An Introduction to Formal Logic, 2nd Edition

Peter Smith's textbook is a rigorous introduction to classical first-order quantification theory, aimed primarily at first-year philosophy undergraduates. It starts from scratch — what an argument is, the distinction between deductive and inductive reasoning, validity vs. soundness — and builds methodically toward formal proof systems for propositional logic (PL) and quantifier logic (QL, then QL with identity). The second edition replaces the first edition's truth-tree chapters with a Fitch-style natural deduction system, though the tree chapters remain available as online supplements.

The book's architecture is careful and cumulative. Early chapters drill the reader on what validity means (necessarily truth-preserving inference), how to spot it, and how to test with the counterexample method, before formalizing anything. A key early principle is the invalidity principle: an inference step with true premises and a false conclusion must be invalid. The formal half opens with propositional logic — syntax, truth-table semantics, tautological entailment, expressive adequacy, and the truth-functional conditional (with extended discussion of how "if" in English maps — and doesn't — to material implication). The quantifier half introduces names, predicates, quantifier-variable notation, QL syntax and semantics, Q-valuations, and Q-validity. Later chapters add identity and definite descriptions, and a final appendix sketches soundness and completeness results.

Smith writes in a conversational but mathematically precise style. He calls out common student mistakes (confusing use with mention, confusing → with English "if"), explains why each formal choice was made (why these three connectives? why natural deduction?), and provides exercises with detailed worked answers online. The book is explicitly opinionated: Smith favours a classical, two-valued conception of logic and takes positions on contested points (e.g., the material conditional is adequate for mathematical reasoning if not for everyday conditionals). Throughout, he's transparent that the book is one path among many into logic — not the final word, but a careful first one.

The freely-available PDF (420 pages) was self-published by Smith at Logic Matters after Cambridge University Press's second edition went out of print. A third edition covering both trees and natural deduction was planned.