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. (shrink)

This book is a defense of modal realism; the thesis that our world is but one of a plurality of worlds, and that the individuals that inhabit our world are only a few out of all the inhabitants of all the worlds. Lewis argues that the philosophical utility of modal realism is a good reason for believing that it is true.

The thesis defended in this article is that by uttering or publishing a great many declarative sentences in assertoric mode, one does not actually assert that their conjunction is true – one rather asserts that the vast majority of these sentences are true. Accordingly, the belief that is expressed thereby is the belief that the vast majority of these sentences are true. In the article, we make this proposal precise, we explain the context-dependency of belief that corresponds to it, we (...) point out why our everyday oral practice of single assertions is not affected by it, and we argue that the proposal leads to a way out of the Paradox of the Preface. (shrink)

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system---ST---that conservatively extends classical (...) logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered. (shrink)

In the general theory of logic built up by Whitehead and Russell to furnish a basis for all mathematics there is a certain subtheory which is unique in its simplicity and precision; and though all other portions of the work have their roots in this subtheory, it itself is completely independent of them. Whereas the complete theory requires for the enunciation of its propositions real and apparent variables, which represent both individuals and propositional functions of different kinds, and as a (...) result necessitates- the introduction of the cumbersome theory of types, this subtheory uses only real variables, and these real variables represent but one kind of entity—which the authors have chosen to call elementary propositions. The most general statements are formed by merely combining these variables by means of the two primitive propositional functions of propositions Negation and Disjunction; and the entire theory is concerned with the process of asserting those combinations which it regards as true propositions, employing for this purpose a few general rules which tell how to assert new combinations from old, and a certain number of primitive assertions from which to begin. (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

The recent development and exploration of mixed metainferential logics is a breakthrough in our understanding of nontransitive and nonreflexive logics. Moreover, this exploration poses a new challenge to theorists like me, who have appealed to similarities to classical logic in defending the logic ST, since some mixed metainferential logics seem to bear even more similarities to classical logic than ST does. There is a whole ST-based hierarchy, of which ST itself is only the first step, that seems to become more (...) and more classical at each level. I think this seeming is misleading: for certain purposes, anyhow, metainferential hierarchies give us no reason to move on from ST. ST is indeed only the first step on a grand metainferential adventure; but one step is enough. This paper aims to explain and defend that claim. Along the way, I take the opportunity also to develop some formal tools and results for thinking about metainferential logics more generally. (shrink)

am going to discuss some issues inspired by a well-known paper ofKeith Donnellan, "Reference and Definite Descriptions,”2 but the interest—to me—of the contrast mentioned in my title goes beyond Donnellan's paper: I think it is of considerable constructive as well as critical importance to the philosophy oflanguage. These applications, however, and even everything I might want to say relative to Donnellan’s paper, cannot be discussed in full here because of problems of length. Moreover, although I have a considerable interest in (...) the substantive issues raised by Donnellan’s paper, and by related literature, my own conclusions will be methodological, not substantive. I can put the matter this way: Donnellan’s paper claims to give decisive objections both to Russell’s theory of definite descriptions (taken as a theory about English) and to Strawson’s. My concem is not primarily with the question; is Donnellan right, or is Russell (or Strawson)? Rather, it is with the question: do the considerations in Donneilarfs paper refute Russell’s theory (or Strawson’s)? For definiteness, I will concentrate on Donnellan versus Russell, leaving Strawson aside. And about this issue I will draw a definite conclusion, one which I think will illuminate a few methodological maxims about language. Namely, I will conclude that the considerations in Donnellan’s paper, by themselves, do not refute Russell’s theory. Any conclusions about Russell’s views per se, or Donnellan’s, must be tentative, IfI were to be asked for a tentative stab about Russell, I would say that although his theory does a far better job of handling ordinary discourse than many have thought, and although many popular arguments against it are inconclusive, probably it ultimately fails. The considerations I have in mind have to do with the existence of “improper” definite descriptions, such as “the table," where uniquely specifying conditions are not contained in the description itself.. (shrink)

By means of an example, shows the possibility of beliefs that are separately rational whilst together inconsistent.

This paper consider Prior's connective Tonk from a particular bilateralist perspective. I show that there is a natural perspective from which we can see Tonk and its ilk as perfectly well-defined pieces of vocabulary; there is no need for restrictions to bar things like Tonk.

Die Grundlagen der Arithmetik. Eine Ionisch mathematische UoterciicboDn über den Begriff der Zahl Dr. 0. Frege, ao Profeuor an der Univer»ität Jena. -. ...

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, (...) which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

am going to discuss some issues inspired by a well-known paper ofKeith Donnellan, "Reference and Definite Descriptions,”2 but the interest—to me—of the contrast mentioned in my title goes beyond Donnellan's paper: I think it is of considerable constructive as well as critical importance to the philosophy oflanguage. These applications, however, and even everything I might want to say relative to Donnellan’s paper, cannot be discussed in full here because of problems of length. Moreover, although I have a considerable interest in (...) the substantive issues raised by Donnellan’s paper, and by related literature, my own conclusions will be methodological, not substantive. I can put the matter this way: Donnellan’s paper claims to give decisive objections both to Russell’s theory of definite descriptions (taken as a theory about English) and to Strawson’s. My concem is not primarily with the question; is Donnellan right, or is Russell (or Strawson)? Rather, it is with the question: do the considerations in Donneilarfs paper refute Russell’s theory (or Strawson’s)? For definiteness, I will concentrate on Donnellan versus Russell, leaving Strawson aside. And about this issue I will draw a definite conclusion, one which I think will illuminate a few methodological maxims about language. Namely, I will conclude that the considerations in Donnellan’s paper, by themselves, do not refute Russell’s theory. Any conclusions about Russell’s views per se, or Donnellan’s, must be tentative, IfI were to be asked for a tentative stab about Russell, I would say that although his theory does a far better job of handling ordinary discourse than many have thought, and although many popular arguments against it are inconclusive, probably it ultimately fails. The considerations I have in mind have to do with the existence of “improper” definite descriptions, such as “the table," where uniquely specifying conditions are not contained in the description itself.. (shrink)

This comprehensive monograph is a cornerstone in the area of mathematical logic and related fields. Focusing on Gentzen-type proof theory, the book presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

The topic of vagueness re-emerged in the twentieth century from relative obscurity. It deals with the phenomenon in natural language that manifests itself in apparent semantic indeterminacy - the indeterminacy, for example, that arises when asked to draw the line between the tall and non-tall, or the drunk and the sober. Vagueness, Logic and Ontology explores various responses to the philosophical problems generated by vagueness and its associated paradox - the sorites paradox. Hyde argues that the theoretical space in which (...) vagueness is sometimes ontologically grounded and modelled by a truth-functional logic affords a coherent response to the problems posed by vagueness. Showing how the concept of vagueness can be applied to the world, Hyde's ontological account proposes a substantial revision of orthodox semantics, metaphysics and logic. (shrink)

Vagueness, Logic and Ontology explores various responses to the philosophical problems generated by vagueness and its associated paradox - the sorites paradox. Hyde argues that the theoretical space in which vagueness is sometimes ontologically grounded and modelled by a truth-functional logic affords a coherent response to the problems posed by vagueness. Showing how the concept of vagueness can be applied to the world, Hyde's ontological account proposes a substantial revision of orthodox semantics, metaphysics and logic. This book will be of (...) particular interest to readers in philosophy, linguistics, cognitive science and geographic information systems. (shrink)

In “The Runabout Inference Ticket” AN Prior (1960) examines the idea that logical connectives can be given a meaning solely in virtue of the stipulation of a set of rules governing them, and thus that logical truth/consequence.

It is part of the current wisdom that the Liar and similar semantic paradoxes can be taken care of by the use of certain non-classical multivalued logics. In this paper I want to suggest that bivalent logic can do just as well. This is accomplished by using a non-deterministic matrix to define the negation connective. I show that the systems obtained in this way support a transparent truth predicate. The paper also contains some remarks on the conceptual interest of such (...) systems. (shrink)