This chapter provides a concise survey on hybrid logics, including their expressive power, axiomatization, decidability and computational complexity. It also sketches a historical overview of the development of these logics.

MOTIVATION

Suppose that a modal logic M is defined over the signature 〈prop, R, R−〉, where prop is a denumerable set of propositional variables {p1, p2, …} and R− is the inverse of the relation R. In fact, we can consider M as a temporal logic, so R is a strict total order without endpoints. We can express the thought “in the future it will be cold” by a formula:

Fφ,

where φ stands for “it is cold”. As mentioned in Section 2.1, modal formulas do not distinguish between worlds satisfying formulas under the scope of a modal operator. Consequently, we do not know whether the F sends us three days or three years ahead. By means of the standard modal logic (or standard temporal logic), we cannot express the thought “on John's birthday it will be cold” or “on September 15th it will be cold” since these sentences contain direct pointers that distinguish particular worlds in a model (“John's birthday”, “September 15th”) and, as such, have some kind of global flavour. If we introduced an additional sort of expressions each of which would label exactly one point in a model, we would be able to express all the thoughts quoted above.

Hybrid logic introduces such expressions and calls them nominals. Each of them indeed holds at exactly one world in a model (although many of them can hold at the same world). Additionally, it augments the basic modal language with the so-called satisfaction operators @i that enable us to retrieve all information from a world labelled by i.