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The γ/em>-admissibility is one of the most important problems in the realm of relevant logics. To prove the γ-admissibility, either the method of normal models or the method using metavaluations may be employed. The γ-admissibility of a wide class of relevant modal logics has been discussed in Part I based on a former method, but the γ-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist axioms are well known as a means of expressing generalized forms of formulas with modal operators. This paper shows that γ is admissible for relevant modal logics with restricted Sahlqvist axioms in terms of the method using metavaluations.

By $$ {\user1{\mathcal{L}}} $$ we denote the set of all propositional formulas. Let $$ {\user1{\mathcal{C}}} $$ be the set of all clauses. Define $$ {\user1{\mathcal{C}}}_{0} = {\user1{\mathcal{C}}} \cup {\left\{ {\neg L\eta :\eta \in {\user1{\mathcal{C}}}} \right\}} $$ . In Sec. 2 of this paper we prove that for normal modal logics $$ {\user1{\mathcal{S}}} $$ , the notions of $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}_{0} } \right)} $$ -expansions and $$ {\user1{\mathcal{S}}} $$ -expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of $$ {\user1{\mathcal{S}}} $$ -expansions for I and $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{C}}}} \right)} $$ -expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of $$ {\user1{\mathcal{S}}} $$ -expansions for I is the same as that of $$ {\left( {{\user1{\mathcal{S}}},{\user1{\mathcal{L}}}} \right)} $$ -expansions for I.

This paper deals with modal Horn formulas. It contains a characterization of the classes of models definable by modal universal Horn formulas as well as a preservation result for modal universal Horn formulas.

In this note, a fully modal proof is given of some conservation results proved in a previous paper by arithmetic means. The proof is based on the extendability of Kripke models.

We consider modal logics whose intermediate fragments lie between the logic of infinite problems [20] and the Medvedev logic of finite problems [15]. There is continuum of such logics [19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from [12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).

We present, in terms of Kripke models, the characteristics of all the known extensions of the modal system S4.3. Such a semantic description makes it possible to give a complete picture of the whole class of systems extending S4.3. We obtain answers to some unsolved problems.

This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.

We show that neither the descending chain property nor the finite model property is a necessary condition for a model logic having no minimal proper extension. This answers in the negative two questions raised by G. E. Hughes.