Binary Modal Companions for Subintuitionistic Logics

The weak subintuitionistic logic WF, for which no standard unary modal companion is known, is found to have a strict implication logic as its binary modal companion. It is also shown that for all modal logics extending the weak logic EN, classical modal logic with necessitation, a strict implication logic exists which is essentially equivalent to it. This logic extends a basic strict implication logic plus an axiom U, and conversely each such logic corresponds to a modal logic extending EN. Among other things this means that any subintuitionistic logic which has a modal companion has a strict implication companion as well.