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.

A modal logic with n primitive unary modal operators ${\displaystyle \Box _{i},i\in \{1,\ldots ,n\}}${\displaystyle \Box _{i},i\in \{1,\ldots ,n\}} is called an n-modal logic. Given these operators and negation, one can always add ${\displaystyle \Diamond _{i}}${\displaystyle \Diamond _{i}} modal operators defined as ${\displaystyle \Diamond _{i}P}${\displaystyle \Diamond _{i}P} if and only if ${\displaystyle \lnot \Box _{i}\lnot P}${\displaystyle \lnot \Box _{i}\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^[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 ${\displaystyle \Box }${\displaystyle \Box } must be capable of bookkeeping the cognition of each agent, thus ${\displaystyle \Box _{i}}${\displaystyle \Box _{i}} must be indexed on the set of the agents. The motivation is that ${\displaystyle \Box _{i}\alpha }${\displaystyle \Box _{i}\alpha } should assert "The subject i has knowledge about ${\displaystyle \alpha }${\displaystyle \alpha } being true". But it can be used also for formalizing "the subject i believes ${\displaystyle \alpha }${\displaystyle \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.^[2]

• Ferenczi, Miklós (2002). Matematikai logika (in Hungarian). Budapest: Műszaki könyvkiadó. ISBN 963-16-2870-1.
• Dov M. Gabbay, Agi Kurucz, Frank Wolter, Michael Zakharyaschev (2003). Many-dimensional modal logics: theory and applications. Elsevier. ISBN 978-0-444-50826-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Walter Carnielli; Claudio Pizzi (2008). Modalities and Multimodalities. Springer. ISBN 978-1-4020-8589-5.

• "Modal Logic" entry by James Garson in the Stanford Encyclopedia of Philosophy

• Modal logic

• CS1 Hungarian-language sources (hu)
• CS1 maint: multiple names: authors list
• Articles with Stanford Encyclopedia of Philosophy links