ABSTRACT

A calculus institutes, over an artificial language, an exact relation of inference and a set of theorems. A system of formal semantics supplements an artificial language by a semantical system which institutes, over it, a relation of consequence and a set of tautologies. If we have, for a language, both a calculus and a formal semantics, then there is the question whether the relation of inference coincides with the relation of consequence. And here there is room for concepts copying those concerning the relationship of the calculus to natural language (interpretation M, validity M, soundness M, completeness M), only now concerning its relationship to its formal semantics (hence the subscript “F”): interpretation F, validity F, soundness F, completeness F. While the notion of formal semantics is not delimited so sharply that we could say that there is a calculus without a matching semantics, we can show that there is a system of semantics without a matching calculus (for example, the standard semantics for second-order logic). This is also connected with the problem of categoricity.