Book:Ian Chiswell/Mathematical Logic

Ian Chiswell and Wilfrid Hodges: Mathematical Logic

Published $\text {2007}$, Oxford University Press

ISBN 978-0-19-857100-1

Volume 3 in the Oxford Texts in Logic series.

Subject Matter

Mathematical Logic

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 $\mathrm{LP}$, 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