Supracompact inference operations

Abstract

When a proposition α is cumulatively entailed by a finite set A of premisses, there exists, trivially, a finite subset B of A such that B ∪ B′ entails α for all finite subsets B′ that are entailed by A. This property is no longer valid when A is taken to be an arbitrary infinite set, even when the considered inference operation is supposed to be compact. This leads to a refinement of the classical definition of compactness. We call supracompact the inference operations that satisfy the non-finitary analogue of the above property. We show that for any arbitrary cumulative operation C, there exists a supracompact cumulative operation K(C) which is smaller than C and agrees with C on finite sets. Moreover, K(C) inherits most of the properties that C may enjoy, like monotonicity or distributivity. The main part of the paper concerns distributive supracompact operations. We prove that such operations satisfy a simple functional equation, and that there exists a representation theorem which provides a semantic characterization of this family of operations.