Multimodal logic

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.
Overview.
A modal logic with "n" primitive unary modal operators formula_1 is called an "n"-modal logic. Given these operators and negation, one can always add formula_2 modal operators defined as formula_3 if and only if formula_4.
Perhaps the first substantive example of a two-modal logic is Arthur Prior's tense logic, with two modalities, F and P, corresponding to "sometime in the future" and "sometime in the past". A logic with infinitely many modalities is dynamic logic, introduced by Vaughan Pratt in 1976 and having a separate modal operator for every regular expression. A version of temporal logic introduced in 1977 and intended for program verification has two modalities, corresponding to dynamic logic's ["A"] and ["A"*] modalities for a single program "A", understood as the whole universe taking one step forwards in time. The term "multimodal logic" itself was not introduced until 1980. Another example of a multimodal logic is the Hennessy–Milner logic, itself a fragment of the more expressive modal μ-calculus, which is also a fixed-point logic.
Multimodal logic can be used also to formalize a kind of knowledge representation: the motivation of epistemic logic is allowing several agents (they are regarded as subjects capable of forming beliefs, knowledge); and managing the belief or knowledge of each agent, so that epistemic assertions can be formed about them. The modal operator formula_5 must be capable of bookkeeping the cognition of each agent, thus formula_6 must be indexed on the set of the agents. The motivation is that formula_7 should assert "The subject "i" has knowledge about formula_8 being true". But it can be used also for formalizing "the subject "i" believes formula_8". For formalization of meaning based on the possible world semantics approach, a multimodal generalization of Kripke semantics can be used: instead of a single "common" accessibility relation, there is a series of them indexed on the set of agents.