Archetypal Forms of Inference

摘要

A form (or pattern) of inference, let us say, explicitlysubsumes just such particular inferences as are instances of the form, and implicitly subsumes thoseinferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instanceof the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form ofinference is archetypal if it implicitly subsumes every correct inference. A precise definition (Section 1)of these concepts relativizes them to logics, since different logics classify different inferences ascorrect, as well as ruling differently on the matter of logical equivalence which entered into the definitionof implicit subsumption. When relativized to classical propositional logic, we find (Section 2) thatall but a handful of `degenerate' inference forms turn out to be archetypal, whereas matters are verydifferent in this respect for the case of intuitionistic propositional logic (Sections 3 and 4), and an interestingstructure emerges in this case (the poset of equivalence classes of inference forms, with respect tothe equivalence relation of implicitly subsuming the same inferences). Thus a more accurate, if excessivelylong-winded title would be 'Archetypal and Non-Archetypal Forms of Inference in Classical andIntuitionistic Propositional Logic'. Some left-overs are postponed for a final discussion (Section 5).The overall intention is to introduce a new subject matter rather than to have the last word on thequestions it raises; indeed several significant questions are left as open problems.