Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

The Handbook of Metaphysics and Ontology reflects the conviction that the history of metaphysics and current work in metaphysics and ontology can each throw valuable light on the other. Thus it is designed to serve both äs a means of making more widely accessible the results of recent scholarship in the history of philosophy, and also äs a unique work of reference in reladon to the metaphysical themes at the centre of much current debate in analyüc philosophy. The work contains (...) more man 450 shorter articies on specific themes and methods in metaphysics and ontology, together with biographical essays summarizing the thinking of the most important philosophers within the traditions dealt with. In addition, it contains about 20 longer survey arücles mapping out both historically and systemadcally the main areas of metaphysics and ontology. AU contributions have been speciaüy commissioned for this project from leading authorities in their respective fields. A special feature of the work is the pardcular attention that is paid to topics and issues of relevance to contemporary philosophical debates. The work includes also a series of arücles on the logical and mathematical tools of metaphysics and on the applications of metaphysical ideas within linguisdcs, psychology, cognitive science, and mathemadcs. (shrink)

« Je considère ici le sens et le rôle de l'irrationnel dans certaines œuvres humaines, dans certaines créations majeures de l'esprit humain, et tout particulièrement dans les œuvres de la science. Dans cette perspective, je distinguerais trois types significatifs d'irrationnel. Le premier serait l'irrationnel comme obstacle, point de départ d'une reconquête de la rationalité. Le second, l'irrationnel comme recours, moyen de renouveler et de prolonger l'acte créateur. Le troisième, l'irrationnel par renoncement, ou si l'on veut par abandon, est au contraire (...) un véritable rejet du rationnel. C'est cette triple distinction qui servira de fil conducteur à nos réflexions sur l'irrationnel dans les œuvres, dans l'esprit d'un rationalisme ouvert et dynamique, en vue de reconnaître et de délimiter le rôle positif de l'irrationnel. » G. G. G. (shrink)

Professor Charles S. Chihara has criticized the views on the subject of inconsistency which Wittgenstein put forward in his recently published 1939 lectures. Chihara notes that these views are not peculiar to the 1939 lectures, and in fact they are to be found in all Wittgenstein's later writings on mathematics . So these ideas about inconsistency appear not to be just a momentary aberration on Wittgenstein's part. One would therefore expect that he had some good reasons for holding them. But (...) Chihara justly complains that the kind of strong argumentation one would hope for is not forthcoming in these lectures. Instead Wittgenstein's usual procedure is to try to defend his views by producing series of rather unconvincing examples. (shrink)

The views on contradiction and consistency that Wittgenstein expressed in his later writings have met with misunderstanding and almost uniform hositility. In this paper, I trace the roots of these views by attempting to show that, in his early writings, Wittgenstein accorded a ?unique status? to tautologies and contradictions, marking them off logically from genuine propositions. This is integral both to his Tractatus project of furnishing a theory of inference, and to the enterprise of explaining the nature of the Satz (...) (statement, proposition). Wittgenstein mantained that contradictions are not false. In his early writings this surprising thesis is a consequence of his view that contradictions are not statements. In his late writings he continues to advocate the thesis, but for quite different reasons. In these late writings, I contend, Wittgenstein succeeds in making the surprising thesis plausible. (shrink)