Truth without satisfaction
Tarski's definition of truth is given by using the concept of satisfaction. This paper studies systematically the conditions for the possibility of defining truth without satisfaction. The authors state results for the case, where the object language is a classical first- order language L and the meta-language is a many-sorted first-order language \(L_ M\). A meta-theory M in \(L_ M\) is said to be over a theory T in L if every theorem of T is a theorem of M. Theorem 8 says that there exists a metatheory M over T such that M defines truth but not satisfaction (for formulas with n free variables) if and only if T does not admit only finitely many n-types. Such a ''non-Tarskian\(_ n\)'' metatheory which is also finitely axiomatizable exists if T has a complete and axiomatizable extension that does not admit only finitely many n-types (Theorem 20). These results imply that there is a non- Tarskian\(_ n\) metatheory (finitely axiomatizable or not) for the language L if and only if L is not purely monadic, i.e., L contains other non-logical constants then one-place predicates and individual constants (Theorems 12 and 22).
scientific article; zbMATH DE number 3885862