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.

## Contents

* 1 Overview
* 2 Notes
* 3 References
* 4 External links

## Overview

A modal logic with n primitive unary modal operators ◻i,i∈{1,…,n} is called an n-modal logic. Given these operators and negation, one can always add ◊i modal operators defined as ◊iP if and only if ¬◻i¬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 ◻ must be capable of bookkeeping the cognition of each agent, thus ◻i must be indexed on the set of the agents. The motivation is that ◻iα should assert "The subject i has knowledge about α being true". But it can be used also for formalizing "the subject i believes α". 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. pp. 112. ISBN 978-3-642-03753-5. https://books.google.com/books?id=JdyeU7zs4-AC&pg=PA112.
2. ↑ Ferenczi 2002, p. 257.

## References

* Ferenczi, Miklós (2002) (in hu). Matematikai logika. 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

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