MODAL LOGIC

For starters, let us quote three slogans with which Patrick Blackburn, Maarten de Rijke and Yde Venema - the authors of one of the most influential textbooks of modal logic - characterize contemporary modal logic (see [15, 11]):

slogan 1: Modal languages are simple yet expressive languages for talking about relational structures.

slogan 2: Modal languages provide an internal, local perspective on relational structures.

slogan 3: Modal languages are not isolated formal systems.

Yet, since our aim is not to give a versatile overview of modal logics - in fact, the discussion on standard modal logics only serves further considerations on hybrid logics - let us concisely comment on all three slogans.

First, since the ground-breaking paper of Kripke ([63]) appeared in which he exposes his possible world semantics for modal logics, it has been quite clear that models for these logics can be seen as relational structures (or directed graphs). More precisely, a Kripke model for a modal logic M is a triple 〈W, R, V〉, where W is a non-empty set called a universe, R is a binary (or n-ary) relation over W called an accessibility relation, and V is a valuation function mapping propositional formulas to subsets of W. Elements of W are called worlds, states, points or nodes (in places where it does not lead to a confusion, we use all these names interchangeably). Primarily, elements of W were considered to be possible worlds which are accessible to each other through R. However, the contemporary modal logic departs from such strict and narrow alethic understanding of relational structures. In fact, we can interpret W and R arbitrarily: as states of a computer linked by the execution of an appropriate program (the case of the multi-modal logic PDL), as states of knowledge of an agent (the case of epistemic logics), or as obligated and permitted states of affairs (the case of deontic logics).