Logic for Philosophy

Citation

Sider, T. (2010). Logic for Philosophy. Oxford University Press.

Chapter Summary

The book has ten chapters, a set of answers and hints (Appendix A), references and an index, and runs to roughly 360 pages. Exercises are placed inside the chapters rather than at their ends. In the preface, Sider explains that those marked with one asterisk have solutions in Appendix A, those marked with two are harder and have hints, and the exercises on metalogic are the more difficult kind and can be skipped in a more elementary course. The stated aim is "logic literacy" for contemporary philosophy: model theory first, proof theory second, and some elementary philosophy of logic.

Chapter 1: What is Logic?

Sider assumes the reader has taken an introductory symbolic logic course and uses this chapter to say what the rest is for. It treats logical consequence as truth-preservation by virtue of form, explains why logic is studied through formal languages and why that makes metalogic possible, and separates proofs inside a formal system from metalogical proofs about it. Later sections cover the nature of logical consequence, logical constants, extensions, deviations and variations of classical logic, and a long section (1.8, about fifteen pages) of set theory for the rest of the book.

Chapter 2: Propositional Logic

The core technical chapter for propositional logic: grammar, the semantic approach, truth-table style semantics, validity and invalidity, sequent proofs and axiomatic proofs. It then turns to metalogic proper, proving soundness by induction, the deduction theorem and, through maximal consistent sets, completeness.

Chapter 3: Beyond Standard Propositional Logic

Alternatives and additions to the classical system: alternate connectives (the Sheffer stroke, inadequate connective sets), Polish notation, and nonclassical propositional logics. Three-valued logic is treated through Łukasiewicz's and Kleene's systems, Priest's logic of paradox and supervaluationism, and the chapter ends with the proof theory of intuitionistic propositional logic.

Chapter 4: Predicate Logic

Grammar and semantics of standard predicate logic, ways of establishing validity and invalidity, axiomatic proofs, and a short section (about two pages) on the metalogic of predicate logic. The preface states that soundness and completeness for predicate logic are deliberately not proved in the book.

Chapter 5: Beyond Standard Predicate Logic

Extensions that philosophers meet in practice: the identity sign, function symbols, definite descriptions (including how to eliminate function symbols and descriptions), generalized monadic and binary quantifiers, second-order logic, complex predicates, and free logic with its semantics and proof theory.

Chapter 6: Propositional Modal Logic

The longest chapter, at about sixty pages. It gives the grammar and symbolizations of modal propositional logic, Kripke-model semantics with validity proofs and countermodels, and the axiomatic systems K, D, T, B, S4 and S5. It closes with soundness proofs and a completeness proof by canonical models.

Chapter 7: Beyond Standard MPL

Applications and variants of the modal framework: deontic logic, epistemic logic, propositional tense logic (with a discussion of the metaphysics of time and formal constraints on the earlier-than relation), and the semantics of intuitionistic propositional logic, complementing the proof theory of Chapter 3.

Chapter 8: Counterfactuals

Begins with the logical behaviour of natural-language counterfactuals (they can be contingent; augmentation and contraposition fail; they depend on context). It then presents the Lewis/Stalnaker approach, Stalnaker's system SC with validity proofs and countermodels, Lewis's criticisms and his own system, and the problem of disjunctive antecedents.

Chapter 9: Quantified Modal Logic

Brings predicate logic and modal logic together: the de re/de dicto distinction, a simple semantics for quantified modal logic, countermodels and validity proofs, philosophical questions about that semantics (the necessity of identity and of existence), variable domains, and axiomatization.

Chapter 10: Two-dimensional modal logic

Introduces an actuality operator with its semantics (10.1), a further operator treated with two-dimensional semantics (10.2), and "fixedly" (10.3), and closes with a section on necessity and a priority.

Key Concepts

Chapters 1–3: Foundations and propositional logic

• Logical consequence: truth-preservation by virtue of form, the notion that formal systems are built to model.
• Metalogic: proving facts about formal systems, such as soundness (everything provable is valid) and completeness (everything valid is provable).
• Nonclassical logics: systems with more than two truth values or different proof rules, such as three-valued logics, the logic of paradox and intuitionistic logic.

Chapters 4–5: Predicate logic and its extensions

• Semantics for quantifiers: interpretations of predicate-logic sentences that settle validity and invalidity.
• Definite descriptions and free logic: two treatments of terms that may fail to denote.
• Second-order logic and generalized quantifiers: quantification beyond "all" and "some" over individuals.

Chapters 6–10: Modal logic and counterfactuals

• Kripke models: worlds with an accessibility relation, the semantic tool behind the whole modal part of the book.
• Systems K to S5: axiom systems distinguished by the conditions placed on accessibility.
• Canonical models: the construction behind completeness proofs for modal systems.
• De re and de dicto: whether a modal claim concerns a particular thing or a whole proposition.
• Variable domains: letting the set of existing things differ from world to world, which bears on contingent existence.
• Actuality and two-dimensional semantics: evaluating sentences at pairs of worlds to model "actually" and the link between necessity and a priority.

Critical Analysis

Strengths

• A clear scope and a stated rationale. The preface presents the book as a one-course alternative to an intensive metalogic course, and the contents follow that plan: wide coverage of the extensions philosophers use, with metalogic included only where it shows what metalogic is.
• Breadth that matches philosophical literature. Counterfactuals, tense and deontic logic, intuitionistic and paraconsistent systems, free logic, second-order logic and two-dimensional semantics all have their own sections.
• Exercises with solutions and hints. Starred exercises have answers in Appendix A and double-starred ones have hints, which makes the book usable for self-study.
• Modal logic treated at length. Chapters 6 to 10 carry most of the book's weight, including the soundness and completeness proofs, so a reader can follow modal arguments in the literature rather than only recognise the notation.

Limitations

• Metalogic is partial by design. Completeness is proved for propositional logic and propositional modal logic, but not for predicate logic, and computability and Gödel's incompleteness theorems are outside the book. Readers who need those results should use a dedicated metalogic text.
• An introductory course is assumed. Chapter 1 opens by taking symbolization, truth tables and some proof theory for granted, so the book is not a first logic course.
• Informality about notation. The preface says use and mention are treated loosely where it does not matter, which some readers will find imprecise.
• A compressed set-theory section. Section 1.8 is the longest in the first chapter, and readers without prior set theory will find it dense.

Where It Fits in the Course

Within the logic list, Sider is the second-course text. Language, Proof and Logic and the open-licensed forall x: Calgary are first-order introductions with natural deduction (the latter ends with brief modal logic), and Copi's Introduction to Logic is a long-established general textbook that also covers informal reasoning and syllogisms. Any of these prepares for Sider.

Boolos, Burgess and Jeffrey's Computability and Logic is the metalogic text the preface names as the usual basis for the one-course alternative: it proves the results Sider omits, namely completeness for predicate logic, undecidability and incompleteness. A student who needs philosophical breadth should choose Sider; a student heading for the mathematics of logic should choose Boolos et al., or take both in either order.

How to Use This Book

The book suits advanced undergraduates and beginning graduate students in philosophy who have finished a first logic course and want to read modal, counterfactual and nonclassical logic in journal articles.

A practical plan for one semester:

• Core: Chapter 1 (skim 1.8 if set theory is familiar), Chapter 2 through the completeness proof, Chapter 4, and Chapters 6, 8 and 9, which hold the book's modal and counterfactual material.
• Choose by interest: Chapter 3 for nonclassical logic, Chapter 5 for descriptions, free logic and second-order logic, Chapter 7 for deontic, epistemic and tense logic, Chapter 10 for two-dimensional semantics.
• Metalogical sections: the soundness and completeness proofs in 2.7 to 2.9 and 6.5 to 6.6 are the difficult parts; a course that wants less metalogic can read the statements and skip the proofs, as the preface allows.

Do the exercises as the text goes, check the starred ones against Appendix A, and attempt the double-starred ones with the hints. The author's website offers a sample, the table of contents and errata for the book.