modal logic T
T is valid in a frame iff is reflexive.
First, suppose is not reflexive, say, . Let be a model based on such that , where is a propositional variable. By the construction of , we see that for all such that , we have , so . But since , . This means that .
Conversely, let be a reflexive frame, and any model based on , with a world in . Suppose . Then for all such that , . Since , we get . Therefore, . ∎
As a result,
T is sound in the class of reflexive frames.
Since any theorem in T is deducible from a finite sequence consisting of tautologies, which are valid in any frame, instances of T, which are valid in reflexive frames by the proposition above, and applications of modus ponens and necessitation, both of which preserve validity in any frame, whence the result. ∎
T is complete in the class of reflexive frames.
We show that the canonical frame is reflexive. For any maximally consistent set , if , then . Since T contains , we get that by modus ponens and the fact that is closed under modus ponens. Therefore , or is reflexive. ∎
T properly extends the modal system D, for is not valid in any non-reflexive serial frame, such as the one , where and : just let . So and , or . This means .