Life and Logic Reviewed by Hourya Benis Sinaceur

This is the introduction to the book of the same name. Co-authored with Marianna Antonutti

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.

1986. Alfred Tarski's "What are Logical Notions?" (Edited and introduced by John Corcoran), History and Philosophy of Logic 7, 143–154. MR88b:03010 It is widely assumed that this article has something to say about what is sometimes called the problem of distinguishing logical constants from non-logical constants. But what this problem is taken to be varies from author to author and—within the writings of one author—from publication to publication according to which object language the “constants” occur in, how that object language is interpreted, and other factors, parameters, constraints, and the like. It is time to reread this classic paper, which is based on a lecture by the author to the Buffalo Logic Colloquium attended by the editor, who took notes and on the same day wrote an article on the lecture for his University’s periodical publication. It is time to reexamine the various meanings attached to ‘the problem of distinguishing logical constants from non-logical constants’ and similar expressions.

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.

Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, and collaborator—on his eightieth birthday. This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege–Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.

1998

In (1936), Tarski presented for the first time, in Gemnan, his new semantic definition of logical consequence and logical truth. He starts to motivate his definition by cdtizicing traditional syntactic definitions. He gives two reasons why syntactic definitions are not satisfactory. First, the traditional calculus-based syntactic definitions (defined by a set of axioms and rules, and a recursive notion of proof) are too weak to capture the ordinary notion of logical consequence. Tat'ski gives the example of w-incomplete themies of Pea no arithmetics. Here it may be the case that P(n) is derivable (within the theory) for every natural number n, without having thatl.lxP(x) be dedvable, although the latter sentence intuitively follow (Tarski 1936, p. 41Of). Tarski continues that this is just one aspect of the incompleteness of first order arithmetics, which shows that calculus-based recursive definitions are 77 J. Wale,1ski and E. Kohler (eds.), Alfred Tarski and the Vienna Cire/e. 77~94.