A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.
A modal logic with n primitive unary modal operators
\Boxi,i\in\{1,\ldots,n\}
\Diamondi
\DiamondiP
lnot\BoxilnotP
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[1] 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
\Box
\Boxi
\Boxi\alpha
\alpha
\alpha