This paper is a survey of results concerning the disjunction property, Halldén-completeness, and other related properties of intermediate prepositional logics and normal modal logics containing S4.

This paper is a survey of results concerning embeddings of intuitionistic propositional logic and its extensions into various classical modal systems.

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)

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)

Given an intermediate prepositional logic L, denote by L –d its disjuctionless fragment. We introduce an infinite sequence {J n}n1 of propositional formulas, and prove:(1)For any L: L –d =I –d (I=intuitionistic logic) if and only if J n L for every n 1.

For intermediate logics, there is obtained in the paper an algebraic equivalent of the disjunction propertyDP. It is proved that the logic of finite binary trees is not maximal among intermediate logics withDP. Introduced is a logicND, which has the only maximal extension withDP, namely, the logicML of finite problems.

A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following.Theorem on interpolation. For every intermediate logic L the following statements are equivalent:(i) Craig's interpolation theorem holds in L, (ii) L possesses the generalized interpolation property, (iii) Robinson's consistency statement is true in L.

We give in this paper a sufficient condition, cast in semantic terms, for Hallden-completeness in normal modal logics, a modal logic being said to be Hallden-complete (or Ήallden-reasonable') just in case for any disjunctive formula provable in the logic, where the disjuncts have no propositional variables in common, one or other of those disjuncts is provable in the logic.

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.