1st Edition

# Logic Works A Rigorous Introduction to Formal Logic

666 Pages
by Routledge

666 Pages
by Routledge

666 Pages
by Routledge

Also available as eBook on:

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

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 D6.3: 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-5.1:
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-6.1:
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

### Biography

Lorne Falkenstein is Professor Emeritus at Western University in London, Canada, where he taught symbolic logic for many years. He has published on treatments of spatial representation, temporal awareness, and visual perception in the work of a number of 17th and 18th century philosophers, and continues to do work in that area.

Scott Stapleford is Professor of Philosophy at St. Thomas University in Fredericton, Canada. He is the author of Kant’s Transcendental Arguments: Disciplining Pure Reason (2008), coauthor (with Tyron Goldschmidt) of Berkeley’s Principles: Expanded and Explained (Routledge, 2016) and Hume’s Enquiry: Expanded and Explained (Routledge, 2021), coeditor (with Kevin McCain) of Epistemic Duties: New Arguments, New Angles (Routledge, 2020), and coeditor (with Kevin McCain and Matthias Steup) of Epistemic Dilemmas: New Arguments, New Angles (Routledge, 2021).

Molly Kao is Assistant Professor of Philosophy at the University of Montreal, in Montreal, Canada. Her primary area of research is philosophy of science, having worked on issues in the development of quantum theory as well as methodological questions involving unification and confirmation.

"Logic Works is a thorough treatment of core topics in elementary logic, and of several topics in intermediate logic. Its precision and rigor is a step above typical presentations of this material. It will be an invaluable resource for teachers, as well as for students of logic who want to go beyond the basics."
Fabrizio Cariani, University of Maryland