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? (shrink)

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 such that the-fragment ofMV equals the 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 (abc) (ab)(a c) separable? (shrink)

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = (( ) ) (Scott), andKP = ()()() (Kreisel-Putnam). Using Kripke's semantics, we prove:1) SKP has the finite model property; 2) SKP has the disjunction property. In the last section of the paper we give some results about Scott's logic S = I+S.