A modal logic amalgam of classical and intuitionistic propositional logic

Original · EN

A famous result, conjectured by Gödel in 1932 and proved by McKinsey and Tarski in 1948, says that φ is a theorem of intuitionistic propositional logic IPC iff its Gödel-translation φ' is a theorem of modal logic S4. In this paper, we extend an intuitionistic version of modal logic S1+SP, introduced in our previous paper (S. Lewitzka, Algebraic semantics for a modal logic close to S1, J. Logic and Comp., doi:10.1093/logcom/exu067) to a classical modal logic L and prove the following: a propositional formula φ is a theorem of IPC iff φ is a theorem of L (actually, we show: Φφ iff Φₗφ, for propositional Φ,φ). Thus, the map φφ is an embedding of IPC into L, i.e. L contains a copy of IPC. Moreover, L is a conservative extension of classical propositional logic CPC. In this sense, L is an amalgam of CPC and IPC. We show that L is sound and complete w.r.t. a class of special Heyting algebras with a (non-normal) modal operator.