In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equatio...

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's. First a family of propositional logical systems is considered, which is obtained by gradually addin...

Two-layer modal syntax is based on two propositional languages (lower and upper one) and a modal language (a collection of modalities together with their arities) and features three kinds of formulae: (i) non-modal formulae build using lower propositional language, (ii) atomic modal formulae obtained by applying the modalities to non-modal ones, and (iii) complex modal formulae built from the a...

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axi...

Abstract We translate the three-valued quantum logic into modal logic, and prove 3-equivalence between valuation of a kind Kripke model in regard to this translation. To 3-equivalence, we introduce an observable-dependent which is fragment many-valued logic. Compared Birkhoff von Neumann’s some notions about observables, completeness relation for example, mechanics can be utilized if employed.

A miracle happens In one hand we have a class of marvelously complex theories in predicate logic theories with su cient coding potential like PA Peano Arithmetic or ZF Zermelo Fraenkel Set Theory In the other we have certain modal propositional theories of striking simplicity We translate the modal operators of the modal theories to certain speci c xed de ned predicates of the predicate logical...

We establish that the quantifier alternation hierarchy of formulae of secondorder propositional modal logic (SOPML) induces an infinite corresponding semantic hierarchy over the class of finite directed graphs. This solves an open problem problem of van Benthem (1985) and ten Cate (2006). We also identify modal characterizations of the expressive powers of second-order logic (SO) and monadic se...

Kripke-style models with two accessibility relations, one intuitionistio and the other modal, are given for analogues of the modal system K based on Heyting's propositional logic. I t is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddi...