No abstract

Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, 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 Principiainfluenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Bulletin of Symbolic Logic.Tarski's 1936 paper, "On the concept of logical consequence", is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: (1) terminological and conceptual preliminaries, (2) Tarski's definition of the concept of logical consequence, (3) Tarski's discussion of omega-incomplete theories, and (4) concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question of the criterion for distinguishing between logical and non-logical terms. ?1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P (the premise-set) and a single proposition c (the conclusion). The expression 'c is a [logical] consequence of P' is used with the same meaning as the expression 'c is [logically] implied by P'. The expressions 'is a logical consequence of' and the converse 'implies' are relational. Often, I shall be talking in the same sense of validity of an argument.Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions1.Tarski uses the word 'sentence' to refer to the objects of which truth and falsehood are [coherently] predicable; i.e, to the bearers of truth and falsehood. In a strict sense this use does not seem quite appropriate. In the logical tradition it has been customary to distinguish sentences, which are strings of symbols formed according to rules, from propositions, which are Received October 29, 1996. This paper is supported by research projects PB95-0863 DGES of the Spanish government and Xuga 20506B96 of the Galician government.'This concept of a purely ontic nature contrasts with others of an epistemic nature, such as "argumentation", "deduction", "proof", etc. See Corcoran [8].

Each science has its own domain of investigation, but one and the sam e science can be formalized in diOE erent languages with diOE erent universes of discourse. The concept of the dom ain of a science and the concept of the universe of discours e of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppos itions and im plications of choices of domain and universe , this article discusses the treatm ent of omega argum ents in three very diOEerent form alizations of arithmetic. In

No abstract