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The Law of Non-Contradiction has been high orthodoxy in Western philosophy since Aristotle. The so-called Law has been the subject of radical challenge in recent years by dialetheism, the view that some contradictions are indeed true. Many philosophers have taken the Law to be central to many of our most important philosophical concepts. This book mounts the case against this view. Starting with an analysis of Aristotle on the Law, it discusses the nature of truth, rationality, negation, and logic itself, and argues that the Law is inessential to all of these things. The book develops Priest’s earlier ideas in In Contradiction.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.

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A counterpossible conditional is a counterfactual with an impossible antecedent. Common sense delivers the view that some such conditionals are true, and some are false. In recent publications, Timothy Williamson has defended the view that all are true. In this paper we defend the common sense view against Williamson's objections.

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A logic is paraconsistent if it does not validate the principle that from a pair of contradictory sentences, A and ∼A, everything follows, as most orthodox logics do. If a theory has a paraconsistent underlying logic, it may be inconsistent without being trivial (that is, entailing everything). Sustained work in formal paraconsistent logics started in the early 1960s. A major motivating thought was that there are important naturally occurring inconsistent but non-trivial theories. Some logicians have gone further and claimed that some of these theories may be true. By the mid-1970s, details of the semantics and proof-theories of many paraconsistent logics were well understood. More recent research has focused on the applications of these logics and on their philosophical underpinnings and implications.