modal logic B
(T) , and
In this entry (http://planetmath.org/ModalLogicT), we show that T is valid in a frame iff the frame is reflexive.
B is valid in a frame iff is symmetric.
First, suppose B is valid in a frame , and . Let be a model based on , with , a propositional variable. Since , , and by assumption, for all such that . In particular, , which means there is a such that and . But this means that , so , whence , and is symmetric.
As a result,
B is sound in the class of symmetric frames.
Since any theorem in B is deducible from a finite sequence consisting of tautologies, which are valid in any frame, instances of B, which are valid in symmetric frames by the proposition above, and applications of modus ponens and necessitation, both of which preserve validity in any frame, whence the result. ∎
B is complete in the class of reflexive, symmetric frames.
Since B contains T, its canonical frame is reflexive. We next show that any consistent normal logic containing the schema B is symmetric. Suppose . We want to show that , or that . It is then enough to show that if , then . If , because is maximal, or by modus ponens on B, or by the substitution theorem on , or by the definition of , or since , or , since is maximal, or by the definition of . So is symmetric, and is both reflexive and symmetric. ∎