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Classical and Nonclassical Logics CLASSICAL AND NONCLASSICAL LOGICS CLASSICAL AND NONCLASSICAL LOGICS An Introduction to the Mathematics of Propositions Eric Schechter Princeton University Press Princeton and Oxford Copyright °c 2005 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Wood- stock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Schechter, Eric, 1950{ Classical and nonclassical logics : an introduction to the math- ematics of propositions / Eric Schechter. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-691-12279-3 (acid-free paper) ISBN-10: 0-691-12279-2 (acid-free paper) 1. Mathematics|Philosophy. 2. Proposition (Logic). I. Title. QA9.3.S39 2005 160|dc22 2004066030 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for provid- ing the camera-ready copy from which this book was printed. Printed on acid-free paper. 1 pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents A Preliminaries 1 1 Introduction for teachers 3 ² Purpose and intended audience, 3 ² Topics in the book, 6 ² Why pluralism?, 13 ² Feedback, 18 ² Acknowledgments, 19 2 Introduction for students 20 ² Who should study logic?, 20 ² Formalism and certi¯- cation, 25 ² Language and levels, 34 ² Semantics and syntactics, 39 ² Historical perspective, 49 ² Pluralism, 57 ² Jarden's example (optional), 63 3 Informal set theory 65 ² Sets and their members, 68 ² Russell's paradox, 77 ² Subsets, 79 ² Functions, 84 ² The Axiom of Choice (optional), 92 ² Operations on sets, 94 ² Venn dia- grams, 102 ² Syllogisms (optional), 111 ² In¯nite sets (postponable), 116 4 Topologies and interiors (postponable) 126 ² Topologies, 127 ² Interiors, 133 ² Generated topologies and ¯nite topologies (optional), 139 5 English and informal classical logic 146 ² Language and bias, 146 ² Parts of speech, 150 ² Semantic values, 151 ² Disjunction (or), 152 ² Con- junction (and), 155 ² Negation (not), 156 ² Material implication, 161 ² Cotenability, fusion, and constants vi Contents (postponable), 170 ² Methods of proof, 174 ² Working backwards, 177 ² Quanti¯ers, 183 ² Induction, 195 ² Induction examples (optional), 199 6 De¯nition of a formal language 206 ² The alphabet, 206 ² The grammar, 210 ² Removing parentheses, 215 ² De¯ned symbols, 219 ² Pre¯x notation (optional), 220 ² Variable sharing, 221 ² Formula schemes, 222 ² Order preserving or reversing subformulas (postponable), 228 B Semantics 233 7 De¯nitions for semantics 235 ² Interpretations, 235 ² Functional interpretations, 237 ² Tautology and truth preservation, 240 8 Numerically valued interpretations 245 ² The two-valued interpretation, 245 ² Fuzzy interpre- tations, 251 ² Two integer-valued interpretations, 258 ² More about comparative logic, 262 ² More about Sugihara's interpretation, 263 9 Set-valued interpretations 269 ² Powerset interpretations, 269 ² Hexagon interpretation (optional), 272 ² The crystal interpretation, 273 ² Church's diamond (optional), 277 10 Topological semantics (postponable) 281 ² Topological interpretations, 281 ² Examples, 282 ² Common tautologies, 285 ² Nonredundancy of sym- bols, 286 ² Variable sharing, 289 ² Adequacy of ¯nite topologies (optional), 290 ² Disjunction property (op- tional), 293 Contents vii 11 More advanced topics in semantics 295 ² Common tautologies, 295 ² Images of interpreta- tions, 301 ² Dugundji formulas, 307 C Basic syntactics 311 12 Inference systems 313 13 Basic implication 318 ² Assumptions of basic implication, 319 ² A few easy derivations, 320 ² Lemmaless expansions, 326 ² De- tachmental corollaries, 330 ² Iterated implication (post- ponable), 332 14 Basic logic 336 ² Further assumptions, 336 ² Basic positive logic, 339 ² Basic negation, 341 ² Substitution principles, 343 D One-formula extensions 349 15 Contraction 351 ² Weak contraction, 351 ² Contraction, 355 16 Expansion and positive paradox 357 ² Expansion and mingle, 357 ² Positive paradox (strong expansion), 359 ² Further consequences of positive para- dox, 362 17 Explosion 365 18 Fusion 369 19 Not-elimination 372 ² Not-elimination and contrapositives, 372 ² Interchange- ability results, 373 ² Miscellaneous consequences of not- elimination, 375 viii Contents 20 Relativity 377 E Soundness and major logics 381 21 Soundness 383 22 Constructive axioms: avoiding not-elimination 385 ² Constructive implication, 386 ² Herbrand-Tarski De- duction Principle, 387 ² Basic logic revisited, 393 ² Soundness, 397 ² Nonconstructive axioms and classical logic, 399 ² Glivenko's Principle, 402 23 Relevant axioms: avoiding expansion 405 ² Some syntactic results, 405 ² Relevant deduction principle (optional), 407 ² Soundness, 408 ² Mingle: slightly irrelevant, 411 ² Positive paradox and classical logic, 415 24 Fuzzy axioms: avoiding contraction 417 ² Axioms, 417 ² Meredith's chain proof, 419 ² Addi- tional notations, 421 ² Wajsberg logic, 422 ² Deduction principle for Wajsberg logic, 426 25 Classical logic 430 ² Axioms, 430 ² Soundness results, 431 ² Independence of axioms, 431 26 Abelian logic 437 F Advanced results 441 27 Harrop's principle for constructive logic 443 ² Meyer's valuation, 443 ² Harrop's principle, 448 ² The disjunction property, 451 ² Admissibility, 451 ² Results in other logics, 452 Contents ix 28 Multiple worlds for implications 454 ² Multiple worlds, 454 ² Implication models, 458 ² Soundness, 460 ² Canonical models, 461 ² Complete- ness, 464 29 Completeness via maximality 466 ² Maximal unproving sets, 466 ² Classical logic, 470 ² Wajsberg logic, 477 ² Constructive logic, 479 ² Non-¯nitely-axiomatizable logics, 485 References 487 Symbol list 493 Index 495 Part A Preliminaries Chapter 1 Introduction for teachers Readers with no previous knowledge of formal logic will ¯nd it more useful to begin with Chapter 2. Purpose and intended audience 1.1. CNL (Classical and Nonclassical Logics) is intended as an introduction to mathematical logic. However, we wish to im- mediately caution the reader that the topics in this book are modal 2 3 traditional applied + £ = 2 ª this book predicate 8 9 ª propositional ^ _ : ! & ± classical constructive fuzzy relevant others not the same as those in a conventional introduction to logic. CNL should only be adopted by teachers who are aware of the di®erences and are persuaded of this book's advantages. Chiefly, CNL trades some depth for breadth: ² A traditional introduction to logic covers classical logic only, though possibly at several levels | propositional, predicate, modal, etc. 4 Chapter 1. Introduction for teachers ² CNL is pluralistic, in that it covers classical and several non- classical logics | constructive, quantitative, relevant, etc. | though almost solely at the propositional level. Of course, a logician needs both depth and breadth, but both cannot be acquired in the ¯rst semester. The depth-¯rst approach is prevalent in textbooks, perhaps merely because classical logic developed a few years before other logics. I am convinced that a breadth-¯rst approach would be better for the students, for reasons discussed starting in 1.9. 1.2. Intended audience. This is an introductory textbook. No previous experience with mathematical logic is required. Some experience with algebraic computation and abstract thinking is expected, perhaps at the precalculus level or slightly higher. The exercises in this book are mostly computational and require lit- tle originality; thus CNL may be too elementary for a graduate course. Of course, the book may be used at di®erent levels by di®erent instructors. CNL was written for classroom use; I have been teaching un- dergraduate classes from earlier versions for several years. How- ever, its ¯rst few chapters include su±cient review of prerequisite material to support the book's use also for self-guided study. Those chapters have some overlap with a \transition to higher mathematics" course; CNL might serve as a resource in such a course. I would expect CNL to be used mainly in mathematics depart- ments, but it might be adopted in some philosophy departments as well. Indeed, some philosophers are very mathematically in- clined; many of this book's mathematical theorems originated on the chalkboards of philosophy departments. Colleagues have also informed me that this book will be of some interest to students of computer science, but I am unfamiliar with such connections and have not pursued them in this book. 1.3. In what sense is this new? This book is a work of exposition and pedagogy, not of research. All the main theorems of this book have already appeared, albeit in di®erent form, in research Purpose and intended audience 5 journals or advanced monographs. But those articles and books were written to be read by experts. I believe that the present work is a substantially new selection and reformulation of results, and that it will be more accessible to beginners. 1.4. Avoidance of algebra. Aside from its pluralistic approach (discussed at much greater length later in this chapter), probably CNL's most unusual feature is its attempt to avoid higher algebra. In recent decades, mathematical logic has been freed from its philosophical and psychological origins; the current research literature views di®erent logics simply as di®erent kinds of al- gebraic structures. That viewpoint may be good for research, but it is not a good prescription for motivating undergraduate students,
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