SEMANTICS AS BASED ON INFERENCE

We may say that logic is the study of consequence; and the pioneers of modern formal logic (especially Hilbert, but also, e.g., the early Carnap) hoped to be able to theoretically reconstruct consequence in terms of the relation of derivability (and, consequently, necessary truth in terms of provability or theoremhood – derivability from an empty set of premises). The idea was that the general logical machinery will yield us derivability as the facsimile of the relation of consequence, and once we are able to formulate appropriate axioms of a scientific discipline, the class of resulting theorems will be the facsimile of the class of truths of the discipline. These hopes were largely shattered by the incompleteness proof of Godel (1931): this result appeared to indicate that there was no hope for fine-tuning our axiom systems so that theoremhood would come to align with truth. Tarski (1936) then indicated that there are also relatively independent reasons to doubt that we might ever be able to align derivability with consequence: he argued that whereas intuitively the statement every natural number has the property P follows from the