Truth, semantical
Truth, semantical
Closely connected with the name relation (q.v.) is the property of a propositional formula (sentence) that it expresses a true proposition (or if it has free variables, that it expresses a true proposition for all values of these variables). As in the case of the name relation, a notation for the concept of truth in this sense often cannot be added, with its natural properties, to an (interpreted) logistic system without producing contradiction. A particular system may, however, be made the beginning of a hierarchy of systems each containing the truth concept appropriate to the preceding one.
The notion of truth should be kept distinct from that of a theorem, the true formulas being in general only some among the theorems (in view of Gdel’s result, Logic, formal 6).
The first paper of Tarski cited below is devoted to the problem of finding a definition of semantical truth for a logistic system L, not in L itself but in another system (metasystem) containing notations for the formulas of L and for syntactical relations between them. This is attractive as an alternative to the method of introducing the concept of truth by arbitrarily adding a notation for it, with appropriate new primitive formulas, to the metasystem, but in many important cases it is possible only if the metasystcm is in some essential respect logically stronger than L.
Tarski’s concept of truth, obtained thus by a syntactical definition, is closely related to Carnap’s concept of analyticity. According to Tarski, they are the same in the case that L is a “logical language.” See further semiotic 2. — A.C.
A. Tarski,
Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1935), pp 261-405.
A. Tarski,
On urdecidable statements in enlarged systems of logic and the concept of truth, The Journal of Symbolic Logic, vol. 4 (1939), pp. 105-112.
R. Carnap,
The Logical Syntax of Language, New York and London, 1937.