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 ${\displaystyle \Box _{i},i\in \{1,\ldots ,n\}}$ is called an n-modal logic. Given these operators and negation, one can always add ${\displaystyle \Diamond _{i}}$ modal operators defined as ${\displaystyle \Diamond _{i}P}$ if and only if ${\displaystyle \lnot \Box _{i}\lnot P}$.

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

Notes

1. Sergio Tessaris; Enrico Franconi; Thomas Eiter (2009). Reasoning Web. Semantic Technologies for Information Systems: 5th International Summer School 2009, Brixen-Bressanone, Italy, August 30 – September 4, 2009, Tutorial Lectures. Springer. p. 112. ISBN   978-3-642-03753-5.
2. Ferenczi 2002, p. 257.

References

• 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.
• Walter Carnielli; Claudio Pizzi (2008). Modalities and Multimodalities. Springer. ISBN   978-1-4020-8589-5.

External links

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