Abstract

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