Supracompact inference operations

Michael Freund (1)

1 IHPST - Institut d'Histoire et de Philosophie des Sciences et des Techniques

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a finite subset B of A such that entails α for all finite subsets B′ that are entailed by A. This property is no longer valid when A is taken to be an arbitrary infinite set, even when the considered inference operation is supposed to be compact.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) that is smaller then C and agrees with C on finite sets. Moreover, K(C) inherits most of the properties that C may enjoy, like monotonicity, distributivity or disjunctive rationality. The main part of the paper concerns distributive supracompact operations. These operations satisfy a simple functional equation, and there exists a representation theorem that provides a semantic characterization for this family of operations. We examine finally the case of rational operations and show that they can be represented by a specific kind of model particularly easy to handle.

Mots clés: logic

Domaines: Histoire, Philosophie et Sociologie des sciences

Michael Freund. Supracompact inference operations. Studia Logica, 1993, 52 (3), pp.457-481. ⟨10.1007/BF01057658⟩. ⟨halshs-00792044⟩