mathematics - language proof and logic

LANGUAGE, PROOF AND LOGIC JON BARWISE & JOHN ETCHEMENDY In collaboration with Gerard Allwein Dave Barker-Plummer Albert Liu 7 7 SEVEN BRIDGES PRESS NEW YORK • LONDON Library of Congress Cataloging-in-Publication Data Barwise, Jon. Language, proof and logic / Jon Barwise and John Etchemendy ; in collaboration with Gerard Allwein, Dave Barker-Plummer, and Albert Liu. p. cm. ISBN 1-889119-08-3 (pbk. : alk. paper) I. Etchemendy, John, 1952- II. Allwein, Gerard, 1956- III. Barker-Plummer, Dave. IV. Liu, Albert, 1966- V. Title. IN PROCESS 99-41113 CIP Copyright © 1999 CSLI Publications Center for the Study of Language and Information Leland Stanford Junior University 03 02 01 00 99 5 4 3 2 1 Acknowledgements Our primary debt of gratitude goes to our three main collaborators on this project: Gerry Allwein, Dave Barker-Plummer, and Albert Liu. They have worked with us in designing the entire package, developing and implementing the software, and teaching from and refining the text. Without their intelli- gence, dedication, and hard work, LPL would neither exist nor have most of its other good properties. In addition to the five of us, many people have contributed directly and in- directly to the creation of the package. First, over two dozen programmers have worked on predecessors of the software included with the package, both earlier versions of Tarski’s World and the program Hyperproof, some of whose code has been incorporated into Fitch. We want especially to mention Christopher Fuselier, Mark Greaves, Mike Lenz, Eric Ly, and Rick Wong, whose outstand- ing contributions to the earlier programs provided the foundation of the new software. Second, we thank several people who have helped with the develop- ment of the new software in essential ways: Rick Sanders, Rachel Farber, Jon Russell Barwise, Alex Lau, Brad Dolin, Thomas Robertson, Larry Lemmon, and Daniel Chai. Their contributions have improved the package in a host of ways. Prerelease versions of LPL have been tested at several colleges and uni- versities. In addition, other colleagues have provided excellent advice that we have tried to incorporate into the final package. We thank Selmer Bringsjord, Renssalaer Polytechnic Institute; Tom Burke, University of South Carolina; Robin Cooper, Gothenburg University; James Derden, Humboldt State Uni- versity; Josh Dever, SUNY Albany; Avrom Faderman, University of Rochester; James Garson, University of Houston; Ted Hodgson, Montana State Univer- sity; John Justice, Randolph-Macon Women’s College; Ralph Kennedy, Wake Forest University; Michael O’Rourke, University of Idaho; Greg Ray, Univer- sity of Florida; Cindy Stern, California State University, Northridge; Richard Tieszen, San Jose State University; Saul Traiger, Occidental College; and Lyle Zynda, Indiana University at South Bend. We are particularly grateful to John Justice, Ralph Kennedy, and their students (as well as the students at Stan- ford and Indiana University), for their patience with early versions of the software and for their extensive comments and suggestions. We would also like to thank Stanford’s Center for the Study of Language and Information and Indiana University’s College of Arts and Sciences for iii iv / Acknowledgements their financial support of the project. Finally, we are grateful to our two publishers, Dikran Karagueuzian of CSLI Publications and Clay Glad of Seven Bridges Press, for their skill and enthusiasm about LPL. Acknowledgements Contents Acknowledgements iii Introduction 1 The special role of logic in rational inquiry . . . . . . . . . . . . . . 1 Why learn an artificial language? . . . . . . . . . . . . . . . . . . . . 2 Consequence and proof . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Instructions about homework exercises (essential! ) . . . . . . . . . . 5 To the instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Web address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 I Propositional Logic 17 1 Atomic Sentences 19 1.1 Individual constants . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 Predicate symbols . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Atomic sentences . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4 General first-order languages . . . . . . . . . . . . . . . . . . . 28 1.5 Function symbols (optional ) . . . . . . . . . . . . . . . . . . . . 31 1.6 The first-order language of set theory (optional) . . . . . . . . 37 1.7 The first-order language of arithmetic (optional) . . . . . . . . 38 1.8 Alternative notation (optional) . . . . . . . . . . . . . . . . . . 40 2 The Logic of Atomic Sentences 41 2.1 Valid and sound arguments . . . . . . . . . . . . . . . . . . . . 41 2.2 Methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Formal proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.4 Constructing proofs in Fitch . . . . . . . . . . . . . . . . . . . . 58 2.5 Demonstrating nonconsequence . . . . . . . . . . . . . . . . . . 63 2.6 Alternative notation (optional) . . . . . . . . . . . . . . . . . . 66 3 The Boolean Connectives 67 3.1 Negation symbol: ¬ . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Conjunction symbol: ∧ . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Disjunction symbol: ∨ . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 Remarks about the game . . . . . . . . . . . . . . . . . . . . . 77 v vi / Contents 3.5 Ambiguity and parentheses . . . . . . . . . . . . . . . . . . . . 79 3.6 Equivalent ways of saying things . . . . . . . . . . . . . . . . . 82 3.7 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.8 Alternative notation (optional) . . . . . . . . . . . . . . . . . . 89 4 The Logic of Boolean Connectives 93 4.1 Tautologies and logical truth . . . . . . . . . . . . . . . . . . . 94 4.2 Logical and tautological equivalence . . . . . . . . . . . . . . . 106 4.3 Logical and tautological consequence . . . . . . . . . . . . . . . 110 4.4 Tautological consequence in Fitch . . . . . . . . . . . . . . . . . 114 4.5 Pushing negation around (optional) . . . . . . . . . . . . . . . 117 4.6 Conjunctive and disjunctive normal forms (optional) . . . . . . 121 5 Methods of Proof for Boolean Logic 127 5.1 Valid inference steps . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2 Proof by cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3 Indirect proof: proof by contradiction . . . . . . . . . . . . . . . 136 5.4 Arguments with inconsistent premises (optional ) . . . . . . . . 140 6 Formal Proofs and Boolean Logic 142 6.1 Conjunction rules . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Disjunction rules . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3 Negation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4 The proper use of subproofs . . . . . . . . . . . . . . . . . . . . 163 6.5 Strategy and tactics . . . . . . . . . . . . . . . . . . . . . . . . 167 6.6 Proofs without premises (optional) . . . . . . . . . . . . . . . . 173 7 Conditionals 176 7.1 Material conditional symbol: → . . . . . . . . . . . . . . . . . . 178 7.2 Biconditional symbol: ↔ . . . . . . . . . . . . . . . . . . . . . . 181 7.3 Conversational implicature . . . . . . . . . . . . . . . . . . . . 187 7.4 Truth-functional completeness (optional) . . . . . . . . . . . . . 190 7.5 Alternative notation (optional) . . . . . . . . . . . . . . . . . . 196 8 The Logic of Conditionals 198 8.1 Informal methods of proof . . . . . . . . . . . . . . . . . . . . . 198 8.2 Formal rules of proof for → and ↔ . . . . . . . . . . . . . . . . 206 8.3 Soundness and completeness (optional) . . . . . . . . . . . . . . 214 8.4 Valid arguments: some review exercises . . . . . . . . . . . . . . 222 Contents Contents / vii II Quantifiers 225 9 Introduction to Quantification 227 9.1 Variables and atomic wffs . . . . . . . . . . . . . . . . . . . . . 228 9.2 The quantifier symbols: ∀, ∃ . . . . . . . . . . . . . . . . . . . . 230 9.3 Wffs and sentences . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.4 Semantics for the quantifiers . . . . . . . . . . . . . . . . . . . . 234 9.5 The four Aristotelian forms . . . . . . . . . . . . . . . . . . . . 239 9.6 Translating complex noun phrases . . . . . . . . . . . . . . . . 243 9.7 Quantifiers and function symbols (optional) . . . . . . . . . . . 251 9.8 Alternative notation (optional) . . . . . . . . . . . . . . . . . . 255 10 The Logic of Quantifiers 257 10.1 Tautologies and quantification . . . . . . . . . . . . . . . . . . . 257 10.2 First-order validity and consequence . . . . . . . . . . . . . . . 266 10.3 First-order equivalence and DeMorgan’s laws . . . . . . . . . . 275 10.4 Other quantifier equivalences (optional ) . . . . . . . . . . . . . 280 10.5 The axiomatic method (optional) . . . . . . . . . . . . . . . . . 283 11 Multiple Quantifiers 289 11.1 Multiple uses of a single quantifier . . . . . . . . . . . . . . . . 289 11.2 Mixed quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . 293 11.3 The step-by-step method of translation . . . . . . . . . . . . . . 298 11.4 Paraphrasing English . . . . . . . . . . . . . . . . . . . . . . . . 300 11.5 Ambiguity and context sensitivity . . . . . . . . . . . . . . . . 304 11.6 Translations using function symbols (optional) . . . . . . . . . 308 11.7 Prenex form (optional ) . . . . . . . . . . . . . . . . . . . . . . . 311 11.8 Some extra translation problems . . . . . . . . . . . . . . . . . 315 12 Methods of Proof for Quantifiers 319 12.1 Valid quantifier steps . . . . . . . . . . . . . . . . . . . . . . . . 319 12.2 The method of existential instantiation . . . . . . . . . . . . . . 322 12.3 The method of general conditional proof . . . . . . . . . . . . . 323 12.4 Proofs involving mixed quantifiers . . . . . . . . . . . . . . . . 329 12.5 Axiomatizing shape (optional ) . . . . . . . . . . . . . . . . . . 338 13 Formal Proofs and Quantifiers 342 13.1 Universal quantifier rules . . . . . . . . . . . . . . . . . . . . . 342 13.2 Existential quantifier rules . . . . . . . . . . . . . . . . . . . . . 347 13.3 Strategy and tactics . . . . . . . . . . . . . . . . . . . . . . . . 352 13.4 Soundness and completeness (optional) . . . . . . . . . . . . . . 361 Contents viii / Contents 13.5 Some review exercises (optional) . . . . . . . . . . . . . . . . . 361 14 More about Quantification (optional) 364 14.1 Numerical quantification . . . . . . . . . . . . . . . . . . . . . . 366 14.2 Proving numerical claims . . . . . . . . . . . . . . . . . . . . . 374 14.3 The, both, and neither . . . . . . . . . . . . . . . . . . . . . . . 379 14.4 Adding other determiners to fol . . . . . . . . . . . . . . . . . 383 14.5 The logic of generalized quantification . . . . . . . . . . . . . . 389 14.6 Other expressive limitations of first-order logic . . . . . . . . . 397 III Applications and Metatheory 403 15 First-order Set Theory 405 15.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 406 15.2 Singletons, the empty set, subsets . . . . . . . . . . . . . . . . . 412 15.3 Intersection and union . . . . . . . . . . . . . . . . . . . . . . . 415 15.4 Sets of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 15.5 Modeling relations in set theory . . . . . . . . . . . . . . . . . . 422 15.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 15.7 The powerset of a set (optional) . . . . . . . . . . . . . . . . . 429 15.8 Russell’s Paradox (optional) . . . . . . . . . . . . . . . . . . . . 432 15.9 Zermelo Frankel set theory zfc (optional) . . . . . . . . . . . . 433 16 Mathematical Induction 442 16.1 Inductive definitions and inductive proofs . . . . . . . . . . . . 443 16.2 Inductive definitions in set theory . . . . . . . . . . . . . . . . . 451 16.3 Induction on the natural numbers . . . . . . . . . . . . . . . . . 453 16.4 Axiomatizing the natural numbers (optional ) . . . . . . . . . . 456 16.5 Proving programs correct (optional ) . . . . . . . . . . . . . . . 458 17 Advanced Topics in Propositional Logic 468 17.1 Truth assignments and truth tables . . . . . . . . . . . . . . . . 468 17.2 Completeness for propositional logic . . . . . . . . . . . . . . . 470 17.3 Horn sentences (optional) . . . . . . . . . . . . . . . . . . . . . 479 17.4 Resolution (optional) . . . . . . . . . . . . . . . . . . . . . . . . 488 18 Advanced Topics in FOL 495 18.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . 495 18.2 Truth and satisfaction, revisited . . . . . . . . . . . . . . . . . . 500 18.3 Soundness for fol . . . . . . . . . . . . . . . . . . . . . . . . . 509 Contents Contents / ix 18.4 The completeness of the shape axioms (optional ) . . . . . . . . 512 18.5 Skolemization (optional) . . . . . . . . . . . . . . . . . . . . . . 514 18.6 Unification of terms (optional ) . . . . . . . . . . . . . . . . . . 516 18.7 Resolution, revisited (optional) . . . . . . . . . . . . . . . . . . 519 19 Completeness and Incompleteness 526 19.1 The Completeness Theorem for fol . . . . . . . . . . . . . . . 527 19.2 Adding witnessing constants . . . . . . . . . . . . . . . . . . . . 529 19.3 The Henkin theory . . . . . . . . . . . . . . . . . . . . . . . . . 531 19.4 The Elimination Theorem . . . . . . . . . . . . . . . . . . . . . 534 19.5 The Henkin Construction . . . . . . . . . . . . . . . . . . . . . 540 19.6 The L¨owenheim-Skolem Theorem . . . . . . . . . . . . . . . . . 546 19.7 The Compactness Theorem . . . . . . . . . . . . . . . . . . . . 548 19.8 The G¨odel Incompleteness Theorem . . . . . . . . . . . . . . . 552 Summary of Formal Proof Rules 557 Propositional rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 First-order rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Inference Procedures (Con Rules) . . . . . . . . . . . . . . . . . . . 561 Glossary 562 General Index 573 Exercise Files Index 585 Contents [...]... intelligence logic and computer science Why learn an artificial language? 4 / Introduction artificial languages logic and ordinary language logical basis of Prolog a bit in Part III of this book Fol serves as the prototypical example of what is known as an artificial language These are languages that were designed for special purposes, and are contrasted with so-called natural languages, languages like English and. .. artificial language that is precise yet rich enough to program computers was inspired by this language In addition, all extant programming languages borrow some notions from one or another dialect of fol Finally, there are so-called logic programming languages, like Prolog, whose programs are sequences of sentences in a certain dialect of fol We will discuss the logic and philosophy logic and artificial... rational inquiry, like medicine and finance, it is also a valuable tool for understanding the principles of rationality underlying these disciplines as well The language goes by various names: the lower predicate calculus, the functional calculus, the language of first-order logic, and fol The last of Why learn an artificial language? / 3 these is pronounced ef–oh–el, not fall, and is the name we will use... there both are and are not nine planets The importance of logic has been recognized since antiquity After all, no 1 logic and rational inquiry logic and convention 2 / Introduction laws of logic goals of the book science can be any more certain than its weakest link If there is something arbitrary about logic, then the same must hold of all rational inquiry Thus it becomes crucial to understand just what... what the laws of logic are, and even more important, why they are laws of logic These are the questions that one takes up when one studies logic itself To study logic is to use the methods of rational inquiry on rationality itself Over the past century the study of logic has undergone rapid and important advances Spurred on by logical problems in that most deductive of disciplines, mathematics, it developed... put the language to more sophisticated uses As we said earlier, besides teaching the language fol, we also discuss basic methods of proof and how to use them In this regard, too, our approach is somewhat unusual We emphasize both informal and formal methods of proof We first discuss and analyze informal reasoning methods, the kind used in everyday life, and then formalize these using a Fitch-style natural... subject-predicate sentence It consists of the subject Max followed by the predicate likes Claire In fol, by contrast, we view this as a claim involving two “logical subjects,” the names Max and Claire, and a 1 There is, however, a variant of first-order logic called free logic in which this assumption is relaxed In free logic, there can be individual constants without referents This yields a language. .. fol In this book, we will explore methods of proof how we can prove that one claim is a logical consequence of another and also methods for showing that a claim is not a consequence of others In addition to the language fol itself, these two methods, the method of proof and the method of counterexample, form the principal subject matter of this book proof and counterexample Essential instructions about... particularly, we have two main aims The first is to help you learn a new language, the language of first-order logic The second is to help you learn about the notion of logical consequence, and about how one goes about establishing whether some claim is or is not a logical consequence of other accepted claims While there is much more to logic than we can even hint at in this book, or than any one person... predicates like Cube, Larger, and Between Some examples of atomic sentences in this language are Cube(b), Larger(c, f), and Between(b, c, d) These sentences say, respectively, that b is a cube, that c is larger than f , and that b is between c and d Later in this chapter, we will look at the atomic sentences used in two other versions of fol, the first-order languages of set theory and arithmetic In the next . Cataloging-in-Publication Data Barwise, Jon. Language, proof and logic / Jon Barwise and John Etchemendy ; in collaboration with Gerard Allwein, Dave Barker-Plummer, and Albert Liu. p. cm. ISBN 1-8 8911 9-0 8-3 . cm. ISBN 1-8 8911 9-0 8-3 (pbk. : alk. paper) I. Etchemendy, John, 195 2- II. Allwein, Gerard, 195 6- III. Barker-Plummer, Dave. IV. Liu, Albert, 196 6- V. Title. IN PROCESS 9 9-4 1113 CIP Copyright. The Logic of Boolean Connectives 93 4.1 Tautologies and logical truth . . . . . . . . . . . . . . . . . . . 94 4.2 Logical and tautological equivalence . . . . . . . . . . . . . . . 106 4.3 Logical