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Mathematical Logic
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IAN Chiswell & WILFRID Hodges
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Oxford
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1 Edition
ÆäÀÌÁö¼ö
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250
ISBN
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9780199215621
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Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in Logic, Mathematics, Philosophy, and Computer Science.
Prelude 1(4)
What is mathematics? 1(2)
Pronunciation guide 3(2)
Informal natural deduction 5(26)
Proofs and sequents 6(3)
Arguments introducing `and' 9(5)
Arguments eliminating `and' 14(2)
Arguments using `if' 16(6)
Arguments using `if and only if' 22(2)
Arguments using `not' 24(3)
Arguments using `or' 27(4)
Propositional logic 31(66)
LP, the language of propositions 32(6)
Parsing trees 38(7)
Propositional formulas 45(8)
Propositional natural deduction 53(9)
Truth tables 62(7)
Logical equivalence 69(3)
Substitution 72(6)
Disjunctive and conjunctive normal forms 78(7)
Soundness for propositional logic 85(4)
Completeness for propositional logic 89(8)
First interlude: Wason's selection task 97(4)
Quantifier-free logic 101(56)
Terms 101(4)
Relations and functions 105(6)
The language of first-order logic 111(10)
Proof rules for equality 121(7)
Interpreting signatures 128(6)
Closed terms and sentences 134(5)
Satisfaction 139(4)
Diophantine sets and relations 143(5)
Soundness for qf sentences 148(2)
Adequacy and completeness for qf sentences 150(7)
Second interlude: the Linda problem 157(2)
First-order logic 159(54)
Quantifiers 159(4)
Scope and freedom 163(6)
Semantics of first-order logic 169(8)
Natural deduction for first-order logic 177(9)
Proof and truth in arithmetic 186(3)
Soundness and completeness for first-order logic 189(5)
First-order theories 194(5)
Cardinality 199(7)
Things that first-order logic cannot do 206(7)
Postlude 213(4)
Appendix A The natural deduction rules 217(6)
Appendix B Denotational semantics 223(6)
Appendix C Solutions to some exercises 229(16)
Index 245
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