A CONCEPTION OF TARSKIAN LOGIC

Abstract

A CONCEPTION OF TARSKIAN LOGIC* BY GILA SHER Which logic is the right logic? In a paper so titled Leslie Tharp’ poses the question: What properties should a logical system have? In particular: ls standard lst-order logic the right logic? The question asked in this paper is somewhat less general: Which logic is Tarski’s logic? More precisely: Are the basic principles of Tarskian logic exhausted by the standard lst- ordcr s_vstetn or does it take a new, extended logic, to fully realize them? (By 'Tarskian logic‘ I here understand the modern semantic conception of logic as it evolved out of Tarski’s theory.) To answer questions on the adequacy of a system of logic, Tharp says, it is essential that we acquire first an idea of “the role logic is expected to play.” I think Tharp’s point is important, and with this guideline in mind I will turn to Tarski's early work on the foundations of semantics.’ I. The Task of Logic. the Origins of Semantics ln q1 he Concept of Truth in Formalized Languagesq, “On the Concept of Logical Consequence” and other writings‘ Tarski presents logical semantics as providing (i) a definition of the general concept of truth for formalized languages, and (ii) definitions of the logical concepts ‘logical truth‘, ‘logical consequence‘, ‘consistency’, etc., for such languages. The main purpose of (i) is to secure metalogic against semantic paradoxes. Tarski worried lest the uncritical use of semantic concepts prior to his work concealed an inconsistency: a hidden fallacy would undermine the entire venture. He therefore sought precise, materially as well as formally correct. definitions for ‘truth’ and related notions which would serve as a hedge against paradox. This aspect of Tarski’s work is well known. In qModel Theory Before 1945'‘, Robert Vaugltt5 puts Tarski‘s enterprise in a slightly different light: Work in model theory ,’tl('I:/l(' I’III/(mt/)Iti(‘tII QmIrI(’rI_t' 70 ( I989) 34 I -363 0279-0750/89/0400-034 I -$02.80 Lopvright kt) I989 by University of Southern California