The present chapter is more extensive than the previous one since the reported results are mostly new. They previously occurred in two papers by the author of the book: [102] and [104]

MOTIVATION

Ordinary modal logics do not distinguish between worlds satisfying the same formulas. It means that whenever a modal formula φ is satisfied by a model M, it is also satisfied by a model M′ which collapses identical states from M′ to only one state. If we reason about concrete domains represented by modal models and we want to express the fact that a particular formula (representing, e.g., a particular state of affairs) occurs in a domain, we need to augment the ordinary modal language with some additional operators. Introducing graded modalities of the form ◊>n, ◊<n, ◊=n allowed to express the cardinality of a set of a particular world's successors that satisfy a certain formula φ. However, graded modalities do not enable making global numerical restrictions. Only introducing global counting operators of the form E>n, E<n, and E=n fills this gap. The syntax and semantics of a language enriched with global counting modalities is presented in Section 5.2. In Section 5.3, it is shown that taking into account modal logics with global counting operators in the book devoted to hybrid logics is not a whim and that it is fully justified by the expressive power of such logics.

A few words of comment should be devoted to the problem of coding numerical subscripts in counting operators. Subscript n coded in unary can be regarded as a sequence of n consecutive 1s. The size of this sequence equals n. Subscript n coded in binary is a sequence of consecutive bits which can take values from the set {0, 1}, whose binary value equals n.