Window operator

In modal logic, the window operator ${\displaystyle \triangle }$ is a modal operator with the following semantic definition:

${\displaystyle M,w\models \triangle \phi \iff \forall u,M,u\models \phi \Rightarrow Rwu}$

for ${\displaystyle M=(W,R,f)}$ a Kripke model and ${\displaystyle w,u\in W}$. 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 '${\displaystyle \square }$', or its existential dual, often denoted by '${\displaystyle \Diamond }$'). 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.