Modality

A property of a judgement characterizing its degree of certainty. Different modalities and the interelations between them are studied in modal logic. The modalities "necessary" and "possible" have been studied in logic since Aristotle (4th century B.C.) who, however, did not impart a precise meaning to them. These modalities are called fundamental and are denoted by the symbols $\Box$ and $\Diamond$ (or $L$ and $M$), respectively. Various combinations of the fundamental modalities and the negation $\neg$ are also modalities. The dual of a modality $Q$ is the modality $\overline Q$ obtained from $Q$ by replacing each occurrence of $\Box$ by $\Diamond$, and conversely. In most systems of modal logic, for a modality $Q$ and its dual $\overline Q$ the equivalence

$$Q\neg A\Leftrightarrow\neg\overline QA\label{*}$$

holds.

In principle, it is possible to form an infinite number of combinations of $\Box$, $\Diamond$ and $\neg$; however, in concrete systems the number of pairwise inequivalent modalities often remains bounded (in view of \eqref{*} and the presence of axioms simplifying certain modalities, or reducing one modality to another). For example, in the system S3 there are exactly 40 different modalities, and in S4 there are only 12:

$$\Box A,\quad\Box\Diamond A,\quad\Box\Diamond\Box A,\quad\neg\Box A,\quad\neg\Box\Diamond A,\quad\neg\Box\Diamond\Box A,$$

and their duals. In S5 there are only 4 modalities: $\Box A$, $\Diamond A$, $\neg\Box A$, $\neg\Diamond A$. On the other hand, in the system of modal logic T, and also in S1 and S2, the number of modalities is infinite and, even more, there are no reductions of modalities; that is, two positive (not containing $\neg$) modalities, $Q_1$ and $Q_2$ are equivalent if and only if $Q_1=Q_2$.

Sometimes the term modality refers to such notions (formalized in corresponding theories) as "truth" , "provability" , "disprovability" , and is also connected with the temporal "will be" , "always was" , etc.

For references see Modal logic.

How to Cite This Entry:
Modality. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Modality&oldid=43486
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article