Classically Archetypal Rules

Abstract

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence (according to that relation) of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion.

Supervenience, Dependence, Disjunction

“…Notice in this case that the inference from this ϕ to this ψ is not only valid (in classical logic) but archetypally so, in the sense that whenever a formula ϕ ′ has a formula ψ ′ as a classical consequence, the transition from ϕ ′ to ψ ′ can be subsumed under the transition from ϕ to ψ: there is some substitution s with ϕ ′ and ψ ′ classically equivalent to s(ϕ) and s(ψ). (More on this may be found in [Humberstone, 2004b] and [Połacik and Humberstone, 2018].) In particular, we want to choose s in such a way as to subsume the transition from p ∧ q to q (which represents, as it happens, another archetypal inference form) under the ϕ to ψ transition.…”