Window operator

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

$M,w\models \mathrm{△}\varphi ⟺\mathrm{\forall }u,M,u\models \varphi \Rightarrow Rwu$

for $M=(W,R,f)$ a Kripke model and $w,u\in W$. 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 '$◻$', or its existential dual, often denoted by '$◊$'). 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.