modal logic S5
(T) , and
S5 is also denoted by KT5, where T and 5 correspond to the schemas T and 5 respectively.
In this entry (http://planetmath.org/ModalLogicT), we show that T is valid in a frame iff the frame is reflexive.
5 is valid in a frame iff is Euclidean.
First, let be a frame validating 5. Suppose and . Let be a model based on , with . Since , we have , and so , or for all such that . In particular, . So there is a such that and . But this means , whence , meaning is Euclidean.
Conversely, suppose is a Euclidean frame, and a model based on . Suppose . Then there is a such that and . Now, for any with , we have since is Euclidean. So . Since is arbitrary, , and therefore . ∎
This also shows that
where B is the schema , valid in any symmetric frame (see here (http://planetmath.org/ModalLogicB)), and 4 is the schema , valid in any transitive frame (see here (http://planetmath.org/ModalLogicS4)). It is also not hard to show that
S5 KDB4 KDB5,
where is the schema , valid in any serial frame (see here (http://planetmath.org/ModalLogicD)).
As a result,
S5 is sound in the class of equivalence frames.
Since any theorem in S5 is deducible from a finite sequence consisting of tautologies, which are valid in any frame, instances of T, which are valid in reflexive frames, instances of 5, which are valid in Euclidean frames by the proposition above, and applications of modus ponens and necessitation, both of which preserve validity in any frame, is valid in any frame which is both reflexive and Euclidean, and hence an equivalence frame. ∎
S5 is complete in the class of equivalence frames.
By the discussion above, it is enough to show that the canonical frame of S5 is reflexive, symmetric, and transitive. Since S5 contains T, B, and 4, is reflexive, symmetric, and transitive respectively, the proofs of which can be found in the corresponding entries on T, B, and S4. ∎
Remark. Alternatively, one can also show that the canonical frame of the consistent normal logic containing 5 must be Euclidean.
Let be such a logic. Suppose and . We want to show that , or . Let be any wff. If , since , so by the definition of , or since is maximal, or by substitution theorem on , or by modus ponens on 5 and the fact that is closed under modus ponens. This means that by the definition of , or since , so that by the substitution theorem on , which means since is maximal, or by the definition of . ∎
|Title||modal logic S5|
|Date of creation||2013-03-22 19:34:04|
|Last modified on||2013-03-22 19:34:04|
|Last modified by||CWoo (3771)|