How Can We Avoid Popper's Collapsing Problem and Have Craig Interpolation?

Abstract

This paper establishes the Craig interpolation for the logic FOC+J, studied in [14, 43]. This logic is known as a logic that “combines” first-order intuitionistic and classical logic, i.e., that is conservative over the both logics. As a corollary, the Robinson Joint Consistency is established. From this property we obtain a sufficient condition for consistently combining two consistent theories, described in the syntax of a combination of first-order intuitionistic and classical logic. Preceding to the Craig interpolation and the Robinson joint consistency, this paper describes Popper's [29] original formulation of the collapsing problem, a problem occurring when we try to combined two logics, and briefly explains how FOC+J avoids this problem.