Philosophy:Window operator

Short description: Operator in modal logic

In modal logic, the window operator [math]\displaystyle{ \triangle }[/math] is a modal operator with the following semantic definition:

[math]\displaystyle{ M,w\models\triangle\phi \iff \forall u, M,u\models\phi\Rightarrow Rwu }[/math]

for [math]\displaystyle{ M=(W,R,f) }[/math] a Kripke model and [math]\displaystyle{ w,u\in W }[/math]. Informally, it says that w "sees" every φ-world (or every φ-world is seen by w). This operator is not definable in the basic modal logic (i.e. some propositional non-modal language together with a single primitive "necessity" (universal) operator, often denoted by '[math]\displaystyle{ \square }[/math]', or its existential dual, often denoted by '[math]\displaystyle{ \Diamond }[/math]'). Notice that its truth condition is the converse of the truth condition for the standard "necessity" operator.

For references to some of its applications, see the References section.

References

• Blackburn, P; de Rijke, M; Venema, Y (2002). Modal Logic. Cambridge University Press.