What languages have Tarski truth definitions? Original Research Article

Tarskiʹs model-theoretic truth definition of the 1950s differs from his 1930s truth definition by allowing the language to have a set of parameters that are interpreted by means of structures. The paper traces how the model-theoretic theorems that Tarski and others were proving in the period between these two truth definitions became increasingly difficult to fit into the framework of the earlier truth definition, making the later one more or less inevitable. The paper also maintains that neither recursiveness nor satisfaction are essential features of the truth definition (in either version). The recursive form was a ‘practical’ step towards the explicit definition that was Tarskiʹs target. There are model-theoretic languages for which satisfaction is provably too crude a notion for expressing those properties of formulas that are needed for determining the truth of sentences.