In this textbook, John MacFarlane presents a comprehensive and accessible overview of philosophical logic suitable for undergraduate students. The book’s impressive scope is one of its most notable strengths, as it covers a wide range of topics that reflect the inherent ambiguity of the term “philosophical logic.” The book’s overall organization is exceptionally well-planned, with each subsequent subject smoothly transitioning from the previous one (especially Chapters 5 and 6). Moreover, the recommended readings provided throughout the chapters serve as excellent references for anyone wishing to delve deeper on the topics covered in this book.

Chapter 1 starts with a self-contained introduction to first-order logic that is later extended to include identity. Function symbols are not considered in the definition of terms. The adoption of a natural deduction system in Fitch-style makes the proofs easy to understand. The chapter ends with a discussion of use and mention where Quine’s corner quotes, which are used extensively throughout the book, are introduced.

Chapter 2 begins by delving into the theory of generalized quantifiers, which is motivated with a discussion of the limitations of unary quantifiers. The reader is introduced to definite descriptions through a series of examples. While the inclusion of “ $2 + 6$ ” is legitimate if one thinks of it as “the sum of $2$ and $6$ ” as MacFarlane suggests, it is worth noting that this term is most naturally expressed using a function symbol. The treatment of second-order quantifiers is well-balanced, including both technical and philosophical considerations. The examination of the expressive limitations of first-order logic is particularly noteworthy for its clarity and thoroughness. Although it is understandable that a book with such a wide scope may not be able to discuss all topics, it is unfortunate that the semantics of polyadic second-order logic and its additional proof-theoretic considerations, such as the failure of completeness for second-order logic, are not included. Incompleteness is only briefly mentioned much later in Chapter 5 for higher-order logics. Next, MacFarlane analyses the substitutional interpretation of quantifiers and explores its various applications. A reference to free logic in the exposition of the problem of nonexistent objects would have been appreciated.

Chapter 3 covers modal logic in both its propositional and predicate forms. Again, MacFarlane is careful to connect technical aspects to broader philosophical debates. The technical discussion of modal propositional logic is particularly satisfying, and the exposition of possible world semantics for systems K to S5 is complemented with a consideration of Fitch-style natural deduction, though only for T, S4, and S5. MacFarlane then briefly notes that the technical development of a semantics for modal predicate logic is complicated by a variety of difficult design choices. Rather than pushing these aside, he decides to focus on a comprehensive examination of Quine’s philosophical objections against quantified modal logic as well as Kripke’s defense of de re modality.

Chapter 4 is devoted to the topic of conditionals and addresses the initial concerns commonly encountered by students when first introduced to truth tables. It deals with questions of truth-functionality, but it is only later in Chapter 7 that the reader will find a full discussion of the paradoxes of material implication, given that issues of relevance and explosion are deliberately left out at this point. In my view the chapter’s highlight lies in the investigation of McGee’s attacks on the validity of modus ponens and the possible solutions presented by Edgington’s theory and Stalnaker’s semantics.

Chapters 5 and 6 provide a double analysis of the concept of logical consequence from the model- and proof-theoretic perspectives respectively. There appears to be an interchangeable usage of the terms “logical consequence” and “inference” at the beginning of Section 5.1.1. In particular, the criterion of necessary truth preservation is discussed for logical consequences but actually proposed for inferences. MacFarlane’s usage of the terms as interchangeable seems to follow Tarski, whose model-theoretic account of logical consequence occupies a significant portion of Chapter 5. (For more on the difference between logical consequence and inference, see Sundholm, Göran. “Inference versus consequence” revisited: inference, consequence, conditional, implication. Synthese 187 (2012) : 943–956). To motivate a short introduction to intuitionism in Chapter 6, MacFarlane suggests that Prawitz’s account of consequence yields intuitionistic logic. This claim is misleading. Prawitz’s treatment of atomic formulas (a complication wisely omitted in the book, see Chapter 6, footnote 2) leads to incompleteness with respect to intuitionistic logic, as evidenced by the validation of Harrop’s rule $(A \to (B \lor C)) \to ((A \to B)\lor (A \to C))$ . But a minor modification to one of Prawitz’s definitions of proof-theoretic notion of validity results in completeness for intuitionistic propositional logic, as proved in Stafford, Will and Nascimento, Victor. Following all the rules: Intuitionistic completeness for generalised proof-theoretic validity. Analysis (forthcoming). MacFarlane’s exposition of propositional intuitionistic logic via Kripke models is very accessible, and the chapter even covers the double-negation translation and touches on logical pluralism.

In Chapter 7, relevance logic and the exclusion of ex falso rule are thoroughly examined. The extensive discussion of Lewis’s argument is particularly informative. Despite the introduction of Priest’s dialetheism, it is regrettable that paraconsistent logic, a philosophically rich development in nonclassical logic, is only briefly discussed in a footnote to Priest, given how the chapter is opened by an examination of the rejection of ex falso. This does little to dispel the common misconception that paraconsistent logic is synonymous with dialetheism.

Chapter 8 discusses vagueness and possible solutions to the sorites paradox using three-valued logics, fuzzy logic, and supervaluations, concluding with a philosophical discussion of the nature of vagueness following Evans.

To conclude, MacFarlane has written a superb introduction to philosophical logic. The text is well-written, informative, and easily comprehensible, making it an outstanding resource for those looking to gain a deeper understanding of logic and its philosophical implications.