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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 <math>\Box_i, i\in \{1,\ldots, n\}</math> is called an ''n''-modal logic. Given these operators and [[negation]], one can always add <math>\Diamond_i</math> modal operators defined as <math>\Diamond_i P</math> if and only if <math>\lnot \Box_i \lnot P</math>.

The first substantive example of a 2-modal logic is perhaps [[Arthur Prior]]'s [[tense logic]], with two modalities, F and P, corresponding to “sometime in the future” and “sometime in the past”. A logic<ref name="TessarisFranconi2009">{{cite book|author1=Sergio Tessaris|author2=Enrico Franconi|author3=Thomas Eiter|title=Reasoning Web. Semantic Technologies for Information Systems: 5th International Summer School 2009, Brixen-Bressanone, Italy, August 30 - September 4, 2009, Tutorial Lectures|url=http://books.google.com/books?id=JdyeU7zs4-AC&pg=PA112|year=2009|publisher=Springer|isbn=978-3-642-03753-5|pages=112}}</ref> with infinitely many modalities is (propositional) [[Dynamic logic (modal logic)|dynamic logic]], introduced 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 additionally 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 [[Subject (philosophy)|subject]]s 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 <math>\Box</math> must be capable of bookkeeping the cognition of each agent,  thus <math>\Box_i</math> must be indexed on the set of the agents. The motivation is that <math>\Box_i \alpha</math> should assert "The subject ''i'' has knowledge about <math>\alpha</math> being true". But it can be used also for formalizing "the subject ''i'' believes <math>\alpha</math>". For formalization of meaning based on the [[possible world semantics]] approach, a [[Kripke semantics#Multimodal logics|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.<ref>Ferenczi 2002: 257</ref>

== Notes ==
<references/>

== References ==
* {{cite book |last=Ferenczi |first=Miklós |title=Matematikai logika |publisher=Műszaki könyvkiadó |location=Budapest |year=2002 |language=Hungarian |isbn=963-16-2870-1|lang=hu}}
* {{cite book|author=[[Dov M. Gabbay]], A. Kurucz, F. Wolter, M. Zakharyaschev|title=Many-dimensional modal logics: theory and applications|year=2003|publisher=Elsevier|isbn=978-0-444-50826-3}}
* {{cite book|author1=Walter Carnielli|author2=Claudio Pizzi|title=Modalities and Multimodalities|year=2008|publisher=Springer|isbn=978-1-4020-8589-5}}

== External links ==
*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/logic-modal Modal logic]" – by [[James Garson]].

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[[Category:Modal logic]]