Window operator

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

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

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 '$$\square$$', or its existential dual, often denoted by '$$\Diamond$$'). 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.