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Alan Ross Anderson and Nuel D.Belnap, Jr. Entailment the Logic

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Alan Ross Anderson and Nuel D.Belnap, Jr. Entailment the Logic (", , ENTAILMENT ,l THE LOGIC OF RELEVANCE AND NECESSITY by ALAN ROSS ANDERSON and I i--' NUEL D. BELNAP, JR. wUh contributions by J. MICHAEL DUNN j ROBERT K. MEYER Ii l and further contributions by JOHN R. CHIDGEY STORRS MCCALL J. ALBERTO COPPA ZANE PARKS DOROTHY L. GROYER GARREL POTTINGER BAS YAN FRAASSEN RICHARD ROUTLEY HUGUES LEBLANC ALASDAIR URQUHART ROBERT G. WOLF i J VOLUME I PRINCETON UNIVERSITY PRESS Dedicated to the memory of WILHELM ACKERMANN (1896-1962) whose insights in BegrUndung einer strengen Implikatiol1 (Journal of symbolic logic, 1956) provided the impetus for this enterprise COPYRIGHT © 1975 BY PRINCETON UNIVERSITY PRESS Published by Princeton University Press Princeton and London All Rights Reserved LCC: 72-14016 ISBN: G-691-07192-6 Library of Congress cataloging in Publication Data will be found on the last printed page of this book Printed in the United States of America by Princeton University Press Princeton, New Jersey CONTENTS VOLUME I Analytical Table of Contents IX Preface xxi Acknowledgments xxix I. THE PURE CALCULUS OF ENTAILMENT 3 II. ENTAILMENT AND NEGATION 107 III. ENTAILMENT BETWEEN TRUTH FUNCTIONS 150 IV. THE CALCULUS E OF ENTAILMENT 231 V. NEIGHBORS OF E 339 Appendix: Grammatical propaedeutic 473 Bibliography for Volume I 493 Indices to Volume I 517 VOLUME II (tentative) VI. THE THEORY OF ENTAILMENT VII. INDIVIDUAL QUANTIFICATION VIII. ACKERMANN'S Strengen Implikation IX. SEMANTIC ANALYSIS OF RELEVANCE LOGICS X. ASSORTED TOPICS Comprehensive Bibliography (by Robert G. Wolf) Combined Indices vii ANALYTICAL TABLE OF CONTENTS VOLUME I I. THE PURE CALCULUS OF ENTAILMENT §1. The heart of logic 3 § 1.1. "If ... then -" and the paradoxes 3 § 1.2. Program 5 §1.3. Natural deduction 6 §IA. Intuitionistic implication 10 §2. Necessity: strict implication 14 §3. Relevance: relevant implication 17 §4. Necessity and relevance: entailment 23 §4.1. The pure calculus of entailment: natural deduction formulation 23 §4.2. A strong and natural list of valid entailments 26 §4.3. That A is necessary P .A-+A-+A 27 §5. Fallacies 30 §5.1. Fallacies of relevance 30 §5.1.1. Subscripting (in 30 §5.1.2. Variable-sharing (in 32 ,i §5.2. Fallacies of modality 35 §5.2.1. Propositional variables entailing entailments (in 37 §5.2.2. Use of propositional variables in establishing entailments (in 40 §6. Ticket entailment 41 §7. Gentzen consecution calculuses 50 § 7.1. Perspectives in the philosophy of logic 50 § 7 .2. Consecution, elimination, and merge 51 §7.3. Merge formulations 57 I §7A. Elimination theorem 62 §7.5. Equivalence 67 I §8. Miscellany 69 §8.1. An analysis of subordinate proofs 70 §8.2. Ackermann's "strengen Implikation" and the rule (0) 72 §8.3. Axiom-chopping 75 §8.3.1. Terminology for derived rules of inference 75 §8.3.2. Alternative formulations of 76 ix :'[ { x Analytical table of contents Analytical table of contents xi §8.3.3. Alternative formulations of 77 §14.1.1. Alternative formulations of 139 §8.3.4. Alternative formulations of 79 §14.1.2. Alternative formulations of E.. 142 §8.4. Independence 80 §14.1.3. Alternative formulations of 142 §8.4.1. Matrices 84 §14.2. Independence (by John R. Chidgey) 143 §8.4.2. Independent axioms for 87 §14.2.1. Matrices 143 §8.4.3. Independent axioms for 87 §14.2.2. Independent axioms for T.. 144 §8.4.4. Independent axioms for 88 §14.2.3. Independent axioms for E.. 144 §8.5. Single-axiom formulations 88 §14.2.4. Independent axioms for R" 144 §8.5.1. Problem 89 §14.3. Negation with das Fa/sche 145 §8.5.2. Solution for L (by Zane Parks) 89 §14.4. Conservative extensions 145 §8.6. Transitivity 90 §14.5. and R" with co-entailment 147 §8.7. Co-entailment 91 §14.6. Paradox regained 147 §8.S. Antecedent and consequent parts 93 §14.7. Mingle again 148 §S.9. Replacement theorem 93 §S.IO. is not the intersection of and 94 III. ENTAILMENT BETWEEN TRUTH FUNCTIONS ISO §8.11. Minimal logic 94 §S.12. Converse Ackermann property 95 §15. Tautological entailments 150 §S.13. Converse of contraction 96 §15.1. 'Tautological entailments 151 §S.14. Weakest and strongest formulas 96 §15.2. A formalization of tautological entailments (Efde) ISS §S.15. Mingle 97 §15.3. Characteristic matrix 161 §8.16. without subscripts 99 §16. Fallacies 162 §S.17. No finite characteristic matrix 99 §16.1. The Lewis argument 163 §8.IS. Indefinability of necessity in (by Zane Parks) 99 §16.2. Distinguished and undistinguished normal forms 167 §S.19. Necessity in 100 §16.2.1. Set-ups 169 §8.20. The Cr systems: an irenic theory of implications (by Garrel §16.2.2. Facts, and some philosophical animadversions 171 Pottinger) 101 §16.2.3. A special case of the disjunctive syllogism 174 §8.20.1. The systems FCr and Cr 101 §16.3. A remark on intensional disjunction and subjunctive con- §8.20.2. Some theorems 103 ditionals 176 §S.21. Fogelin's restriction 106 §17. Gentzen consecution calculuses 177 " §18. Intensional algebras (Efd,) (by J. Michael Dunn) ISO § 18.1. Preliminary definitions 190 II. ENTAILMENT AND NEGATION 107 §18.2. Intensional lattices 193 §9. Preliminaries 107 §IS.3 The existence of truth filters 194 • §1O. Modalities 110 §IS.4. Homomorphisms of intensional lattices 197 §II. Necessity: historical remarks lIS §IS.5. An embedding theorem 200 §12. Fallacies 119 §18.6. Intensional lattices as models 202 §13. Gentzen consecution calculuses: decision procedure 124 §IS.7. The Lindenbaum algebra of Efd, 202 § 13.1. Calculuses 124 §IS.S. An algebraic completeness theorem for Efd, 204 §13.2. Completing the circle 126 §19 First degree formulas Efdf 206 §13.3. Decision procedure 136 §19.1. Semantics 206 §14. Miscellany 139 §19.2. Axiomatization 207 §14.1. Axiom-chopping 139 §19.3. Consistency 209 xii Analytical table of contents Analytical table of contents xiii §19.4. Facts 209 §24.1.2. Two valued logic is a fragment of E 283 §19.5. Completeness 212 §24.2. E and first degree entailments 285 §20. Miscellany 215 §24.3. E and first degree formulas 285 §20.1. The von Wright-Geach-Smiley criterion for entailment 215 §24.4. E and its positive fragment 286 §20.1.1. The intensional WGS criterion 217 §24.4.1. E+: the positive fragment of E 287 §20.1.2. The extensional WGS criterion 218 §24.4.2. On conserving positive logics I (by Robert K. §20.2. A howler 220 Meyer) 288 §20.3. Facts and tautological entailments (by Bas van §24.5. E and its pure entailment fragment 296 F raassen) 221 §25. The disjunctive syllogism 296 §20.3.1. Facts 221 §25.1. The Dog 296 §20.3.2. And tautological entailments 226 §25.2. The admissibility of (,,) in E; first proof (by Robert K. Meyer and J. Michael Dunn) 300 IV. THE CALCULUS E OF ENTAILMENT 231 §25.2.1. E-theories 300 §25.2.2. Semantics 303 §21. E E"+E/d,, 231 §2S.2.3. Generalizations 311 §21.1. Axiomatic formulation of E 231 §25.3. Meyer-Dunn theorem; second proof 314 §21.2. Choice of axioms 232 §25.3.1. Definitions 315 §21.2.1. Conjunction 233 §25.3.2. Abstract properties 316 §21.2.2. Necessity 235 §25.3.3. Facts 318 §22. Fallacies 236 §25.3.4. Punch line 319 §22.1. Formal fallacies 237 §26. Miscellany 321 §22.1.1. Ackermann-Maksimova modal fallacies 237 §26.1. Axiom-chopping 321 §22.1.2. Fallacies of modality (by J. Alberto Coffa) 244 §26.2. Independence (by John R. Chidgey) 322 §22.1.3. Fallacies of relevance 252 §26.3. Intensional conjunctive and disjunctive normal forms 323 §22.2. Material fallacies 255 §26.4. Negative formulas; decision procedure 325 §22.2.1. The Official deduction theorem 256 §26.5. Negative implication formulas 326 §22.2.2. Fallacies of exportation 261 §26.6. Further philosophical ruminations on implications 328 §22.2.3. Christine Ladd-Franklin 262 §26.6.1. Facetious 329 • §22.3. On coherence in modal logics (by Robert K. Meyer) 263 §26.6.2. Serious 330 §22.3.1. Coherence 264 §26. 7. A --. B, C --.D, and A"'''':--.--C;;B---.-.''''CC;---.-;C;:D 33 3 §22.3.2. Regular modal logics 265 §26.8. Material "implication" is sometimes implication 334 §22.3.3. Regularity and relevance 268 §26.9. Sugihara's characterization of paradox, his system, and his §23. Natural deduction 271 matrix. 334 §23.1. Conjunction 271 §23.2. Disjunction 272 §23.3. Distribution of conjunction over disjunction 273 V. NEIGHBORS OF E 339 §23.4. Necessity and conjunction 274 §27. A survey of neighbors of E 339 §23.5. Equivalence of FE and E 276 §27.1. Axiomatic survey 339 §23.6. The Entailment theorem 277 §27.1.1. Neighbors with same vocabulary: T, E, R, EM, and §24. Fragments of E 279 RM 339 §24.1. E and zero degree formulas 280 §27.1.2. Neighbors with propositional constants: Rand E with §24.1.1. The two valued calculus (TV) 280 t,j, w, w', T, and F 342 xiv Analytical table of contents Analytical table of contents xv §27.1.3. Neighbors with necessity as primitive: RD and §29.6.1. Parry's analytic implication 430 ED 343 §29.6.2. Dunn's analytic deduction and completeness §27.1.4. R with intensional disjunction and co-tenability as theorems 432 primitive 344 §29.7. Co-entailment again 434 §27.2. Natural deduction survey: FR, FE, FT, FRM, and §29.8. Connexive implication (by Storrs McCall) 434 FEM 346 §29.8.1.
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