This entry is meant to be about the flavor of modal logic known as S5 modal logic with several () modal operators and their duals.
The axiom denoted , (see Kracht 1999 for indication of possible reasons) is
The interpretation is the ‘if is true then agent knows that it is possible that is true.’
Another axiom that is relevant here is:
Axiom (5) interprets as saying ‘If agent does not know that holds, then (s)he knows that (s)he does not know’. This is termed ‘negative introspection’.
- this logic is obtained from by adding the axiom .
starting from the logic S4(m), add, for each the axiom .
In either case the same result can be obtained by adding in or in place of the corresponding .
With , the models corresponded to frames where each relation was reflexive. With , the frames needed to be transitive as well. Here we consider the class of models with Kripke frames, where each is an equivalence relation. These are sometimes called equivalence frames.
(Soundness of )
(We show this for .) We have already shown (here in the logic S4(m)) that the older axioms and hold so it remains to show if we have a frame, , where is an equivalence relation on then .
We suppose the we have a state so that . Now we need to find out if , so we note that
if and only if with , , but
that holds if with , there is some with and .
However whatever we have with , we have as is symmetric, and we know, by assumption, that , so we have what we need.
General books on modal logics which treat these logics thoroughly in the general context include e
Patrick Blackburn, M. de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.
Marcus Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.
Application to artificial intelligence:
Last revised on July 16, 2023 at 09:22:43. See the history of this page for a list of all contributions to it.