INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject.

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.

Keine ausführliche Beschreibung für "Die dreiwertige Logik der Sprache" verfügbar.

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...) fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

We consider the logical form of a natural language sentence to be a formal object which determines both the logical properties of the sentence and, more generally, the ways the sentence is logically related to other sentences. Thus if some NL sentence logically entails another, this fact must follow, given the logical forms of the two sentences. The power of a theory of logical forms of natural language then lies first in what logical properties and relations it can define, and (...) second, in which NL sentences it can show to be logically related to which others. (shrink)

The set-theoretic and individual-theoretic analyses of collections make the same mistake: their ontologies are far removed from language. Our basic notion, the collector, has a clear linguistic counterpart: the definite article. Our theory furnishes a theory of definite descriptions as a special case. We use 3-valued valued logic in order to come to grips with the existential presuppositions of the collections.

The Hangman Paradox has a simple solution. The amazing refutation of the judge's decree rests on the axiom of knowledge-conservation. This axiom is false under unfavourable conditions. You can have a perfect piece of knowledge in the ordinary sense, i.e. a true justified conviction, and yet be unable to conserve it. More interesting than its solution is the element of self-reference, connecting the Hangman via Moore's Paradox and Buridan's Epistemic Paradox with the Liar. This one, I think, has also a (...) natural solution, but less simple. The basic idea is given here, but the technical treatment goes far beyond this paper. It requires a strong, but conservative, extension of classical and three-valued logic to a six-valued logic with infinitely many levels of reflection. (shrink)