Inference as an explication and as a counterpart of consequence

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..
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