Truth and Satisfaction by the Empty Sequence

Abstract

Tarski (1933) selected the concept of satisfaction as the fundamental semantic notion serving as the basis for defining other categories of semantics, and in particular of truth. This choice has a deep intuitive motivation. According to the normal use of the word “satisfies”, we have

$$Object\;a\;satisfies\;the\;formula\;Fx\;if\;and\;only\;if\;the\;formula\;Fa\;is\;true.$$

(1)

For example, Warsaw satisfies the formula “x is the capital of Poland” if and only if the sentence “Warsaw is the capital of Poland” is true, or, more simply, if and only if Warsaw is the capital of Poland. However, (1) does not explain why satisfaction is conceptually prior to truth. Two reasons can be given to justify the priority of satisfaction in semantic constructions (see Tarski, 1944; Woleński, 1999). First, satisfaction is a more general concept. Intuitively, satisfaction is applied to open formulas, that is, formulas with free variables, but truth to sentences, that is, formulas in which all variables are bound by quantifiers or in which no variable occurs at all (this means that a formula consists of individual constant predicates and sentential connectives). Let (S) QνA be a general scheme for formulas, where “Q” stands for a quantifier (universal or existential) and the letter A denotes an arbitrary well-formed formula of first-order predicate logic. We do not know in advance whether A contains variables other than ν. If it does, (S) yields an open formula, but if it does not, we have a sentence. Thus, we can define formulas as properly constructed inscriptions that consist of quantifiers, variables, and other building blocks, and sentences as inscriptions without free variables.

This paper is a corrected and enlarged version of (Woleński, 2000).