Editing Modal logic (section)

=== Topological semantics ===

Modal logic has also been interpreted using topological structures. For instance, the ''Interior Semantics'' interprets formulas of modal logic as follows.

A ''topological model'' is a tuple <math> \Chi = \langle X, \tau, V \rangle </math> where <math> \langle X, \tau \rangle</math> is a [[topological space]] and <math>V</math> is a valuation function which maps each atomic formula to some subset of <math>X</math>. The basic interior semantics interprets formulas of modal logic as follows:

* <math> \Chi, x \models P </math> iff <math> x \in V(P) </math>
* <math> \Chi, x \models \neg \phi </math> iff <math> \Chi, x \not\models \phi </math>
* <math> \Chi, x \models \phi \land \chi </math> iff <math> \Chi, x \models \phi</math> and <math>\Chi, x \models \chi </math>
* <math> \Chi, x \models \Box \phi </math> iff for some <math> U \in \tau </math> we have both that <math> x \in U </math> and also that <math> \Chi, y \models \phi </math> for all <math> y \in U </math>

Topological approaches subsume relational ones, allowing [[non-normal modal logic]]s. The extra structure they provide also allows a transparent way of modeling certain concepts such as the evidence or justification one has for one's beliefs. Topological semantics is widely used in recent work in formal epistemology and has antecedents in earlier work such as [[David Lewis (philosopher)|David Lewis]] and [[Angelika Kratzer]]'s logics for [[counterfactuals]].