Mathematical Logic.
Title:
Mathematical Logic.
Author:
Chiswell, Ian.
ISBN:
9780191524806
Personal Author:
Physical Description:
1 online resource (259 pages)
Series:
Oxford Texts in Logic ; v.No. 3
Oxford Texts in Logic
Contents:
Contents -- 1 Prelude -- 1.1 What is mathematics? -- 1.2 Pronunciation guide -- 2 Informal natural deduction -- 2.1 Proofs and sequents -- 2.2 Arguments introducing 'and' -- 2.3 Arguments eliminating 'and' -- 2.4 Arguments using 'if' -- 2.5 Arguments using 'if and only if' -- 2.6 Arguments using 'not' -- 2.7 Arguments using 'or' -- 3 Propositional logic -- 3.1 LP, the language of propositions -- 3.2 Parsing trees -- 3.3 Propositional formulas -- 3.4 Propositional natural deduction -- 3.5 Truth tables -- 3.6 Logical equivalence -- 3.7 Substitution -- 3.8 Disjunctive and conjunctive normal forms -- 3.9 Soundness for propositional logic -- 3.10 Completeness for propositional logic -- 4 First interlude: Wason's selection task -- 5 Quantifier-free logic -- 5.1 Terms -- 5.2 Relations and functions -- 5.3 The language of first-order logic -- 5.4 Proof rules for equality -- 5.5 Interpreting signatures -- 5.6 Closed terms and sentences -- 5.7 Satisfaction -- 5.8 Diophantine sets and relations -- 5.9 Soundness for qf sentences -- 5.10 Adequacy and completeness for qf sentences -- 6 Second interlude: the Linda problem -- 7 First-order logic -- 7.1 Quantifiers -- 7.2 Scope and freedom -- 7.3 Semantics of first-order logic -- 7.4 Natural deduction for first-order logic -- 7.5 Proof and truth in arithmetic -- 7.6 Soundness and completeness for first-order logic -- 7.7 First-order theories -- 7.8 Cardinality -- 7.9 Things that first-order logic cannot do -- 8 Postlude -- Appendix A: The natural deduction rules -- Appendix B: Denotational semantics -- Appendix C: Solutions to some exercises -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.
Abstract:
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. - ;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. -
Mathematical Logic is crisply written and is a pleasure to read. ..Chiswell and Hodges' book is at the very top of the reading list. - Michael Berg, MAA Online;The text is clearly laid out and written in an easy-to-read free-flowing style. - Times Higher Education Supplement.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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