Multimodal logic

 ( Modal Logic ) 

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.

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Overview A modal logic with n primitive unary modal operators $\backslash Box\_i,\; i\backslash in\; \backslash \{1,\backslash ldots,\; n\backslash \}$ is called an n-modal logic. Given these operators and negation, one can always add $\backslash Diamond\_i$ modal operators defined as $\backslash Diamond\_i\; P$ if and only if $\backslash lnot\; \backslash Box\_i\; \backslash lnot\; P$, to give a classical multimodal logic if it is in addition stable under necessitation (or "possibilization", therefore) of both members of provable equivalences.

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 $\backslash Box$ must be capable of bookkeeping the cognition of each agent, thus $\backslash Box\_i$ must be indexed on the set of the agents. The motivation is that $\backslash Box\_i\; \backslash alpha$ should assert "The subject i has knowledge about $\backslash alpha$ being true". But it can be used also for formalizing "the subject i believes $\backslash alpha$". 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.

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Notes

• Miklós Ferenczi (2025). 9789631628708, Műszaki könyvkiadó. ISBN 9789631628708
• Dov M. Gabbay, Agi Kurucz, Frank Wolter, Michael Zakharyaschev (2025). 9780444508263, Elsevier. ISBN 9780444508263
• (2025). 9781402085895, Springer. ISBN 9781402085895

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