ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that ‘modal logic’ (...) as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet. (shrink)

The paper deals with polymodal languages combined with standard semantics defined by means of some conditions on the frames. So, a notion of "polymodal base" arises which provides various enrichments of the classical modal language. One of these enrichments, viz. the base £(R,-R), with modalities over a relation and over its complement, is the paper's main paradigm. The modal definability (in the spirit of van Benthem's correspondence theory) of arbitrary and ~-elementary classes of frames in this base and in some (...) of its extensions, e.g., £(R,-R,R-1 ,_R-1), £(R,-R,=I=) etc., is described, and numerous examples of conditions definable there, as well as undefinable ones, are adduced. (shrink)

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).

Intermediate prepositional logics we consider here describe the setI() of regular informational types introduced by Yu. T. Medvedev [7]. He showed thatI() is a Heyting algebra. This algebra gives rise to the logic of infinite problems from [13] denoted here asLM 1. Some other definitions of negation inI() lead to logicsLM n (n ). We study inclusions between these and other systems, proveLM n to be non-finitely axiomatizable (n ) and recursively axiomatizable (n ). We also show that formulas in (...) one variable do not separateLM from Heyting's logicH, andLM n (n ) from Scott's logic (H+S). (shrink)

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).

The logic of 'elsewhere,' i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions, as well as in the case in which they represent moments of time. This logic is applied here to the alethic modal case, in which (...) the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers under the name 'modal agnosticism.' In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position - its neutrality as between modal realism and modal anti-realism. (shrink)

We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...) iff Vy(y ≠ x → y ⊧ φ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒ $_{c}$ . Strong completeness of the normal ℒ $_{c}$ logics is proved with respect to models in which all worlds are named. Every ℒ $_{c}$ -logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒ $_{c}$ are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. (shrink)