Abstract

This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzen-style proof theory in which there is only alimited possibility to use structural rules. Following the literture, we use an operator to mark formulas to which the extra structural rules may be applied. New in our approach is that we do not see this as a modality, but rather as themeet of the marked formula with a special typeQ. In this way we can make the specific structural behaviour of marked formulas more explicit.The main motivation for our approach is that we can provide a nice, intuitive semantics for hybrid substructural logics. Soundness and completeness for this semantics are proved; besides this we consider some proof-theoretical aspects like cut-elimination and embeddings of the strong system in the hybrid one.