Lesson plan
Window operator
In modal logic, the window operator \triangle is a modal operator with the following semantic definition in the language of predicate logic: 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 (The following can be read as "The satisfiability of the window operator delta applied to phi in a world w for Model m is only met if for all u, if phi is satisfied for a world u in model M, then w is related to u.) 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(which make sure cases satisfying phi are within accessible realms, which sets a possible upper bound for phi ) is the converse of the truth condition for the standard "necessity" operator because necessity operators are about making sure all accessible worlds satisfy phi, which secures phi as a minimum condition to be accessible. For references to some of its applications, see the References section.
Article summary
In modal logic, the window operator \triangle is a modal operator with the following semantic definition in the language of predicate logic: 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 (The following can be read as "The satisfiability of the window operator delta applied to phi in a world w for Model m is only met if for all u, if phi is satisfied for a world u in model M, then w is related to u.) 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(which make sure cases satisfying phi are within accessible realms, which sets a possible upper bound for phi ) is the converse of the truth condition for the standard "necessity" operator because necessity operators are about making sure all accessible worlds satisfy phi, which secures phi as a minimum condition to be accessible. For references to some of its applications, see the References section.
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.