Gila Sher Truth, Logical Structure, and Compositionality

In this paper I examine a cluster of concepts relevant to the methodology of truth theories: ‘informative definition’, ‘recursive method’, ‘semantic structure’, ‘logical form’, ‘compositionality’, etc. The interrelations between these concepts, I will try to show, are more intricate and multi-dimensional than commonly assumed. 1. TRUTH AND RECURSION In his 1933 paper, “The Concept of Truth in Formalized Languages”, Tarski drew a tentative connection between the definition of truth (for a given language) and the recursive method: If the language investigated only contained a finite number of sentences fixed from the beginning, and if we could enumerate all these sentences, then the problem of the construction of a correct definition of truth would present no difficulties. For this purpose it would suffice to complete the following scheme: x ∈ T r if and only if either x = x1 and p1, or x = x2 and p2, . . . or x = xn and pn, the symbols ‘x1’, ‘x2’, . . . , ‘xn’ being replaced by structural-descriptive names of all the sentences of the language investigated and ‘p1’, ‘p2’, . . . , ‘pn’ by the corresponding translation of these sentences into the metalanguage. But the situation is not like this. Whenever a language contains infinitely many sentences, the definition constructed automatically according to the above scheme would have to consist of infinitely many words, and such sentences cannot be formulated either in the metalanguage or in any other language. Our task is thus greatly complicated. The idea of using the recursive method suggests itself. [188–189] Tarski’s idea involves, however, an indirect use of the recursive method: Among the sentences of a language we find expressions of rather varied kinds from the point of view of logical structure, some quite elementary, others more or less complicated. It would thus be a question of first giving all the operations by which simple sentences are combined into composite ones and then determining the way in which the truth or falsity of composite sentences depends on the truth or falsity of the simpler ones contained in them. Moreover, certain elementary sentences could be selected, from which, with the help of the operations mentioned, all the sentences of the language could be constructed; these selected sentences could be explicitly divided into true and false, by means, for example, of partial definitions of the type described above. In attempting to realize this idea we are however confronted with a serious obstacle. Even a superficial analysis of Defs. 10–12 of Synthese 126: 195–219, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.