Tarski’s Logic

This paper is more a series of notes than a scholarly treatise. It focuses on certain achievements of Aristotle, Boole and Tarski. The notes presented here using concepts introduced or formalized by Tarski contribute toward two main goals: comparing Aristotle’s system with one Boole constructed intending to broaden and to justify Aristotle’s, and giving a modern perspective to both logics. Choice of these three logicians has other advantages. In history of logic, Aristotle is the best representative of the earliest period, Boole the best of the transitional period, and Tarski the best of the most recent period. In philosophy of logic, all three were amazingly successful in having their ideas incorporated into mainstream logical theory. This last fact makes them hard to describe to a modern logician who must be continually reminded that many of the concepts, principles, and methods that are taken to be “natural” or “intuitive” today were all at one time discoveries. Keywords: Counterargument, countermodel, formal epistemology, formal ontology, many‑sorted, metalogic, one‑sorted, proof, range‑indicator, reinterpretation.

The paper starts with an examination and critique of Tarski's wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L ∞,∞ . Also characterized similarly is a natural generalization of Tarski's thesis, due to Sher, in terms of bijections between domains. My main objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M 0 of individuals and {T, F } at their base) is introduced to accomplish the latter. The main result is that an operation is definable from the first-order predicate calculus without equality just in case it is definable from homomorphism invariant monadic operations, where definability in both cases is taken in the sense of the λ-calculus. The paper concludes with a discussion of the significance of the results described for the views of Tarski and Boolos on logicism.

2007

Alfred Tarski (1901–1983) is one of the two greatest logicians of the twentieth century, the other being Kurt Gödel (1906–1978). Each began his career in Europe, respectively in Warsaw and Vienna, and came to America shortly before the Second World War. In contrast to the otherworldly Gödel, Tarski was ambitious and practical. He strove for, and succeeded at, building a school of logic at the University of California, Berkeley, that attracted students and distinguished researchers from all over the world. Tarski was the leader of the " semantic turn " in mathematical logic. This means that he achieved a shift from a view focused on formal systems, axioms, and rules of deduction to a view focusing on the relations between formal systems and their possible interpretations by usual mathematical theories such as real numbers or Cartesian geometry. Hence he gave precise definitions of semantic concepts that had been used informally before. The most important of those concepts a...

2010

Analytic philosophy and modern logic are intimately connected, both historically and systematically. Thinkers such as Frege, Russell, and Wittgenstein were major contributors to the early development of both; and the fruitful use of modern logic in addressing philosophical problems was, and still is, definitive for large parts of the analytic tradition. More specifically, Frege's analysis of the concept of number, Russell's theory of descriptions, and Wittgenstein's notion of tautology have long been seen as paradigmatic pieces of philosophy in this tradition. This close connection remained beyond what is now often called "early analytic philosophy", i.e., the tradition's first phase. In the present chapter I will consider three thinkers who played equally important and formative roles in analytic philosophy's second phase, the period from the 1920s to the 1950s: Rudolf Carnap, Kurt Gödel, and Alfred Tarski.

1987

INTRODUCING TARSKI'S 1983 LSM.docx Editor's introduction revised edition. Logic, Semantics, Metamathematics. Alfred Tarski, pages xv–xxvii. I wish to express here my most genuine and cordial gratefulness to Professor John Corcoran for his work related to the publication of the present volume. The new edition of Logic, Semantics, Metamathematics is almost exclusively due to his initiative. In serving as the editor, he fulfilled his task with great fervor and enthusiasm. We collaborated in searching out and trying to remove various defects and weak spots of earlier texts. The new indices at the end of this volume, prepared by Corcoran, are much superior to those in the first edition and hopefully will facilitate the study and the use of the work. What is more valuable, he has provided the volume with his own introduction.--ALFRED TARSKI, Berkeley, California, January 1982 Corcoran’s 1983 edition of Tarski’s LOGIC, SEMANTICS, METAMATHEMATICS is available free. http://library1.org/_ads/DE3C6E8E30D380DC39D1057DB4BD8EE1 CORCORAN ON TARSKI https://www.academia.edu/s/179246c7cd/corcoran-on-tarski-december-2016-pdf?source=link Corcoran’s publications that interpret, criticize, explain, or build on Alfred Tarski’s theories and results.

Annals of Pure and Applied Logic, 2004

The article recalls shortly the early story of cooperation between the already existing Lvov philosophical school, headed by Twardowski, and the just then establishing Warsaw mathematical school, headed by Sierpià nski. After that recollection the article proceeds to contributions made by men in uenced by the two schools. Most prominent of them was Alfred Tarski whose work in those times, concentrated mainly upon the theory of deduction, axiom of choice, cardinal arithmetic, and measure problem, has been described in some detail.

Published, 1993