Consequence & Inference

Logic is usually considered to be the study of logical consequence – of the most basic laws governing how a statement’s truth depends on the truth of other statements. Some of the pioneers of modern formal logic, notably Hilbert and Carnap, assumed that the only way to get hold of the relation of consequence was to reconstruct it as a relation of inference within a formal system built upon explicit inferential rules. Even Alfred Tarski in 1930 seemed to foresee no kind of consequence other than one induced by a set of inference rules: "Let A be an arbitrary set of sentences of a particular discipline. With the help of certain operations, the so-called rules of inference, new sentences are derived from the set A, called the consequences of the set A. To establish these rules of inference, and with their help to define exactly the concept of consequence, is again a task of special metadisciplines; in the usual terminology of set theory the schema of such definition can be formulated as follows: The set of all consequences of the set A is the intersection of all sets which contain the set A and are closed under the given rules of inference." (p. 63) Thereby also the concept of truth came to be reconstructed as inferability from the empty set of premises. (More precisely, this holds only for non-empirical, necessary truth; but of course logic never set itself the task of studying empirical truth.) From this viewpoint, logic came to look as the enterprise of explication of consequence in terms of inference. This view was soon shattered by the incompleteness proof of Kurt Gödel and by the arguments of Tarski himself, which seemed to indicate that inference can never be a fully satisfying explication of consequence, and that hence we must find a more direct way of dealing with consequence. And model theory, established by Tarski, appeared to be the requisite tool. Consequence and inference thus started to appear as two things to be confronted and one of the main issues of logic became the investigation of the results of various instances of the confrontations – various kinds of soundness and completeness theorems. However, as Bencivenga (1999, 16) duly points out, "completeness theorems are a dime a dozen, but there is hardly any discussion of what it means to prove such a theorem."