View source for Modality

A property of a judgement characterizing its degree of certainty. Different modalities and the interelations between them are studied in [[Modal logic|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643302.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643304.png" />), respectively. Various combinations of the fundamental modalities and the negation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643305.png" /> are also modalities. The dual of a modality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643306.png" /> is the modality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643307.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643308.png" /> by replacing each occurrence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m0643309.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433010.png" />, and conversely. In most systems of modal logic, for a modality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433011.png" /> and its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433012.png" /> the equivalence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

holds.

In principle, it is possible to form an infinite number of combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433016.png" />; however, in concrete systems the number of pairwise inequivalent modalities often remains bounded (in view of (*) 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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433017.png" /></td> </tr></table>

and their duals. In S5 there are only 4 modalities: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433021.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433022.png" />) modalities, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433024.png" /> are equivalent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064330/m06433025.png" />.

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|Modal logic]].