Syntax, Semantics and Tarski’s Truth Definition

Until Tarski’s semantic truth definition, the concept of truth was used informally in metalogic (metamathematics) or even proposed to be eliminated in favour of syntactic concepts, as in Rudolf Carnap’s early programme of philosophy via logical syntax. Tarski demonstrated that the concept of truth can be defined using precise mathematical devices. If L is a language for which the truth definition is given, it must be done in the metalanguage ML. According to this construction, semantics for L must be performed in ML. The most important example concerns the arithmetic of natural numbers. According to Tarski’s theorem of undefinability, the set of truths of this theory cannot be defined in it – such a definition can be formulated in the metatheory. This fact illuminates the relation between syntax and semantics. If Th is a rich theory and presented as a syntactic theory (a calculus), its semantics is not reducible to its syntax. According to Tarski’s view, related to his work in the simple theory of types, semantics for L can be always constructed in the morphology of ML, provided that L is of the finite order. Two problems arise: what does the word “morphology” mean and how to formulate these ideas, when the framework is based on the distinction between first‑order logic and higher‑order logic. As far as the issue concerns morphology, it is possible to consider it as an extended syntax, i.e. vocabulary which does not refer to semantic concepts or defines such notions by not‑semantic, e.g. set‑theoretical, categories. If the hierarchy of logical types is replaced by the distinction of logics of various orders, in particular between first‑order and higher‑order (it is sufficient to use second‑order), it is possible to show that semantics of first‑order rich theories cannot be defined inside them.