Semantics for Analytic Containment

In 1977, R. B. Angell presented a logic for analytic containment, a notion of “relevant” implication stronger than Anderson and Belnap’s entailment. In this paper I provide for the first time the logic of first degree analytic containment, as presented in [2] and [3], with a semantical characterization—leaving higher degree systems for future investigations. The semantical framework I introduce for this purpose involves a special sort of truth–predicates, which apply to pairs of collections of formulas instead of individual formulas, and which behave in some respects like Gentzen’s sequents. This semantics captures very general properties of the truth–functional connectives, and for that reason it may be used to model a vast range of logics. I briefly illustrate the point with classical consequence and Anderson and Belnap’s “tautological entailments”.