This entry is meant to be about the flavor of modal logic known as S5 modal logic with several ($m \in \mathbb{N}$) modal operators and their duals.
The axiom denoted $B_i$, (see Kracht 1999 for indication of possible reasons) is
The interpretation is the ‘if $p$ is true then agent $i$ knows that it is possible that $p$ is true.’
Another axiom that is relevant here is:
Axiom (5) interprets as saying ‘If agent $i$ does not know that $p$ holds, then (s)he knows that (s)he does not know’. This is termed ‘negative introspection’.
$S5$ - this logic is obtained from $S4$ by adding the axiom $B$.
$S5_{(m)}$ starting from $S4_{(m)}$ the logic S4(m), add, for each $i = 1,\ldots, m$ the axiom $B_i$.
In either case the same result can be obtained by adding in $5$ or $(5)_i$ in place of the corresponding $B$.
With $T_{(m)}$, the models corresponded to frames where each relation $R_i$ was reflexive. With $S4_{(m)}$, the frames needed to be transitive as well. Here we consider the class $\mathcal{S}5(m)$ of models with Kripke frames, where each $R_i$ is an equivalence relation. These are sometimes called equivalence frames.
(Soundness of $S5_{(m)}$)
$S5_{(m)}\vdash \phi \Rightarrow \mathcal{S}5(m)\models \phi.$
(We show this for $m = 1$.) We have already shown (here in the logic S4(m)) that the older axioms $T$ and $(4)$ hold so it remains to show if we have a frame, $\mathfrak{M}= ((W,R),V)$, where $R$ is an equivalence relation on $W$ then $\mathfrak{M}\models B$.
We suppose the we have a state $w$ so that $\mathfrak{W},w \models p$. Now we need to find out if $\mathfrak{W},w \models K M p$, so we note that
$\mathfrak{W},w \models K M p$ if and only if $\forall u$ with $R w u$, $\mathfrak{W},u \models M p$, but
that holds if $\forall u$ with $R w u$, there is some $v$ with $R u v$ and $\mathfrak{W},v \models p$.
However whatever $u$ we have with $R w u$, we have $R u w$ as $R$ is symmetric, and we know, by assumption, that $\mathfrak{W},w \models p$, 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.