Abstract

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectives Φ such that {→, ∨, ⅂}  Φ (m=subseteq) {→, ∧, ∨, ⅂}, the Φ-fragment of MV equals the Φ-fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (⅂a→b ∨ c) → (⅂a→b) ∨ (⅂a→c) separable?