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)

I ON THE PRIMITIVE TERM OF LOGISTICf IN this article I propose to establish a theorem belonging to logistic concerning some connexions, not widely known, ...

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1973.

This study provides a unified theory of properties, relations, and propositions (PRPs). Two conceptions of PRPs have emerged in the history of philosophy. The author explores both of these traditional conceptions and shows how they can be captured by a single theory.

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.

Bringing together powerful new tools from set theory and the philosophy of language, this book proposes a solution to one of the few unresolved paradoxes from antiquity, the Paradox of the Liar. Treating truth as a property of propositions, not sentences, the authors model two distinct conceptions of propositions: one based on the standard notion used by Bertrand Russell, among others, and the other based on J.L. Austin's work on truth. Comparing these two accounts, the authors show that while the (...) Russellian conception of the relation between sentences, propositions, and truth is crucially flawed in limiting cases, the Austinian perspective has fruitful applications to the analysis of semantic paradox. In the course of their study of a language admitting circular reference and containing its own truth predicate, Barwise and Etchemendy also develop a wide range of model-theoretic techniques--based on a new set-theoretic tool, Peter Aczel's theory of hypersets--that open up new avenues in logical and formal semantics. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

We start with a nearly arbitrary standard classical first order "language" $C\sb{o},$ which is expanded to $C\sb{M,T}$ = "$C\sb{o}+M+T$", where for any variable x, M and T are unary formulas. We start also with a model ${\cal T}\sb{o},$ which together with $C\sb{o}$ represents a fixed non-problematic interpreted first order language. For each $\mu,\tau\subseteq U\sb{o},$ the universe of discourse for ${\cal T}\sb{o},$ the model ${\cal T}\sb{\mu,\tau}$ over $C\sb{M,T}$ is given so that its reduct to $C\sb{o}$ is just ${\cal T}\sb{o},$ and so (...) that for any $C\sb{M,T}$-variable x, the set-extensions of M and T are $\mu$ and $\tau$ respectively. For each $C\sb{M,T}$-expression e, $\vec e$ is its distinctly chosen representative in $U\sb{o},$ and $\lbrack\!\lbrack e\rbrack\!\rbrack$ is a constant $C\sb{o}$-term which names $\vec e.$ ;The goal is to find $\mu\sb*$ and $\tau\sb*,$ such that $\mu\sb*$ and $\tau\sb*$ plausibly and "self-referentially" approximate meaningfulness and truth respectively for any interpreted language represented by ${\cal T}\sb{\mu\sb*,\tau\sb*},$ by satisfying at least the following conditions. ; $\tau\sb*\subseteq\mu\sb*,$ and for each $C\sb{M,T}$-sentence $\sigma$ such that ${\cal T}\sb{\mu\sb*,\tau\sb*}\models M\lbrack\!\lbrack\sigma\rbrack\!\rbrack,$ $$\eqalign{&\ {\cal T}\sb{\mu\sb*,\tau\sb*} \models\sigma\Longleftrightarrow{\cal T}\sb{\mu\sb*,\tau\sb*}\models T\lbrack\!\lbrack\sigma\rbrack\!\rbrack,\ {\rm and}\cr&\ {\cal T}\sb{\mu\sb*,\tau\sb*}\models T\lbrack\!\lbrack\sigma\\leftrightarrow T\lbrack\!\lbrack\sigma\rbrack\!\rbrack\rbrack\!\rbrack.\cr}$$ ; There is a class of expression representatives $\mu\sp{D}\subseteq\mu\sb*$ containing every $C\sb{o}$-expression representative, such that the set $\tau\sp{D}=\mu\sp{D}\ \cap\ \tau\sb*$ itself satisfies the following properties. ;First, for each $\vec\sigma$ such that $\sigma$ is a $C\sb{M,T}$-sentence, if for all $\mu,\tau\subseteq U\sb{o}$ satisfying $\mu\supseteq\mu\sb*$ and $\tau\supseteq\tau\sp{D},$ it is the case that ${\cal T}\sb{\mu,\tau}\models\sigma,$ then $\vec\sigma\in\tau\sp{D}.$ ;Second, whenever $\vec\sigma\in\tau\sp{D},$ then also $\sp{\vec{\enspace}}\ \in\tau\sp{D}.$ ;Third, $\tau\sp{D}$ is "upwardly closed" with respect to the strong-Kleene rules for the truth-value evaluation of Boolean combinations of formulas. For example, for $C\sb{M,T}$-sentences $e\sb1$ and $e\sb2$: $$\eqalign{&\ \vec e\sb1\in\tau\sp{D}\Longleftrightarrow)\sp{\vec{\enspace}}\ \in\tau\sp{D}.\cr&\ {\rm If}\ \vec e\sb1\in\tau\sp{D}\ {\rm or}\ \vec e\sb2\in\tau\sp{D},\ {\rm then\ so\ is}\ \sp{\vec{\enspace}}\ \in\tau\sp{D}.\cr&\ {\rm If}\ \sp{\vec{\enspace}}\ \in\tau\sp{D}\ {\rm and}\ \sp{\vec{\enspace}}\ \in\tau\sp{D},\ {\rm then}\ )\sp{\vec{\enspace}}\ \in\tau\sp{D}.\cr}$$ ; For each $C\sb{M,T}$-expression e not involving T, ${\cal T}\sb{\mu\sb*,\tau\sb*}\models M\lbrack\!\lbrack e\rbrack\!\rbrack$; in particular, for any $C\sb{M,T}$-expression e whatsoever, ${\cal T}\sb{\mu\sb*,\tau\sb*}\models M\lbrack\!\lbrack\\neg M\lbrack\!\lbrack e\rbrack\!\rbrack\rbrack\!\rbrack.$ ; ;In this dissertation, it is shown for the first time that there are in fact $\mu\sb*$ and $\tau\sb*$ which simultaneously satisfy all three conditions , , and , even when the consequent of the Godel fixed point theorem must hold. (shrink)

In this rigorous investigation into the logic of truth Anil Gupta and Nuel Belnap explain how the concept of truth works in both ordinary and pathological..

" There are 31 chapters in 5 parts and approximately 320 exercises marked by difficulty and whether or not they are necessary for further work in the book.

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.

This book traces the development of formal logic from its origins in ancient Greece to the present day. The authors first discuss the work of logicians from Aristotle to Frege, showing how they were influenced by the philosophical or mathematical ideas of their time. They then examine developments in the present century.

Gupta’s Rule of Revision theory of truth builds on insights to be found in Martin and Woodruff and Kripke in order to permanently deepen our understanding of truth, of paradox, and of how we work our language while our language is working us. His concept of a predicate deriving its meaning by way of a Rule of Revision ought to impact significantly on the philosophy of language. Still, fortunately, he has left me something to.