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 paper argues for two theses: a) there are certain truths which are analytic; b) these are true by convention. Much of the paper deals with quine's arguments against these claims. The paper starts by accepting quine's network theory of belief and arguing that this presupposes a certain concept of rule following. This may be used to define analyticity. The paper then discusses the conventional nature of rule following and argues that this implies the conventional truth of analytic truths. Quine's (...) arguments against truth by convention are then disarmed. Finally, The paper shows that the account of analyticity given gives rise to a modal logic in the order of s4, Disarming another argument against conventionalism. (shrink)

This article is mainly concerned to summarise a fairly well articulated position on the philosophy of science which may be extracted from scattered passages in quine's "word and object." (1) there is no sharp line between philosophy and science, Or between science and mathematics, Or between science and common sense. (2) abstract mathematical entities are theoretical posits just as electrons are. (3) epistemology is a branch of biology. (4) quine's earlier instrumentalism has given way to a scientific realism. (5) quine's (...) philosophy of space-Time is briefly discussed. (6) an extensional language is adequate for science. (shrink)

Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...) logic might one day solve major problems in set theory, arithmetic, linguistics, physics, and more. It circulated in typescript in the late 1970s before appearing as the Appendix to Exploring Meinong's Jungle and Beyond. With engaging, forceful prose, unsparing criticism of entrenched institutions, and many tantalizing proof sketches, Ultralogic? has had a major influence on the development of paraconsistent and relevant logic. This new edition makes this work available for a modern audience, newly typeset and corrected, along with extensive notes, and new commentary essays. (shrink)