Subintuitionistic logics and their modal companions: a nested approach.

In the present paper we deal with subintuitionistic logics and their modal companions. In particular, we introduce nested calculi for subintuitionistic systems and for modal logics in the $S5$ modal cube ranging from $K$ to $S4$ . The latter calculi differ from standard nested systems, as there are multiple rules handling the modal operator. As an upshot, we get a purely syntactic proof of the Gödel-McKinsey-Tarski embedding which preserves the structure and the height of the derivations. Finally, we obtain a conservativity result for classical logic over a weak subintuitionistic system.