This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving (...) lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted. (shrink)

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.

Both in Kripke's Theory of Truth KTT [8] and Russell's Ramified Type Theory RTT [16, 9] we are confronted with some hierarchy. In RTT, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made (...) in Lm where m)n. Kripke finds a fixed point for his hierarchy . We investigate in this paper the similarities of both hierarchies: At level n of KTT the truth or falsehood of all order-n-propositions of RTT can be established. Moreover, there are order-n-propositions that get a truth value at an earlier stage in KTT. Furthermore, we show that wrr is more restrictive than KTT, as some type restrictions are not needed in KTT and more formula., can be expressed in the latter.Looking back at the double hierarchy of Russell, Ramsey [11] and Hilbert and Ackermann [7] considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Church's Simple Type Theory STT [1] We show however that orders in RTT correspond to levels of truth in KTT. Hence, KTT can be regarded as the dual of STT where types have been removed and orders are maintained. As RTT is more restrictive than KTT, we can conclude that it is the combination of types and orders that was the restrictive factor in RTT. (shrink)

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...) audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens / von Dr. Gottlob Frege,...Date de l'edition originale : 1879Ce livre est la reproduction fidele d'une oeuvre publiee avant 1920 et fait partie d'une collection de livres reimprimes a la demande editee par Hachette Livre, dans le cadre d'un partenariat avec la Bibliotheque nationale de France, offrant l'opportunite d'acceder a des ouvrages anciens et souvent rares issus des fonds patrimoniaux de la BnF.Les oeuvres faisant partie de cette collection ont ete numerisees (...) par la BnF et sont presentes sur Gallica, sa bibliotheque numerique.En entreprenant de redonner vie a ces ouvrages au travers d'une collection de livres reimprimes a la demande, nous leur donnons la possibilite de rencontrer un public elargi et participons a la transmission de connaissances et de savoirs parfois difficilement accessibles.Nous avons cherche a concilier la reproduction fidele d'un livre ancien a partir de sa version numerisee avec le souci d'un confort de lecture optimal. Nous esperons que les ouvrages de cette nouvelle collection vous apporteront entiere satisfaction.Pour plus d'informations, rendez-vous sur www.hachettebnf.frhttp://gallica.bnf.fr/ark:/12148/bpt6k65658c. (shrink)