The Taming of the True poses a broad challenge to realist views of meaning and truth that have been prominent in recent philosophy. Neil Tennant argues compellingly that every truth is knowable, and that an effective logical system can be based on this principle. He lays the foundations for global semantic anti-realism and extends its consequences from the philosophy of mathematics and logic to the theory of meaning, metaphysics, and epistemology.

We investigate the semantics of the logical systems obtained by introducing the modalities and into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in (...) a previous paper. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of category (...) theory' specifically in 'its alleged role as providing a foundation' (p. 130). I will also focus on the foundational question. Page numbers in parentheses are page references to Hellman [2003]. (shrink)

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...) an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic. (shrink)

On several occasions I have pointed out that the lesson taught us by recent developments in physics regarding the necessity of a constant extension of the frame of concepts appropriate for the classification of new experiences leads us to a general epistemological attitude which might help us to avoid apparent conceptual difficulties in other fields of science as well. Since, however, the opinion has been expressed from various sides that this attitude would appear to involve a mysticism incompatible with the (...) true spirit of science, I am very glad to use the present opportunity of addressing this assembly of scientists working in quite different fields but united in their striving to find a common ground for our knowledge, to come back to this question, and above all to try to clear up the misunderstandings which have arisen. (shrink)

In the presence of chance moves in a semantical game, the existence of pure optimal strategies does not guarantee the existence of winning ones. This fact provides a basis for constructing supervaluational semantical games with a chance move. Additional possibilities of using chance moves in game-theoretical semantics are also discussed.

The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds) I will show how to lift the dialogical (...) conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail. (shrink)

In this paper we discuss the notion of “possible worlds” contrasting a philosophical idea due to Malebranche with a mathematical concept of modern logic due to Kripke from which many pseudo-philosophical discussions have arisen.

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...) modern structural approach in mathematics. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

An affirmative answer is given to the question quoted in the title.

Attempts to indentify the fundamental concepts of language, argues that the study of language reveals hidden facts about the mind, and looks at the impact of propaganda.

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

In this paper non-normal worlds semantics is presented as a basic, general, and unifying approach to epistemic logic. The semantical framework of non-normal worlds is compared to the model theories of several logics for knowledge and belief that were recently developed in Artificial Intelligence (AI). It is shown that every model for implicit and explicit belief (Levesque), for awareness, general awareness, and local reasoning (Fagin and Halpern), and for awareness and principles (van der Hoek and Meyer) induces a non-normal worlds (...) model validating precisely the same formulas (of the language in question). (shrink)

Bertrand Russell is concerned in this book with the foundations of knowledge. He approaches his subject through a discussion of language, the relationships of truth to experience and an investigation into how knowledge of the structure of language helps our understanding of the structure of the world. This edition includes a new introduction by Thomas Baldwin, Clare College, Cambridge.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for ...

This is the first elementary book to employ the concept of infinitesimals.