Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, the ordered set Q of rational numbers) are characterized syntactically.

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).

If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser (...) rule are derivable, admissible but nonderivable, or not admissible. (shrink)

In this paper we study the admissible rules of intermediate logics. We establish some general results on extensions of models and sets of formulas. These general results are then employed to provide a basis for the admissible rules of the Gabbay–de Jongh logics and to show that these logics have finitary unification type.

We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

We show that some common varieties of modal K4-algebras have finitary unification type, thus providing effective best solutions for equations in free algebras. Applications to admissible inference rules are immediate.

A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all (...) normal modal logics L containing S4.3 are almost structurally complete, i.e. all admissible rules having unifiable premises are derivable in L. Moreover, L is structurally complete iff L contains McKinsey axiom M. (shrink)

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known (...) results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 is countable and distributive and it forms a Heyting algebra. (shrink)

Dummett's logic LC quantified, Q-LC, is shown to be characterized by the extended frame Q+, ,D, where Q+ is the set of non-negative rational numbers, is the numerical relation less or equal then and D is the domain function such that for all v, w Q+, Dv and if v w, then D v . D v D w . Moreover, simple completeness proofs of extensions of Q-LC are given.

Unification was originally introduced in automated deduction and term rewriting, but has recently also found applications in other fields. In this article, we give a survey of the results on unification obtained in two closely related, yet different, application areas of unification: description logics and modal logics.