This paper examines from a historical perspective Tarski's 1936 essay, "On the concept of logical consequence." I focus on two main aims. The primary aim is to show how Tarski's definition of logical consequence satisfies two desiderata he himself sets forth for it: (1) it must declare logically correct certain formalizations of the -rule and (2) it must allow for variation of the individual domain in the test for logical consequence. My arguments provide a refutation of some interpreters of Tarski, (...) and notably John Etchemendy, who have claimed that his definition does not satisfy those desiderata. A secondary aim of the paper is to offer some basic elements for an understanding of Tarski's definition in the historical logico-philosophical context in which it was proposed. Such historical understanding provides useful insights on Tarski's informal ideas on logical consequence and their internal cohesion. (shrink)

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

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, ...

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

Of course we all know now that mathematics has proved that logic doesn't really make sense, but Etchemendy (philosophy, Stanford Univ.) goes further and challenges the received view of the conceptual underpinnings of modern logic by arguing that Tarski's model-theoretic analysis of logical consequences is wrong. He may have found the soft underbelly of the dead horse. Annotation copyrighted by Book News, Inc., Portland, OR.

Published with the aid of a grant from the National Endowment for the Humanities. Contains the only complete English-language text of The Concept of Truth in Formalized Languages. Tarski made extensive corrections and revisions of the original translations for this edition, along with new historical remarks. It includes a new preface and a new analytical index for use by philosophers and linguists as well as by historians of mathematics and philosophy.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

In the general theory of logic built up by Whitehead and Russell to furnish a basis for all mathematics there is a certain subtheory which is unique in its simplicity and precision; and though all other portions of the work have their roots in this subtheory, it itself is completely independent of them. Whereas the complete theory requires for the enunciation of its propositions real and apparent variables, which represent both individuals and propositional functions of different kinds, and as a (...) result necessitates- the introduction of the cumbersome theory of types, this subtheory uses only real variables, and these real variables represent but one kind of entity—which the authors have chosen to call elementary propositions. The most general statements are formed by merely combining these variables by means of the two primitive propositional functions of propositions Negation and Disjunction; and the entire theory is concerned with the process of asserting those combinations which it regards as true propositions, employing for this purpose a few general rules which tell how to assert new combinations from old, and a certain number of primitive assertions from which to begin. (shrink)

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which (...) follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentenceσis false. Moreover, … we are concerned here with the concept of logical, i.e.,formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since. (shrink)

The purpose of this paper is to examine some passages of Tarski‘s paper ’On the concept of logical consequence’ and to show that some recent readings of those passages are wrong. John Etchemendy has claimed that in those passages Tarski gave an argument purporting to show that the notion of logical consequence defined by him (as opposed to some pretheoretic notion of logical consequence) possesses certain modal properties. Etchemendy further claims that the argument he attributes to Tarski is fallacious. Some (...) of Etchemendy’s critics have granted him that Tarski did give an argument purporting to show that the defined notion possesses certain modal properties ; but they have claimed that Tarski’s argument was not a fallacious one. I will show that both Etchemendy and his critics are wrong; in the relevant passages, Tarski did not offer (nor did he intend to offer) an argument that the defined notion of logical consequence possesses any modal properties. (shrink)

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

Der Laie spricht manchmal die Ansicht aus, die Mathematik ware heutzutage schon eine tote Wissenschaft: nachdem sie einen ungemein hohen Grad der Entwicklung erreicht hat, sei sie in ihrer steinernen Vollkommenheit erstarrt. Dies ist ein vollig irriges Bild der Situation: nur wenige Wissenschaftsgebiete befinden sich heute in der Phase einer solch intensiven Entwicklung wie die Mathematik. Diese Entwicklung ist dabei auBerordentlich vie1seitig: die Mathematik erweitert ihre Domane nach allen moglichen Richtungen, sie wachst in die Rohe, in die Weite und in (...) die Tiefe. Sie wachst in die Rohe, da auf dem Boden ihrer alten Theorien, denen eine jahrhundert-, ja sogar jahrtausend- lange Entwicklung zugrunde liegt, immer wieder neue Prob1eme auftauchen, immer scharfere und vollkommenere Resultate er- zielt werden; in die Weite, da ihre Methoden andere Wissen- schaftszweige durchdringen, ihr Untersuchungsbereich immer umfangreichere Gebiete von Erscheinungen umfaBt und immer neue Theorien in den groBen Kreis mathematischer Disziplinen einbezogen werden; und schlieBlich in die Tiefe, da ihre Grund- lagen immer mehr gefestigt, die bei ihrem Aufbau angewandten Methoden immer vollkommener werden und ihre Prinzipien an Dauerhaftigkeit gewinnen. In dem vorliegenden Buch wiinschte ich dem Leser, der fiir die gegenwartige Mathematik Interesse aufweist, aber ihr fern- steht, mindestens einen ganz allgemeinen Begriff von dieser dritten Entwicklungslinie der Mathematik, d. i. von ihrer Ent- wicklung in die Tiefe, zu geben. (shrink)

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

The paper demonstrates the necessity of introducing innovative approaches in addressing social problems of development. It proves the significance of recruiting investment in the social sphere. The author looks into the usage of social business for handling economic development problems of countries. There are factors promoting social entrepreneurship identified. Spatial social inequality in Ukraine is analyzed, the role of social entrepreneurship in its overcoming demonstrated.