Abstract. Hybrid logic refers to a group of logics lying between modal and first-order logic in which one can refer to individual states of the Kripke structure. In particular, the hybrid logic HL(@, ↓) is an appealing extension of modal logic that allows one to refer to a state by means of the given names and to dynamically create new names for a state. Unfortunately, as for the richer first-order logic, satisfiability for the hybrid logic

We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language

In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language Lt.

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This chapter introduces modal logic as a tool for talking about graphs, or to use more traditional terminology, as a tool for talking about Kripke models and frames. We want the reader to gain an intuitive appreciation of this perspective, and a firm grasp of the key technical ideas (such as bisimulations) which underly it. We introduce the syntax and semantics of basic modal logic, discuss its expressivity at the level of models, examine its computational properties, and then consider what it can say at the level of frames. We then move beyond the basic modal language, examine the kinds of expressivity offered by a number of richer modal logics, and try to pin down what it is that makes them all ‘modal’. We conclude by discussing an example which brings many of the ideas we discuss into play: games.

Preference is a basic notion in human behaviour, underlying such varied phenomena as individual rationality in the philosophy of action and game theory, obligations in deontic logic (we should aim for the best of all possible worlds), or collective decisions in social choice theory. Also, in a more

We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K, a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

Questions are different from statements, but they are just as important in driving reasoning, communication, and general processes of investigation. The first logical studies merging questions and propositions seem to have come from the Polish tradition: cf. [24]. A forceful modern defender of

For every Kripke complete modal logic L we define its hybrid companion LH . For a reasonable class of logics, we present a satisfiability-preserving translation from LH to L. We prove that for this class of logics, complexity, (uniform) interpolation, finite axiomatization transfer from L to LH .

We present some results concerning definability of classes of topological spaces in hybrid languages. We use language Lt described in [9] to establish notion of “elementarity ” for classes of topological spaces. We use it to prove the analogue of Goldblatt-Thomason theorem in topological spaces for hybrid languages H(E) and H(@). We also prove a theorem that allows to reformulate definability result of Gabelaia ([10]) for modal logic in terms of elementary topological space classes. 1