This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. _An Introduction to Substrucural Logics_ is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both (...) students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading. (shrink)

In Metaphysics III, Chapter 4, Aristotle sets out and defends the Law of Non-Contradiction. The arguments are, however, rather less satisfactory than one might have expected, given the enormous historical influence the text has had. His major argument is a particularly tangled one, and the others are often little more than throw-away remarks. This essay is a commentary on the chapter, but its aim is less to interpret the text , than to see whether there is anything that Aristotle could (...) have meant that would have served his purpose. Whilst other commentators have sometimes attempted this, they have always taken his conclusion to be correct, even if his arguments were not; I do not. The commentary is therefore a confrontation between Aristotle and modern dialetheism. (shrink)

First published in 1952, professor Strawson's highly influential Introductionto Logical Theoryprovides a detailed examination of the relationship between the behaviour of words in common language and the behaviour of symbols in a logical ...

In Contradiction advocates and defends the view that there are true contradictions, a view that flies in the face of orthodoxy in Western philosophy since Aristotle. The book has been at the center 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 over the 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. (shrink)

What is truth. Paul Horwich advocates the controversial theory of minimalism, that is that the nature of truth is entirely captured in the trivial fact that each proposition specifies its own condition for being true, and that truth is therefore an entirely mundane and unpuzzling concept. The first edition of Truth, published in 1980, established itself as the best account of minimalism and as an excellent introduction to the debate for students. For this new edition, Horwich has refined and developed (...) his treatment of the subject in the light of subsequent discussions, while preserving the distinctive format that made the earlier edition so successful. (shrink)

It hath been an old remark, that Geometry is an excellent Logic.

A formal theory of truth, alternative to tarski's 'orthodox' theory, based on truth-value gaps, is presented. the theory is proposed as a fairly plausible model for natural language and as one which allows rigorous definitions to be given for various intuitive concepts, such as those of 'grounded' and 'paradoxical' sentences.

The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. The question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. Three basic modelings emerge: a cancellation model, which leads towards connexivism, an explosion model, appropriate to classical and intuitionistic theories, and a constraint model, which (...) includes relevant theories. These three modelings have been seriously confused in the modern literature: untangling them helps motivate the main themes advanced concerning traditional negation and natural negation. Firstly, the dominant traditional view, except around scholastic times when the explosion view was in ascendency, has been the cancellation view, so that the mainstream negation of much of traditional logic is distinctively nonclassical. Secondly, the primary negation determinable of natural negation is ·relevant negation. In order to picture relevant negation the traditional idea of negation as otherthanness is progressive) refined, to nonexclusive restricted otherthanness. Several pictures result, a reversal picture, a debate model, a record cabinet (or files of the universe) model which help explain relevant negation. Two appendices are attached, one on negation in Hegel and the Marxist tradition, the other on Wittgenstein's treatment of negation and contradiction. (shrink)

Philosophical work on truth covers two streams of inquiry, one concerning the nature (if any) of truth, the other concerning truth-related paradox, especially the Liar. For the most part these streams have proceeded fairly independently of each other. In his "Deflationary Truth and the Liar" (JPL 28:455-488, 1999) Keith Simmons argues that the two streams bear on one another in an important way; specifically, the Liar poses a greater problem for deflationary conceptions of truth than it does for inflationist conceptions. (...) We agree with Simmons on this point; however, we disagree with his main conclusion. In a nutshell, Simmons' main conclusion is that deflationists can solve the Liar only by compromising deflationism. If Simmons is right, then deflationists cannot solve the Liar paradox. In this paper we argue that, pace Simmons, there is an approach to the Liar that is available to deflationists, namely dialetheism. (shrink)

Inference rule deflationism is the thesis that the nature of truth can be explained in terms of the inference rules governing the word "true". This paper argues, first, that, in light of the semantic paradoxes, the inference rule deflationist must reject some of the classical rules of inference. It is argued, secondly, that inference rule deflationism is incompatible with model theoretic approaches to the definition of logical validity. Here the argument focuses on the question whether the number of primitive referring (...) expressions in a natural language is denumerably infinite. Finally, it is argued that these conclusions pertain to T-schema deflationism and Horwich's minimal theory as well. (shrink)

I argue that there is nothing about truth as such that prevents contradictions from being true. I argue this by considering the main standard accounts of truth, and showing that they are quite compatible with the existence of true contradictions. Indeed, in many cases, they are actually friendly to the idea.

First published in 1952, professor’s Strawson’s highly influential _Introduction_ _to Logical Theory_ provides a detailed examination of the relationship between the behaviour of words in common language and the behaviour of symbols in a logical system. He seeks to explain both the exact nature of the discipline known as Formal Logic, and also to reveal something of the intricate logical structure of ordinary unformalised discourse.

Of the various accounts of negation that have been offered by logicians in the history of Western logic, that of negation as cancellation is a very distinctive one, quite different from the explosive accounts of modern "classical" and intuitionist logics, and from the accounts offered in standard relevant and paraconsistent logics. Despite its ancient origin, however, a precise understanding of the notion is still wanting. The first half of this paper offers one. Both conceptually and historically, the account of negation (...) as cancellation is intimately connected with connexivist principles such as ¬( ¬). Despite this, standard connexivist logics incorporate quite different accounts of negation. The second half of the paper shows how the cancellation account of negation of the first part gives rise to a semantics for a simple connexivist logic. (shrink)

My main task here is first to distinguish, and then to map out possibilities. I won’t be concerned to argue for a certain position as much as to argue that various combinations of positions are consistent. In particular, I want to argue that a commitment to minimalism about truth does not bring an automatic commitment to what has been called a minimalist theory of truth-aptitude: the claim that every assertoric sentence which is used in a systematic way will be either (...) true or false. Nor does minimalism about truth bring a commitment to what has been called minimalism about reference: the claim that every word used in a systematic way will refer. (shrink)

I argue that there is nothing about truth as such that prevents contradictions from being true. I argue this by considering the main standard accounts of truth, and showing that they are quite compatible with the existence of true contradictions. Indeed, in many cases, they are actually friendly to the idea.

There has been much debate recently as to whether the notion of truth, as applied to one's home language, is metaphysically neutral, the interesting metaphysical questions arising elsewhere (in relation to such notions as mind-independence or objectivity or existence). ' On one side, the minimalists, as they have come to be known, favour deflationary accounts of truth such as the redundancy or disquotational theories and conclude that the notion of truth is applicable to declarative sentences in general - at least (...) where there are fairly definite standards of appropriateness for the assertion or denial of such sentences - irrespective of the metaphysical commitments of the propositions the sentences express. Opponents have criticized the minimalist account of truth or denied that it follows from this account that truth is metaphysically neutral. My sympathies in this debate lie with the minimalist but I wish to focus on a serious problem for the minimalist - that posed by the Liar paradox. This problem has featured very little in the recent debate but seems to me to yield the strongest objection to minimalism. I develop the objection in the next section, in §2 sketching what I feel to be the minimalist's best response, namely developing the position not in terms of Tarskian classical biconditionals but rather in terms of the naive rules for truth. It emerges, then, that the minimalist will be forced to repudiate classical logic. The last section, §3, deals with objections to the strategy of §2. (shrink)