In Contradiction advocates and defends the view that there are true contradictions (dialetheism), a view that flies in the face of orthodoxy in Western philosophy since Aristotle. The book has been at the centre of the controversies surrounding dialetheism ever since its first publication in 1987. This second edition of the book substantially expands upon the original in various ways, and also contains the author's reflections on developments overthe last two decades. Further aspects of dialetheism are discussed in the companion volume, Doubt Truth to be a Liar, also published by Oxford University Press in 2006.

Is it possible for reality as a whole to be part of itself? Can the world appear within itself without thereby undermining the consistency of our thought and knowledge-claims concerning more local matters of fact? This is a question on which Markus Gabriel and Graham Priest disagree. Gabriel argues that the world cannot exist precisely because it is understood to be an absolutely totality. Priest responds by developing a special form of mereology according to which reality is a single all-encompassing whole, everything, which counts itself among its denizens. Their disagreement results in a debate about everything and nothing: Gabriel argues that we experience nothingness once we overcome our urge to contain reality in an all-encompassing thought, whereas Priest develops an account of nothing according to which it is the ground of absolutely everything. A debate about everything and nothing, but also a reflection on the very possibility of metaphysics.

This is the softcover reprint of the very popular hardcover edition. The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a projected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research. A a oegenetica approach is taken, presenting the subject in the context of its historical development. With hindsight the consequential avenues are pursued and the most elegant or accessible expositions given. With open questions and speculations provided throughout the reader should not only come to appreciate the scope and coherence of the overall enterprise but also become prepared to pursue research in several specific areas by studying the relevant sections.

This is the first single-volume edition and translation of Frege's philosophical writings to include all of his seminal papers and substantial selections from all three of his major works. It is intended to provide the essential primary texts for students of logic, metaphysics and philosophy of language.

It contains, in particular, Frege's four essays 'Function and Concept', 'On Sinn and Bedeutung', 'On Concept and Object' and 'Thought', and new translations of key parts of the Begriffschrift , Grundlagen and Grundgesetze . Additional selections have also been made from his Collected Papers , Posthumous Writings and Correspondence . The editor's introduction provides an overview of the development and significance of Frege's philosophy, highlighting some of the main issues of interpretation. Footnotes, appendices and other editorial material have been supplied to facilitate understanding of the works of one of the central figures in modern philosophy.

Kurt Gödel (1906-1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory and stronger systems, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, the foundations of computation theory, unusual cosmological models, and for the strong individuality of his writings on the philosophy of mathematics. The Collected Works is a landmark resource that draws together a lifetime of creative accomplishment. The first two volumes were devoted to Gödel's publications in full (both in the original and translation). This third volume features a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass , documents that enlarge considerably our appreciation of his scientific and philosophical thought and add a great deal to our understanding of his motivations. Continuing
the format of the earlier volumes, the present volume includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, English translations of material originally written in German (some transcribed from Gabelsberger shorthand), and a complete bibliography. A succeeding volume is to contain a comprehensive selection of Gödel's scientific correspondence and a complete inventory of his Nachlass . The books are designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science.

Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. v. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory can be reduced to one in which all truths are analytic. This book dissects these and other arguments through in-depth investigation of the mathematical facts undergirding them. It presents a systematic, mathematically rigorous account of the key notions arising from such debates, including theory, equivalence, translation, reduction, and model. The result is a far-reaching reconceptualization of the role of formal methods in answering philosophical questions.

This book is an introduction to metamathematics and Gödel's theorems. It covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus. It is based on the Open Logic Project, and available for free download at ic.openlogicproject.org.

This book presents an expanded edition of the author's exploration of the nature and limits of thought. Embracing contradiction and challenging traditional logic, the book engages with issues across philosophical borders, from the historical to the modern, from Eastern to Western, and from the continental to the analytic. This edition of the text includes new chapters on European and Indian philosophy, and reflections on responses to the previous edition of the book.

Praise for previous "a splendid tour de force , one which should be read by every philosopher..."-- Philosophical Quarterly
"[H]ighly entertaining and provocative...an engaging and instructive tour through some of the most perplexing features of our own conceptual finitude..."-- TLS

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy
of mathematics.