An algebraic approach to intuitionistic modal logics in connection with intermediate predicate logics (Q910395)

This paper contains an algebraic study of intermediate predicate logics (IPL's) and their modal counterparts. The semantical tools used for intuitionistic modal logics (IML) are bi-topological pseudo-Boolean algebras. This semantics is a slightly refined version of the original one which was introduced by H. Ono. The author introduces a special construction of bi-topological pseudo-Boolean algebras from algebraic frames for predicate logics and studies relations between the mentioned objects. In this technique, necessary and sufficient conditions for characterizing an IML by such algebras are found. As it was shown by H. Ono, for each intermediate propositional logic \({\mathcal J}\), there is the maximum IPL \({\mathcal J}^*\) whose propositional fragment equals \({\mathcal J}\). The author finds an axiomatization of the modal counterpart of \({\mathcal J}^*\) by the given one for \({\mathcal J}\). Examples of IML's K and IPL's L are given such that \(K\vdash A\) iff \(L\vdash \psi (A)\), where \(\psi\) is a translation of modal propositional formulas into non-modal first-order formulas \((\psi (\square p)=\forall xp^*(x)\), \(\psi (\diamond p)=\exists xp^*(x))\). It is shown that there is no maximum IML whose non-modal part is a given intermediate propositional logic \({\mathcal J}\), if \({\mathcal J}\) is not the classical one.