On the Beth properties of some intuitionistic modal logics

Abstract.

 Let L be one of the intuitionistic modal logics considered in [4]. As in the classical modal case (see [7]), we define two different forms of the Beth property for L, which are denoted by B ₁ and B ₂ ; in this paper we study the relation among B ₁ ,B ₂ and the interpolation properties C ₁ and C ₂ , introduced in [4]. It turns out that C ₁ implies B ₁ , but contrary to the boolean case, is not equivalent to B ₁ . It is shown that B ₂ and C ₂ are independent, and moreover it comes out that, in contrast to classical case, there exists an extension of the intuitionistic modal logic of S ₄ -type, that has not the property B ₂ . Finally we give two algebraic properties, that characterize respectively B ₁ and B ₂ .