modal logic D
The Kripke frames corresponding to D are serial, in the following sense:
D is valid in a frame iff is serial.
First, assume valid in a frame , and . Let be a model based on , with . Then , so that . This means there is a such that , and hence is serial.
Conversely, let be a serial frame, a model based on , and a world in . Then there is a such that . Suppose . Then for all such that , we have . In particular, . Therefore, , whence . ∎
As a result,
D is sound in the class of serial frames.
Since any theorem in D is deducible from a finite sequence consisting of tautologies, which are valid in any frame, instances of D, which are valid in serial frames by the proposition above, and applications of modus ponens and necessitation, both of which preserve validity in any frame, whence the result. ∎
D is complete in the class of serial frames.
We show that the canonical frame is serial. Let be any maximally consistent set containing D. For any , we have , so that by modus ponens on D. This means that since is maximal. As a result, , showing that is consistent, and hence can be enlarged to a maximally consistent set . As a result, , whence . ∎
D is a subsystem of T, for any reflexive relation is serial. As a result, any theorem of D is valid in any serial frame, and therefore in any reflexive frame in particular, and as a result a theorem of T by the completeness of T in reflexive frames.
|Title||modal logic D|
|Date of creation||2013-03-22 19:33:54|
|Last modified on||2013-03-22 19:33:54|
|Last modified by||CWoo (3771)|