Archetypal Forms of Inference

• Published in: Synthese (Dordrecht) Vol. 141; no. 1; pp. 45 - 76
• Main Author: Humberstone, Lloyd
• Format: Journal Article
• Language: English
• Published: Dordrecht Kluwer Academic Publishers 01.07.2004; Springer; Springer Nature B.V
• Subjects: Archetypes; Equivalence relation; Inference; Intuitionistic logic; Logic; Logical givens; Logical theorems; Necessary conditions; Partially ordered sets; Philosophical logics. Philosophy of language; Philosophy; Propositional logic; Propositions; Subsumption; Terminology; Variables; Relation; Propositional Logic; Equivalence; Substitution; Archetype; Inference; Irreversibility; Intuitionistic Logic
• ISSN: 0039-7857; 1573-0964
• DOI: 10.1023/B:SYNT.0000035850.89516.e1

A form (or pattern) of inference, let us say, explicitly subsumes just such particular inferences as are instances of the form, and implicitly subsumes those inferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instance of the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form of inference 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 as correct, as well as ruling differently on the matter of logical equivalence which entered into the definition of implicit subsumption. When relativized to classical propositional logic, we find (Section 2) that all but a handful of 'degenerate' inference forms turn out to be archetypal, whereas matters are very different in this respect for the case of intuitionistic propositional logic (Sections 3 and 4), and an interesting structure emerges in this case (the poset of equivalence classes of inference forms, with respect to the equivalence relation of implicitly subsuming the same inferences). Thus a more accurate, if excessively long-winded title would be 'Archetypal and Non-Archetypal Forms of Inference in Classical and Intuitionistic 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 the questions it raises; indeed several significant questions are left as open problems.