402 §71.1. Terms and proofs 403 §71.2. Relevant abstraction and monadic relevant functions 410 §71.3. Pairing and conjunction 414 §71.4. Polyadic relevant functions 420 §72. Relevant Peano arithmetic 423 §72.1. Postulates for relevant Peano arithmetic 424 §72.2. Strength and weakness of the extensional fragment 426

Analytical table of contents §72.3. Relevant implications or material "implications"? §72.4. Oddments 433 §73. Relevant Robinson arithmetic 434 §73.1. Robinson's axioms 435 §73.2. Q R = Q 435 §73.3. Q R (1) Φ Q(I) 440 §73.4. The relations among R *, QR(0), and Q R (1) 442 §73.5. Remarks and speculations 443 §74. Relevant predication: The formal theory 445 §74.1. Introduction 445 §74.2. Properties (monadic) 447 §74.3. Lambda conversion 448 §74.4. Factor 449 §74.5. Indiscernibility of identicals 450 §74.6. Relevant predication 453 §74.7. Relations (polyadic) 454 §74.8. Formal consequences of the definitions 456 §74.9. Background 464 §74.10. Philosophical applications 468 §74.11. Technical appendix 469 §75. Relevant implication and conditional assertion (by Daniel Cohen) 472 §75.1. Assertivity functions 473 §75.2. Axiomatization 474 §75.3. Semantics 476 §75.4. Soundness 477 §75.5. Completeness 478 §75.6. Quantification 486 XII.

APPLICATIONS AND DISCUSSION

xv 429

488

§80. Entailment and the disjunctive syllogism 488 §80.1. Tautological entailment 488 §80.1.1. Review 488 §80.1.2. The disjunctive syllogism 488 §80.1.3. Relevance logic and relevantism 489 §80.1.4. Our plan 490 §80.2. Boolean negation 490 §80.2.1. Background 490 §80.2.2. A dilemma 492 §80.2.3. Horn 1 494 §80.2.4. Horn 2 495 §80.2.5. A puzzle 497 §80.3. Relevant arguments for the admissibility of the disjunctive syllogism 498 §80.3.1. Readings 498

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§80.3.2. "Equivalent" forms 501 §80.3.3. Extensional admissibility is useless for a relevantist 502 §80.4. The phenomenology of relevantism 502 §80.4.1. I'm all right, Jack 503 §80.4.2. The relevantist/deductivist parallel 503 §80.4.3. The leap of faith 504 §80.4.4. The toe in the water 504 §80.4.5. The true relevantist 505 §81. A useful four-valued logic: How a computer should think §81.1. The computer 506 §81.2. Part 1. Atomic inputs 510 §81.2.1. Atomic sentences and the approximation lattice A4 510 §81.2.2. Compound sentences and the logical lattice L4 513 §81.2.3. Entailment and inference: The four-valued logic 518 §81.2.4. Observations 520 §81.3. Part 2. Compound truth-functional inputs 524 §81.3.1. Epistemic states 524 §81.3.2. More approximation lattices 527 §81.3.3. Formulas as mappings: A new kind of meaning 529 §81.3.4. More observations 531 §81.3.5. Quantifiers again 532 §81.4. Part 3. Implicational inputs and rules 533 §81.4.1. Implicational inputs 534 §81.4.2. Rules and information states 539 §81.4.3. Closure 541 §82. Rescher's hypothetical reasoning: An amended amendment §82.1. HR-consequence 542 §82.2. Objections 544 §82.3. Candidate amendments 546 §82.4. Conjunctive containment 550 §83. Relevance logic in computer science (by Stuart C. Shapiro) §83.1. Use of the proof theory 554 §83.1.1. SWM 555 §83.1.1.1. Rules of inference of SWM 556 §83.1.1.2. Example 559 §83.1.2. Implementations 560 §83.2. Use of the four-valued semantics of R 561

PREFACE

continues the line of investigation into the logic of relevance and necessity—Entailment, we say—commenced by Ackermann in 1956 and reported on by ARA and NDB in Volume I of this book in 1975. At that time what we had planned for Volume II was well in hand, and, rather against the suggestion of our gracious editor, we explicitly and publicly projected the second volume for "about a year" after Volume I, that is, for about 1976 or 1977. In the meantime ARA died, NDB and R. K. Meyer entered an industrious collaboration that in the end did not succeed, and finally NDB and JMD, whose joint work on these topics goes back about a quarter of a century, have completed this volume within at least one year, as Russell would say, of the publication of Volume I. In that volume we passed on our belief that the earliest versions of relevance logic were those of Moh 1950 and Church 1951. Though hardly guilty of a howler in the sense of §20.2, we certainly missed the truth by over two decades: relevance logic was already treated with insight and rigor by Orlov 1928! This we first learned from the engaging report of Dosen 1990. We subsequently learned to our increased chagrin that the work of Orlov had already been brought to light by V. M. Popov in 1978. This fact is recorded for instance in a recommendable 1988 book, previously unknown to us, by E. K. Vojshvillo of Moscow State University: Philosophico-methodological aspects of relevance logic (Russian) (Moscow: Izdatel'stvo Moskovskogo Universiteta). We learned of these matters and much else in the course of a memorable and instructive visit to Moscow in the winter of 1991, hospitably arranged for one of us (NDB) by V. A. Smirnov of the Institute for Philosophy of the Academy of Sciences. Under the leadership of Smirnov and Vojshvillo, the city supports a large and interesting group of relevance logicians. Life is full of delights and hazards. Among the former lies the making of new and valued friends in distant places. Among the latter one must number errors in print. Still, although our history was not accurate, the measure of our debt to Moh and Church is undiminished. Intertwining of the philosophical and mathematical voices continues from Volume I, as does the interspersal of the odd joke, though we regret that there are fewer of the latter than we used to enjoy with ARA. Much of the philosophical argle-bargle depends on that in Volume I, or on good sense, so some readers may be disappointed that we do not answer more recent critics in any concentrated fashion. Our reasons are two: first, we do not THIS VOLUME

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Preface

enjoy finding fault with individual arguments or lodging criticisms of individual writers (in contrast with Movements in the Large, such as the Officers) and we do not take ourselves to be good at it; and, second, tired readers would inevitably fasten on the emotion of such disputes, thus drifting away from what we think constitute our positive contributions. In any event, the mathematics here is almost entirely independent, and it is made clear whenever consultation with Volume I is required. As before, readers are encouraged to listen to the voice of polemic, proof, or pun according to their preferences. We have laid out the chapters and sections in a good order for reading, but within this volume most chapters and even many sections are independent. It is not possible to be much more precise; the text has come from many hands over an extended period and is substantially more like a conversation than like a monologue. For further guidance we have provided an analytical table of contents with helpful titles, and we have caused each section to explain how it fits into the whole. For the reader, however, who has not seen Volume I or any of the related literature, a reading of §35 should give a useful introduction to the scheme of intuitions that lie behind relevance logic and entailment. Policies on cross-references, citations, and notation are the same as for Volume I, except that in a few indicated cases we have let stand without tinkering the choice of fonts and symbols of a contributed manuscript, so as to reduce the chance of introduced errors. We think everything is plain except for discontinuities in our numbering of sections. The cause is this: later chapters were not only executed but even dreamt up over many years, so we early decided to start numbering the first section of these chapters "on the decade" so as to leave plenty of room for intellectual expansion. We are glad we did, and hope that readers will not be disconcerted. We think that there is a large quantity of good stuff in this volume; we are pleased to have carried out some of the research ourselves, and we are proud to present a series of excellent studies by others. Among the latter, we may single out as the most recent and so the most exciting contributions (1) the proof by Urquhart in §65 that the concept of relevance essentially outruns any attempt fully to capture it by mechanical means, and (2) the new and original account by Fine in §53 of what quantifiers "for all" and "there exists" can and must mean in the context of relevance logic. We end this volume with a door-opening survey by Shapiro of the actual and potential impact of relevance logic on present-day computer science.

ACKNOWLEDGMENTS

has been in preparation since 1959, and indeed some of the sections of this volume were composed long before Volume I was sent off to the printers. Three decades make it inevitable that sources of crucial suggestions or ideas for Volume II shall have been forgotten, so that a wretchedly impersonal wholesale expression of apologetic gratitude is required to cover all. Here we must make partial amends with a scattering of notes. Our deepest debt is to the distinguished logicians, listed on the title page, who have given us permission to include their work as part of this volume, some of it taken from other publications and some written especially for this work. An extra measure of our gratitude is due those whose specially commissioned contributions have threatened to molder unpublished as the weary years declined. And although readers will of course award merit to contributions as they see fit, we as authors must single out Robert G. Wolf, who not only compiled the distinguished bibliography that he permitted us to publish as part of this volume, but has read over the entire manuscript, using his vast knowledge in order to provide us with hundreds of corrections and suggestions. We renew and indeed redouble our thanks to Sanford Thatcher of Princeton University Press for much, but especially (1) for his intervolume patience, (2) for giving us access to Leigh Cauman, whom we join one—and probably many—of our friends in labeling "the best editor that ever was," and (3) for turning over to Syntax International Pte, Ltd, and its remarkable crew of craftsmen the task of turning our appalling henscratches into graceful print. Beth Gianfagna carefully supervised the final stages, and Gretchen Oberfranc used her skill and sensitivity in making Wolf's bibliography camera-ready. To Kate Maloy we are obliged for the work exhibited in the frontispiece, which deftly recreates an old and typical fragment of ARA's smile-provoking wit. Publishers and journals who have kindly given their permission for republication of works by our contributors or by us also deserve our thanks. Detailed information appears in the bibliography at the end of this volume under the heading given below; section numbers in parentheses indicate where in this volume (a portion of) the cited material appears. D. Reidel Publishing Company: Dunn 1986 (§§42 and 48); Belnap 1977 (§81); 1979 (§82). The journal of philosophical logic: Fine 1974 (§51); 1988 (§53); Belnap and Gupta and Dunn 1980 (§61); Belnap 1982 (§62); Dunn 1987a THIS BOOK

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Acknowledgments

(§74). The journal of philosophy: Anderson and Belnap 1961 (§35). The journal of symbolic logic: Belnap 1967 (§40); Urquhart 1974 (§62.2); Urquhart 1984 (§65.2). Kluwer Academic Publications: Belnap and Dunn 1981 (§80). Logique et analyse: Urquhart 1983 (§65.l). Mathematische Annalen: Dunn and Belnap 1968a (§40). Philosophical studies: Dunn 1976 (§50). Studia logica: Dunn 1976b (§49); Dunn 1976d (§49); McRobbie and Belnap 1979 (§60); Dunn 1980 (§73). We have a special debt to R. K. Meyer. As indicated in the tentative contents incautiously displayed by ARA and NDB in Volume I, Meyer was scheduled to be an important contributor to Volume II. Later NDB and Meyer collaborated for a period, during which Meyer constructively read over much of what ARA and NDB had produced, and wrote or partially wrote a number of pieces beyond those projected in the tentative contents. Since the close of the NDB-Meyer collaboration these pieces have generally been issued elsewhere by Meyer, but their influence certainly remains, often in the form of sections that we ourselves have written about Meyer's research; and if in the odd phrase or two we have quite unknowingly retained Meyer's actual words, we can only ask the reader's understanding and indulgence. Some of the early portions of the manuscript came from the hands of the ten distinguished secretaries listed with admiration in Volume I; the remainder of Volume II, however, including Wolf's bibliography, is entirely from Collie Henderson, who has been awesome indeed. We don't know anyone wiser or more intelligent, and we feel greatly fortunate that she is on our side. We also wish to thank Susan Quinn and Luzia Martins for their excellent editorial assistance. NDB and JMD spent three critical terms together at Indiana University in the late seventies. NDB was sponsored as Oscar R. Ewing Visiting Professor by the Ewing Fund, which also provided a superb research assistant in Daniel Cohen. This volume has prospered owing to the assistance of a sequence of superb Alan Ross Anderson Research Fellows: Anne Nally, Andrew McCafferty, Steven Hetherington, Philip Kremer, and Aldo Antonelli. These Fellows were supported by the Alan Ross Anderson Memorial Fund, which was commenced by liberal gifts from ARA's friends and colleagues soon after he died and generously fulfilled by his mother, Selma Anderson, one of the dedicatees of this volume. Those in academia will recognize the appropriateness of adding a note of thanks to Jerome Rosenberg, Dean of the Faculty of Arts and Sciences and pillar of the University of Pittsburgh during most of the time when this volume was in the works, and a constant supporter. There are many current and former graduate students whom we would like to thank, but, in addition to those mentioned above, we single out Yang Auh, Monica Holland, and Mitchell Green as especially helpful in matters relating to this volume, and most recently, Andre Chapuis and Laura Reutsche.

Acknowledgments

XXl

Other support, for which we are grateful, came from the Center for Advanced Studies in the Behavioral Sciences, then under the wise guidance of Gardner Lindzey, where NDB spent a rewarding year partially supported by the National Endowment for the Humanities. Dana Scott made available to us some stunning computational facilities at Carnegie Mellon University, and gave us access to John Aronis, who has masterminded the computational side of Wolf's bibliography. Scott is also owed a more personal debt of gratitude from NDB for seminal conversations in 1970 at Oxford (where he and NDB were hosted by the gifted Christopher Strachey, now deceased) and over the years, and from JMD for an exciting sabbatical at the Mathematical Institute of Oxford University; and Robin Gandy is owed a similar debt for a subsequent shorter visit. For Senior Common Room rights, we thank Wadham (JMD) and Wolfson (JMD and NDB) Colleges. Also due our thanks is the Research School of the Social Sciences of the Australian National University, and in particular its Department of Philosophy, which hosted each of JMD and NDB at critical points and even at juncture. We have no doubt that while the entire institution made us welcome, the pleasures of our visits were essentially due to the hospitality of R. K. Meyer and R. Sylvan (then Routley). JMD was supported by a Fulbright from the Australian-American Educational Foundation (Director, B. Farrer), and NDB by a Guggenheim Fellowship. The University of Melbourne deserves our gratitude for an intellectually important visit there by JMD at the friendly instigation of Len Goddard and Michael McRobbie. We acknowledge sabbatical support for JMD from the Indiana University Institute for Advanced Study (then Director, Roger Newton), the University of Pittsburgh Center for the Philosophy of Science (then Director, Nicholas Rescher), and the Philosophy Department of the University of Massachusetts at Amherst (Gary Hardegree and Michael Jubien, hosts). Also, JMD received critical support from Indiana University's Office of Research and Graduate Development, so ably led by Dean Morton Lowengrub, who is now Dean of the College of Arts and Sciences. We thank Ed Robertson, Frank Prosser, and the Indiana University Department of Computer Science on the one hand, and Dana Scott and the Carnegie Mellon Department of Computer Science on the other, for facilitating the electronic aspects of our collaboration. Our final recorded debt is an enduring one to Carolyn W. Anderson, constant and cooperative friend of the enterprise.

SUMMARY REVIEW OF VOLUME I

ALTHOUGH THIS VOLUME is

squarely a continuation of Volume I, still, many of the technical results do not depend thereon. And although we hope that most readers will not be dipping deeply into this volume without covering at least the introductory portions of the first, still, we do not wish to penalize too severely the reader who is patiently working through this volume while floating on a raft in a swimming pool, having left Volume I up at the house. Hence, we have provided the following highly compressed summaries of the grammar, axiomatics, and natural deduction formulations of the most important systems developed in Volume I, and of some of the key concepts used in investigating them. §R1. Grammatical review. We make this resurvey chiefly to emphasize one or two points, and to hedge on one or two others. For further reference, we note that there is a list of special symbols at the back of this volume, and some helpful entries under "notation for" in the Index. AU the systems of Volume I have a denumerable stock of propositional (we also say "sentential") variables, and at least -> as a binary connective. For truth functions we use A&B (sometimes AB) for conjunction, AvB for disjunction, and for negation we use whichever of A or ~ A seems convenient. In §27.1.4 we pretty much settled on "co-tenability" for the connective o, since in R one has A°B ?± ~(A-*~B). But since ° has uses in connection with other systems where this equivalence fails, we follow the growing tendency in the literature to call it "fusion," a term which is happily free of unhappy associations. In connection with E, we generally think of OA as defined by A->A->A (§4.3), but we also want to think of it as added to R as a new primitive, yielding RD. We take t, and sometimes / , as propositional constants. And also sometimes T and F. We waffle a bit as to which connectives and constants are present in which systems, partly as a result of the magnitude of the number of years over which this volume has been in the writing. Mostly it does not matter, except when what is at issue is some delicate question of conservative extension or the like—and on these occasions we make a point of trotting out such connectives as we have in our pocket. As a general rule, however, we think of XXlIl

xxiv

Summary review of volume I

§R

things as being like this:

All the connectives except negation are thought of as positive, and are present in the positive fragment of each system S. Whenever necessity is added, we shall record this explicitly—though is the only case we much discuss. Sometimes we may want a name for the set of formulas. In those circumstances we shall use "SL," thinking of it as having and always, and ° and t depending on context. Then we shall use superscripts and subscripts to indicate additions to and deletions from the vocabulary of a system, as described in the Preface to Volume I. R2. Axiomatic review. For purposes of reference we here lay out the chief systems of propositional logic of Volume I. This resurvey draws on 21.1 and (principally) 27.1. First the axioms (some with multiple names) on which we draw:

§R3

Natural deduction review

xxv

Then there are seven rules: From and A to infer B. From A and B to infer A&B. If A is a theorem, so is A. If ' is a theorem, so is If C is a theorem, so is If A is a theorem, so is If is a theorem, so is A. The systems may now be defined as follows. (Numerous other formulations are to be found in Volume I.) All have the first two rules and &I.

Any of the calculuses may also be conservatively extended to a formulation with governed by and and t governed by ' (in the case of the modal systems E and EM, is an additional explicit postulate). We also define

Each of these calculuses S has a positive fragment obtained by dropping negation from the vocabulary and also dropping all its negation axioms. R3. Natural deduction review. This section does for the natural deduction systems of Volume I, as summarized in 27.2, what R2 did for the axiomatic. We define F-formulations of each E, R, T, RM, and EM, as well as The rules as stated below hold for all systems, unless otherwise indicated.

xxiv

Summary review of volume

I

§R

Structural rules Hyp. A step may be introduced as the hypothesis of a new subproof, and each new hypothesis receives a unit class of numerical subscripts, where k is the rank ( 8.1) of the new subproof. Rep. may be repeated, retaining the relevance indices a. Reit. may be reiterated (retaining subscripts) into hypothetical subproofs in FR and F R M with no proviso, and in the others provided A has the form t or Intensional rules From a proof of k is in a. From

and

From From

max(b)

From max(a). From From From From From From

on the hypothesis

to infer

to infer and

to infer

provided

where for FT,

Mixed rules to

infer

where

for

FT,

and to infer where for F T Extensional rules to infer to infer and to infer to infer From to infer to infer From to infer to infer Mingle rules

From . and to infer required that A have the form

(for F R M and F E M only; for F E M it is

We add natural deduction rules, not to be found in Volume I, for t and From From From

to infer to infer and

and conversely. to infer

These fusion and t rules are ugly because (a) we wanted to give them in forms that would work for any system, and (b) the particular form of natural

Natural deduction review

§R3

xxvu

deduction we have been using is not well suited to fusion or to t. We note that the reader will be able to find more satisfying rules in the context of R. In order to be able to add rules for G, so as to be able to get RD, we follow Fitch 1952 in defining a new kind of subproof: a strict categorical subproof has no hypotheses and is marked with a D:

Π

Then reiteration into strict categorical subproofs is limited to formulas Π A and t. The rules for D: DI. DE.

From a strict categorical subproof with last item A3 to infer OAt. From ΟΑΛ to infer A3.

For each system FS, its positive fragment FS + results from dropping nega­ tion from the vocabulary and dropping all negation rules.

ENTAILMENT

CHAPTER VI

THE THEORY OF ENTAILMENT

§30. Propositional quantifiers. Except for a brief forward reference in §21.2.2, we have made no formal or explicit use or mention of quantifiers, propositional, individual, or other. We have followed standard, somewhat inexact mathematical practice in using free variables in displayed formulas to indicate generality, or universal quantification (using mainly metalinguistic variables ranging over formulas, with occasional other ad hoc devices when it was, or was deemed, necessary to talk about propositional variables). In the future we will be using quantifiers explicitly, and we start by airing a few prejudices about the topic, and straightening out notation. §30.1. Motivation. This section may be skipped by any readers who have no worries about propositional quantifiers: it says nothing special about entailment, and it is not needed for reading the rest of the book. We include it for those readers who think that employment of propositional quantifiers requires a special defense. Propositional quantifiers have been neglected in the classical literature in the interest of exploiting individual quantifiers, which come second for us, both in this book and in rerum natura. We don't have terribly fierce feelings about logical or metaphysical priority when it comes to quantifiers, but to the extent that one takes the "logic of unanalyzed propositions" as somehow more fundamental, or something, than (say) the subject-predicate analysis of English sentences or the Fregean analysis provided by first-order quantification theory—to that extent we think propositional quantifiers precede others. Having started out with propositional or sentential connectives and with free variables which must have the generality interpretation, it seems to us reasonable to begin next to make this interpretation explicit, before going on to finer analysis of the propositions themselves. Although Church 1962 may not agree with us in finding some vague sense of "priority" concerning propositional quantification, we certainly agree with his characterization of the natural and obvious character of propositional quantification, and we quote his (as it seems to us) compelling considerations at some length (though of course we disagree with the account at the end of his second sentence of how analytic truths arise—which disagreement is irrelevant to the point under consideration): 3

4

Propositional quantifiers

Ch. VI

§30

That logic does not therefore consist merely in a metatheory of some object language arises in the following way. It is found that ordinary theo­ ries, and perhaps any satisfactory theory, of deductive reasoning in the form of a metatheory will lead to analytic sentences in the object language, i.e., to sentences which, on the theory in question, are consequences of any arbitrary set of hypotheses, or it may be, of any arbitrary nonempty set of hypotheses. These analytic sentences lead in turn to certain generaliza­ tions; e.g., the infinitely many analytic sentences A ν ~ A, where A ranges over all sentences of the object language, lead to the generalization p v ~ p , or more explicitly (/>)(/>ν ~p); and in similar fashion (F)(y)[(x)F(x) =>F(y)] may arise by generalization from infinitely many analytic sentences of the appropriate form. These generalizations are common to many object lan­ guages on the basis of what is seen to be in some sense the same theory of deductive reasoning for the different languages. Hence they are con­ sidered to belong to logic, as not only is natural but has long been the standard terminology. Against the suggestion, which is sometimes made from a nominalistic motivation, to avoid or omit these generalizations, it must be said that to have, e.g., all of the special cases A ν ~ A and yet not allow the general law (p).pv~p seems to be contrary to the spirit of generality in mathematics, which I would extend to logic as the most fundamental branch of mathe­ matics. Indeed such a situation would be much as if one had in arithmetic 2 + 3 = 3 + 2, 4 + 5 = 5 + 4, and all other particular cases of the commuta­ tive law of addition, yet refused to accept or formulate a general law, {x)(y).x + y = y + x (pp. 181-182). The parallel seems to us clear and obvious; yet it is strikingly noticeable that very little attention has been paid to propositional quantification in the literature. (Happily, this neglect has not been so marked in recent years as previously, and so we note with gratification that the discussion below, drafted some time back, might well take a less stern tone today. That right and justice will triumph in the end, as they always do, may be anticipated, for example, from the good work of Bull 1969, Fine 1970, Grover 1972, 1972a, 1973, and Gabbay 1972a. We note moreover, to anticipate remarks immediately to be made, that it is exactly as the theory of propositional quantifiers has become nontrivial—for instance, in modal and intuitionist contexts—that it has begun to assume its proper role in the literature. Never­ theless, lest wrong and injustice make a comeback, we shall retain our stern tone.) There are probably several reasons, historically, for the neglect of the topic that Russell 1906a calls "the theory of propositions" and that Church 1956 calls "the extended propositional calculus" (see Church 1956, §28, who gives

Motivation

§30.1

5

a history of the matter, including references to the intimations of Russell 1903 and to the work of^ukasiewicz and Tarski 1930); no doubt the principal reason is that the theory is not very interesting. We explain briefly why this is so. If, to some standard formulation of the two-valued calculus with "material modus ponens" as sole rule, we add propositional quantifiers, together with a complete set of axioms and rules for propositional quantification, we can always find a formula F that has "falsehood" as its Bedeutung. This could be done in a variety of ways, e.g., by taking F as primitive, with the axiom F=>A,or else by treating F as short for p8i~p, for some propositional vari­ able p, or else by taking F as short for Vpp, or the like. No matter how this is done, we get the result that F is provably materially "equivalent" to, and hence intersubstitutable with, any contradictory truth function; and letting T be (say) ~F, we have dually that T is intersubstitutable with any tau­ tological truth function. Standard treatments then produce the following theorems: VpX(P) =· A(T)ScA[F) and 3pA(p)=

A(T)WA(F)

(where A(p) is some context about ρ and A(T) is the result of putting T for ρ therein, etc.; we use this notation informally for a short space before intro­ ducing it formally). These have the force of guaranteeing that all the work done by propositional quantifiers can also be done by finite conjunctions and disjunctions—indeed, by short ones—so nothing much is gained except some new notation of doubtful value (for the purely truth-functional case). This lack of novelty is accompanied by some embarrassment in trying to render propositional quantifiers in English or other natural languages. Re­ lative (and other) pronouns provide handy locutions for reading individual quantifiers in English. Thus 3xFx can come out "There is something that has the property ef." Or, for short, "Something is ef." We used to think our mother tongue deficient in providing similar locutions for reading proposi­ tional quantifiers. This, however, was being needlessly uncharitable. We were set straight by Grover 1972, which lays the groundwork for a more unified treatment of propositional and individual quantifiers. In this treatment, fur­ ther developed in Grover, Camp, and Belnap 1975, it turns out that English does have prosentences analogous to pronouns, though these prosentences are not always so readily available or so easy to find. For further details, see the cited papers.

6

Prepositional quantifiers

Ch. VI §30

We shall not worry too much here, however, about the precise degree to which English gracefully adapts itself to propositional quantifiers. Truthfunctionally, 3qq comes to TvF, and if we can bring ourselves to utter "tee or ef," we ought to be able to say "there's a queue such that queue" without feeling too red-faced. And if, as in the interesting non-truth-functional cases we treat in this chapter, there are perhaps infinitely many propositions in our domain, then we can think of Iqq as the disjunction of all of them. So, to the degree that English is thought recalcitrant on this point, we shall treat it as no serious obstacle. For, as Richard Montague put it, near enough, in addressing the American Philosophical Association, why should we be bound to a language that is becoming obsolescent? So we render propositional quantifiers into our mother tongue as best we can, treating the reader with equal charity when he says, "There's an ekks such that ef of ekks, and, moreover, if it gees it aitches." Somewhat more serious than either the notational redundancy or the translational inconvenience, is a third feature of the situation which is forced on us by the intended interpretation of the whole calculus. Natural and obvious axioms and rules for quantifiers lead, as mentioned above, to A{T)8iA{F)=>VpA(p\ which stale custom will probably cause us to read "if both tee and eff satisfy a condition, then all propositions do." Even discounting the blunder in reading the horseshoe, this formula has the effect of saying that T and F are the only propositions recognized by the classical theory. And this fact reinforces what we all knew, but tried to conceal, all along, namely, that the "propositional calculus" is not a calculus of propositions at all. As a calculus it is faultless, but havering about its interpretation—a maundering induced in part by the mind-muddling misnomer "propositional calculus"—has created a host of bogus philosophical problems, some of which we hope to dispel in this chapter. We hereby make a firm resolve to remember in the future to call the calculus by its right name, "the (two-valued) truth value calculus" TV, and refer to the result TVv3p of adding truth value quantifiers as "the extended truth value calculus," or "the theory [as opposed to the (freevariable) calculus] of truth values." How well we can adhere to our resolution remains for us and the reader to see. Old habits die hard, and the difficulty of unlearning a well-entrenched and habitual error may be insurmountable, or virtually so, especially since we want promptly to resuscitate the terms "propositional calculus," "extended propositional calculus," and "theory of propositions" for systems like R, E, T, RV3p, Ev3p, and TV3p, where the free variables and variables of quantification may range over propositions. In this connection we mention one final motivation for the usual neglect of truth value quantifiers. Under the misapprehension that truth value quantifiers range over propositions, these bits of notation have been viewed

§30.2

Notation

7

with apprehension by those who have been touted off propositions by neonominalists. (We apologize if it appears that our preoccupation with this issue amounts to a disease, but it must be remembered that in 1947 Goodman and Quine announced, eloquently and influentially, that the subject we are discussing had been abolished. It is disquieting to be told that righteousness demands allegiance to truth values alone and that propositions are only for the unregenerate who hanker after spooks.) But, in view of the equivalences mentioned earlier, it would seem that truth value quantifiers should be ac­ ceptable even to the most abstemious. We shall consider truth value quan­ tifiers later, but we treat full-blooded propositional quantifiers first. §30.2. Notation. Here we lay out our notation for propositional quan­ tification. 1. Propositional variables. We assume that we have a collection of prop­ ositional (sometimes sentential) variables, at most denumerable, and that an alphabetical order is imposed on them. We use p, q, r, s (perhaps subscripted) as ranging over them. In some contexts it is convenient to assume that the variables are divided into two disjoint series. The principal reason for this is narrowly technical: it is easy to become confused when faced with the necessity of instantiating with a variable in a context in which, in spite of oneself, it can get grabbed by a quantifier (see below). In such cases, we call parameters those variables which never get tied to any quantifier. (They are also called "variables of instantiation," or, following Russell, "real variables.") Parameters, we note, are convenient for calculational purposes, but rarely play any semantic role. The other variables, those which can get tied to a quantifier, we call (when we are making the distinction) simply variables. (They are also called "variables of quantification," or, following Russell, "apparent variables.") Sometimes in proof theory it is convenient to insist that such variables are always bound to a quantifier, but such an insistence tends to get in the way of semantic discussions. 2. Quantifiers. We use V and 3, respectively, for universal and existential quantification. In contexts where the parameter-variable distinction is live, they can bind only variables, never parameters. 3. Formulas are built from propositional variables by connectives, as usual; and, if A is a formula, so are VpA and 3pA. As before, A, B, etc. range over formulas, but we now add to this stock the metalinguistic expressions A(p), A(q), B{r), etc., and, more rarely, B{p, q), C(q, r, s), etc., which are also to be construed, like A, B, etc., as metavariables ranging over formulas (in which the indicated propositional variables may or may not occur). Our next topic is the rather tedious one of characterizing the proper sub­ stitution of a formula B for ρ in A{p), the result of this substitution being (notationally) the formula .4(B). The logically mature reader (and the reader

8

Propositional quantifiers

Ch. VI §30

who just doesn't care) may take it that he knows well enough what the nota­ tion is supposed to mean: put B in for all free occurrences of p, and don't confuse any bound variables. But we have a definite policy as to how we are going to build in the "don't confuse any bound variables" clause in the characterization of proper substitution, and we are determined to state that policy (and to use it officially below, whenever we are forced to return to this depressing topic—which will be as rarely as is consistent with avoiding actual technical mistakes). At any rate, the confident (or indifferent) reader may move on immediately to §31, referring back to our exact explanations below when necessary to supplement what he knowns about proper substitu­ tion of a term ί for a variable χ in A(x) in the analogous classical first-order case, the result of this substitution being A(t). "Proper substitution" as used above is in the sense analogous to that of Kalish and Montague 1964; but the same notion, down to matters of minor technical import and termino­ logical practice, is that which the reader will have learned from any good text (e.g., Kleene 1952, Curry 1963) applied here—as in Church 1956—to propositional quantification. One case is simple; if B contains no variables, but only parameters, then A(B) is just the result of replacing all free occurrences of ρ with B in A(p). Otherwise, we need a policy such that no undesirable confusion of free and bound variables results, as it might, for example, if we were to put 3r(r->q) for ρ in Vq(pvq), where the resulting formula Vq(3r(r-»q)vq) has all occur­ rences of q bound, though q was free in 3r(r->q). Difficulty arises only when the substituend B contains a free occurrence of a variable q, and A(p) contains a part of the form VqC in which ρ occurs free; in this case any free occurrence of q in B will become bound on substitution, a procedure which may lead us from a formula that is valid (let A(p) be 3q(q?±p)) to one that is not (since putting ~ q for ρ gives 3q(q?±~q)). We therefore say that A(p) is ready for substitution of B for ρ iff A(p) has no part VqC such that ρ is free in VqC and q is free in B. (Kleene 1952, §18, would say "B is free for ρ in A(p).") We then fix our notation so that, if A(p) is not ready, we first get it ready, and then substitute. We accomplish this as follows (adapting a device of Curry and Feys 1958 and Kripke 1959a). Suppose A(p) is not ready for substitution of B for p; let \/qC be the leftmost troublesome part of A(p). Then replace every occurrence of q in VqC by the first variable in alphabetic order which does not occur in either A(p) or B (let this variable be r), getting VrC; then replace VqC in A(p) by ^rC, getting A(P)1. If -4(P)1 is still not ready, we repeat the procedure, until we arrive at a formula A(p)n such that A(p)„ is ready for substitution of B for p. Then conventions for A(B) are as follows: (i) If ρ does not occur free in A(p), A(B) is A(p); (H) if ρ occurs free in A(p), and A(p) is ready for substitution of B for p, then A(B) is the result of replacing all free occurrences of ρ in A(p) by B; and

§31

3

Natural deduction: FE* "

9

(iii) If ρ occurs free in A(p), and A(p) is not ready for substitution of B for p, then A(B) is the result of replacing all free occurences of ρ in A(p)„ by B, where A(p)„ is as above. It may appear that we are smuggling a rule for alphabetic change of bound variables into our notation. With respect to what we intend, we have to plead guilty. Where C results from C by rewriting the bound variable ρ as the bound variable q (in the sense of mere alphabetic change), we take C and C to be mere notational variants, differing not at all in logical content. So we always anticipate having admissible rules that permit alphabetic change (in the usual sense) without restriction, and we care not a whit if our notational conventions facilitate tricky proofs of the admissibility of a rule which is always wanted in the systems considered in this book. However, it is to be noted that we have not yet smuggled in alphabetic change, even in part, and cannot do so until we have stated some axioms and rules. Thus far, it is just a notational convention that, if A(p) is Vq(pvq) and B is 3r(r->ij), then, where s is the next variable in alphabetic order, A(B) is Vs(3r(r->q)vs). That is, we are merely, at this point, finding notation to avoid repeated statement of the condition that A(p) either be ready or be gotten ready before we make substi­ tutions. So, although perhaps we have made it easier, the rule for alphabetic change of bound variables will still have to be derived if it is to be available. §31. Natural deduction: FE V 3 p . The theory of entailment is that theory E obtained from the (free variable) propositional calculus E of entailment upon addition of propositional quantifiers. We will also discuss the analogous quantificational extensions of R, T, RM, and EM. All the differences in these systems come from their implication fragments, so we will be able to treat their truth-functional and propositional quantificational aspects wholesale. We will also consider various formulations of these systems, and we begin by describing the formulation that is most useful for carrying out proofs: the Fitch-style natural deduction formulation F E v 3 p . Before embarking on this project, however, we interject a remark about the intended interpretation of the quantifiers. In order to motivate discus­ sion thereof, we shall from time to time rely on the familiar analogy with in­ dividual quantifiers, according to which VxF(x) is thought of as a (possibly infinite) conjunction, and 3xFx as a (possibly infinite) disjunction. That is, we think of VpA(p) as A(P1)SLA(P2)SC ... &A(pt)& . . . , where the p ; run through all the (possibly infinitely many) propositions recognized by the theory in question; and similarly for 3pA(p). We are going to use this interpretation in these sections only to provide heuristic guidelines in explaining why entail­ ments hold or fail, particularly when infinite cases would, according to rea­ sonable expectations, be simply generalizations of finite cases. Nevertheless, it is perhaps worth pointing out that we do in fact take the interpretation seriously. What makes this easy is having available propositions as well as V3p

10

Natural deduction: FE

Y3p

Ch. VI §31

sentences. For although talk of an infinite conjunction of sentences, in a sense in which the conjuncts are recoverable parts of the conjunction, is with respect to any usual language sheer nonsense (although logicians have in fact devised mathematical creations which can be described in such terms), still there is nothing very peculiar in speaking of infinite conjunctions of propositions. Reason: although the concept of sentential conjunction sug­ gests that the conjuncts be literal "parts," so that infinity becomes unbear­ able, with propositions we need have no such suggestion. Propositions have no parts; so we are naturally led to another understanding of conjunction— and one which is naturally indifferent as to how many propositions are being conjoined. We think of propositions as being ordered by an implication or entailment relation, and, given a set of propositions, we take their conjunc­ tion to be the greatest lower bound of the set. Which is to say, the conjunction of a set of propositions is the logically weakest propositions that is strong enough to entail each member of the set. And this makes sense regardless of whether or not the set is finite. With this understanding, and without being fussy, we would then say that A&B expresses the propositional conjunction of the set of propositions ex­ pressed by A and by B, while VxF(x) expresses the propositional conjunction of the set of propositions expressed by F{x) as the value of χ runs through the domain of individuals, and Vp^4(p) expresses the propositional conjunction of the set of propositions expressed by A(p) as ρ runs through the appropriate domain of propositions. But, to repeat, we shall be using the view that universal quantification is a generalization of conjunction only heuristically. Indeed, we learn in §§52-53 below that there is another and more sophisticated view which appears even more coherent with the spirit of relevance logic. We assume all the apparatus of §23 for FE (summarized in §R3), and, as before, we wish to have an introduction and an elimination rule for the universal and for the existential quantifier (both of which we take as primi­ V3p tive, so as to be able more readily to consider the positive part of F E subsequently). §31.1. Universal quantification. The elimination rule for the universal quantifier, VE, is obvious: from VpA(p)a to infer A(B)3, for arbitrary B. For VI it is most convenient to introduce a new style of subproof which we call, following Fitch, "general categorical." General categorical subproofs are distinguished from the hypothetical subproofs we have used earlier in three ways: (a) They have no hypotheses. Under one restriction (to be stated below), any step may be reiterated into such proofs. Such proofs may, by way of re­ iteration, use hypotheses of hypothetical proofs to which they are subor­ dinate, but they have no hypotheses of their own.

§31.1

Universal quantification

11

(b) They are flagged with a variable, thus: P

The purpose of the flagging variable is, intuitively, to indicate that the proof is general with respect to the flagged variable, in the sense that nothing inside the proof depends on the choice of p. This condition would be violated unless we restricted reiteration in the following way: (c) Aa may be reiterated into a categorical proof that is general with respect to ρ only if ρ fails to occur free in A. This condition guarantees that, as it is common to say, ρ is an arbitrary specimen; nothing previously established about ρ can be used in the subproof anyhow, so it might as well be anybody. (Note: we may reiterate into such proofs formulas containing bound occurrences of the flagging variable.) We can now state the rule VI for introduction of the universal quantifier: from a categorical subproof which is general with respect to p, having A(p\ as its final step, to infer VpA(p)a. We give some examples. 1 _ VpVqA(p, q)w 2 VpVqA(p, q)w q 3 VpVqA(p, q)w P 4 VqA(p, q)w 5 M.P,