The Language of First-Order Logic

by Barwise, Jon; Etchemendy, John

• ISBN13: 9780937073995
• ISBN10: 0937073997
• Edition: 3rd
• Format: Paperback
• Copyright: 1993-08-01
• Publisher: Stanford Univ Center for the Study

Summary

The Language of First-order Logic is a complete introduction to first-order symbolic logic, consisting of a computer program and a text. The program, an aid to learning and using symbolic notation, allows one to construct symbolic sentences and possible worlds, and verify that a sentence is well-formed. The truth or falsity of a sentence can be determined by playing a deductive game with the computer. The text covers four areas: propositional logic, quantifier logic, the application of first-order logic to set theory and induction, and advanced topics. Hundreds of exercises and problems are included. This revised and expanded edition includes an extensive series of translation exercises to and from ordinary language, and proof exercises using ordinary language. The new release of the Macintosh software, Tarski's World 4.O, has a new sentence window and includes a "verify all" command for quickly checking a set of sentences in a world.

Author Biography

Jon Barwise (1942-2000) was a mathematician, philosopher and logician. He taught at the Universities of Yale and Wisconsin before becoming the first director of the Center for the Study of Language and Information at Stanford University. From 1990 until his death he was professor of philosophy at Indiana University. John Etchemendy has been professor of philosophy at Stanford since 1983. In 2000 he became Provost of Stanford University.

Table of Contents

Preface	vii
Introduction	1	(6)
The special role of logic in rational inquiry	1	(1)
Why learn an artificial language?	2	(2)
About this book	4	(3)
Part I Propositional Logic	7	(106)
Atomic Sentences	9	(26)
Individual constants	9	(1)
Predicate symbols	10	(2)
Atomic sentences	12	(4)
The first-order language of set theory	16	(1)
Function symbols	17	(3)
The first-order language of arithmetic	20	(1)
General first-order languages	21	(3)
Methods of proof	24	(6)
Formal proofs	30	(4)
Alternative notation	34	(1)
Conjunctions, Disjunctions, and Negations	35	(56)
Negation symbol (¬)	36	(1)
Conjunction symbol (∧)	37	(2)
Disjunction symbol (∨)	39	(3)
Ambiguity and parentheses	42	(2)
Logical equivalence	44	(3)
Translation	47	(4)
Satisfiability and logical truth	51	(7)
Methods of proof involving ¬, ∧, and ∨	58	(8)
Formal proofs	66	(16)
Conjunctive and disjunctive normal forms	82	(3)
Truth-functional completeness	85	(3)
Alternative notation	88	(3)
Conditionals and Biconditionals	91	(22)
Material conditional symbol (→)	91	(2)
Biconditional symbol (↔)	93	(4)
Conversational implicature	97	(2)
Methods of proof involving → and ↔	99	(5)
Formal proofs	104	(4)
F': Speeding up system F	108	(4)
Alternative notation	112	(1)
Part II Quantifiers	113	(92)
Introduction to Quantification	115	(39)
Variables	115	(1)
Atomic wffs	115	(1)
Quantifiers	116	(1)
Wffs and sentences	117	(2)
Semantics for the quantifiers	119	(2)
Game rules for the quantifiers	121	(2)
The four Aristotelian forms	123	(2)
Translating complex noun phrases	125	(7)
Logical equivalences involving negation and quantifiers	132	(3)
Methods of proof involving ∀ and ∃	135	(9)
Formal proofs	144	(6)
Function symbols, revisited	150	(2)
Alternative notation	152	(2)
Sentences with More than One Quantifier	154	(31)
Multiple uses of a single quantifier symbol	154	(3)
Mixed quantifiers	157	(3)
The step-by-step method of translation	160	(2)
Paraphrasing English	162	(5)
Ambiguity and context sensitivity	167	(3)
Translations using function symbols	170	(2)
Methods of proof involving mixed quantifiers	172	(5)
Formal proofs	177	(4)
Prenex form	181	(4)
Some Specific Uses of Quantifiers	185	(20)
Making numerical claims	185	(4)
Definite descriptions	189	(1)
Methods of proof involving numerical claims	190	(4)
Some review problems	194	(7)
Expressive limitations of first-order logic	201	(4)
Part III Applications of First-order Logic	205	(36)
First-order Set Theory	207	(20)
Cantor's set theory	208	(3)
Singletons and the empty set	211	(1)
The subset relation	212	(2)
Intersection and union	214	(2)
Sets of sets	216	(2)
The powerset of a set	218	(2)
Russell's Paradox	220	(2)
Modern set theory	222	(5)
Induction	227	(14)
Some examples of induction	228	(6)
Inductive definitions in set theory	234	(1)
Induction on the natural numbers	235	(6)
Part IV Advanced Topics	241	(42)
Advanced Topics in Propositional Logic	243	(15)
Horn sentences	243	(4)
Satisfiable sentences and logically true sentences, revisited	247	(3)
Resolution	250	(5)
The conditional form of Horn sentences	255	(3)
Advanced Topics in FOL	258	(25)
First-order structures	258	(4)
Spurious structures	262	(1)
Truth and satisfaction, revisited	263	(6)
Skolemization	269	(1)
Unification of terms	270	(3)
Resolution, revisited	273	(5)
Completeness and incompleteness	278	(5)
Reference Material
A How to Use Tarski's World	283	(13)
A.1 The basics	283	(3)
Starting Tarski's World	283	(1)
The three main windows	284	(1)
Opening saved files	284	(1)
Starting new files	285	(1)
Saving a file	285	(1)
Saving a file on another disk	286	(1)
Quitting	286	(1)
A.2 Using the World Window	286	(2)
Adding objects	286	(1)
Naming objects	286	(1)
Moving objects	287	(1)
Sizing and shaping objects	287	(1)
Deleting objects	287	(1)
Hiding labels	287	(1)
2-D view	287	(1)
Rotating Worlds	288	(1)
A.3 Using the Keyboard and Sentence Windows	288	(4)
Writing formulas	288	(1)
Commenting your sentences	289	(1)
Creating a list of sentences	289	(1)
Moving from sentence to sentence	289	(1)
Deleting sentences	290	(1)
Editing sentences	290	(1)
Typing symbols from the keyboard	290	(1)
Cutting, copying, and pasting	291	(1)
Moving and sizing windows	292	(1)
Printing	292	(1)
A.4 Using the Evaluation Box	292	(1)
Checking syntax	292	(1)
Verifying truth	293	(1)
Verifying all sentences	293	(1)
A.5 Playing the game	293	(2)
Picking objects and sentences	294	(1)
Backing up and giving up	294	(1)
When to play the game	295	(1)
A.6 Using the Grader	295	(1)
B Macintosh Terminology	296	(3)
C Summary of Proof Rules	299	(5)
C.1 System F	299	(2)
C.2 System F'	301	(3)
General Index	304	(7)
Tarski's World Index	311	(1)
Exercise Files Index	312

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