(Note that $S5 = S5_{(1)}$.)
Although better than $K$ (resp. $K(m)$), and $T$ (resp. $T(m)$), $S4$ (resp. $S4(m)$) is still not considered adequate for knowledge representation. Because of this further axioms have been put forward, too many to be mentioned here. We will limit ourselves, (at this stage in the development of these entries, at least) to introducing one more axiom, that is a bit more contentious, (but is nice from the nPOV.)
The axiom denoted $B_i$, (and again refer to Kracht for some 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 $S4_{(m)}$, 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 $B$.
This is highly doubtful as a property of knowledge when applied to human beings. If an agent is ignorant of the truth of an assertion, it is very often unlikely that (s)he knows this ignorance.
Tim Porter It would be good to have some discussion on this axiom. (Donald Rumsfeld’s known unknowns has been worked to death, (although sometimes quite well done as here), so please… something worth saying :-)). This might stray over to a new entry as I would hope to see what there is to say especially looking to models of ‘why’ an agent ‘knows’ something. My query is whether that aspect has been explored and if so where. My feeling is that in S5 (and elsewhere) the reasons may give a groupoid-like structure (see below about the Kripke semantics) for the geometric semantics.
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 frames, where each $R_i$ is an equivalence relation. These are sometimes called equivalence frames.
$S5_{(m)}\vdash \phi \Rightarrow \mathcal{S}5(m)\models \phi.$
(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.
More on $S5$, $S5_{(m)}$ and their applications in Artificial Intelligence can be found in
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.