Description
Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines.
The book covers classical first-order logic and alternatives, including intuitionistic, free, and many-valued logic. It also considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, independent study as a course text, it covers more material than is typically covered in an introductory course. It also covers this material at greater length and in more depth with the purpose of making it accessible to those with no prior training in logic or formal systems.
Online support material includes a detailed student solutions manual with a running commentary on all starred exercises, and a set of editable slide presentations for course lectures.
Key Features
- Introduces an unusually broad range of topics, allowing instructors to craft courses to meet a range of various objectives
- Adopts a critical attitude to certain classical doctrines, exposing students to alternative ways to answer philosophical questions about logic
- Carefully considers the ways natural language both resists and lends itself to formalization
- Makes objectual semantics for quantified logic easy, with an incremental, rule-governed approach assisted by numerous simple exercises
- Makes important metatheoretical results accessible to introductory students through a discursive presentation of those results and by using simple case studies
Table of Contents
Preface
Symbol Summary
1. Introduction to the study of logic
1.1 Demonstration and interpretation 1.2 Deductive and inductive demonstrations 1.3 The principle of noncontradiction 1.4 Abstraction, variables, and formalization; logical and nonlogical elements; formal contradiction 1.5 A fundamental problem 1.6 Outline of forthcoming chapters Appendix: Elements of a theory of demonstrative logic
Part I: Sentential Logic
2. Vocabulary and syntax
2.1: Introduction Syntax 2.2: Conventions 2.3: Syntactic demonstrations and trees 2.4: Scope; named forms 2.5: Formal properties
3. Semantics
3.1: Semantics for ⊥ and the sentence letters 3.2: Semantics for the connectives 3.3: Semantics for compound sentences 3.4: Intensional concepts Appendix: Expressive adequacy; disjunctive normal form; the lean language
4. Formalization
4.1: Looseness of fit 4.2: Conditional sentences of English 4.3: Necessary conditions 4.4: Sufficient conditions 4.5: Necessary and sufficient conditions; the principle of charity 4.6: Formalizing necessary and sufficient conditions 4.7: Exceptions and strong exceptions 4.8: Disjunction 4.9: Negations and conjunctions 4.10: Punctuation 4.11: Limits of formalization 4.12: Formalizing demonstrations
5. Working with SL semantics
5.1: Identifying and verifying interpretations 5.2: Demonstrating that there is no interpretation 5.3: Demonstrating general principles 5.4: Falsifying general claims 5.5: Relations between intensional concepts; models; entailment Appendix: Alternatives to bivalence
A-1. Advanced topics concerning SL semantics
A-1.1: Mathematical induction A-1.2: Bivalence A-1.3: Extensionality A-1.4: Compactness
6. Derivations
6.1: DL: a lean derivation system 6.2: Strategies for doing derivations in DL 6.3: Ds: a derivation system for SL 6.4: Strategies for doing derivations in Ds 6.5: Extensions of Ds; bracket free notation 6.6: Intuition and "Intuitionism"; derivation in intuitionistic logic
A-2. Advanced topics concerning the soundness and completeness of Ds
A-2.1: Soundness A-2.2: Corollary results; consistency and extensionality A-2.3: Henkin completeness A-2.4: Proof of the Lindenbaum lemma A-2.5: Proof of lemma 2 A-2.6: Proof of lemma 3 A-2.7: Corollary results A-2.8: Post / Hilbert-Ackermann completeness
7. Reduction Trees
7.1: Method and strategies 7.2: Using trees to determine derivability 7.3: Theory and definitions Appendix: Trees for three valued and paraconsistent logic
A-3: Advanced topics concerning the soundness and completeness of Ts
A-3.1: Soundness of Ts A-3.2: Completeness of Ts A-3.3: Decidability of Ts A-3.4: Converting trees to derivations; using Ts to prove the completeness of Ds
Part II: Modal sentential logic
8. Vocabulary, syntax, formalization and derivations
8.1: Vocabulary and syntax 8.2: Formalization 8.3: Derivations
9. Semantics and Trees for Modal and Intuitionistic Sentential Logic
9.1: Semantics for MSL 9.2: Reduction trees for MSL 9.3: Semantics for ISL 9.4: Reduction trees for ISL
A-4: Advanced Topics concerning the "soundness" and "completeness" of Dm and Tm
A-4.1: "Soundness" of the modal derivation systems A-4.2: Completeness of Tm A-4.3: Tree conversions A-4.4: Adequacy of Dm and Tm
Part III: Predicate sentential logic
10. Vocabulary, syntax, formalization, and derivations
10.1: English predication 10.2: Simple terms 10.3: Complex terms
11. Semantics and trees
11.1: Interpretations 11.2: Valuation rules 11.3: Working with the semantics 11.4: Tp 11.5: Semantics for functional terms 11.6: Tpf 11.7: Semantics for definite descriptions
A-5: Advanced topics for PSL
A-5.1: Extensionality and variance A-5.2: Soundness of Dp A-5.3: Completeness of Tp A-5.4: Tree conversion; soundness of Tp; completeness of Dp
Part IV: Quantified Predicate Logic
12. Vocabulary, syntax, and formalization
12.1: Informal vocabulary and syntax 12.2: Formal vocabulary and syntax 12.3: Formalizing English sentences in QPL
13. Derivations
13.1: Dq 13.2: Extensions of Dq
14. Trees and tree model semantics for QPL
14.1: Rules 14.2: Method 14.3: Tree model semantics 14.4: Extensions of Tq
15. Semantics for QPL without mixed multiple quantification
15.1: Objectual semantics 15.2: Denotation 15.3: Satisfaction 15.4: Truth 15.5: Working with the semantics 15.6: Demonstrating general principles
16. Semantics for QPL with mixed multiple quantification
16.1: Variants on variable assignments; denotation of variables 16.2: Satisfaction conditions for quantified formulas 16.3: (P) and (=) applications 16.4: Truth 16.5: Working with the semantics Appendix: Demonstration of the exclusivity principle
A-6: Advanced topics for QPL
A-6.1: Extensionality and variance A-6.2: Soundness of Dq A-6.3: Completeness of Tq A-6.4: Tree conversions; soundness of Tp; Completeness of Dp Appendix: Quantified modal logic
17. Higher order logic
17.1 Vocabulary and syntax 17.2: Formalization; definitions of higher order predicates 17.3: Syntax II: instances 17.4: Derivations 17.5: Semantics 17.6: Trees and incompleteness
Rule summaries
1. Foundational definitions
2. Intensional concepts
3. Formation rules
4. Sentential valuation rules
5. Formulaic and free valuation rules
6. Derivation rules
7. Tree rules
