nLab S5(m) modal logic

Idea

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 axioms $B_i$

The axiom denoted $B_i$, (see Kracht 1999 for indication of possible reasons) is

• $p \to K_i M_i p$.

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:

The axioms $(5)_i$

• $\neg K_i p \to K_i \neg K_i p$

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’.

The logics $S5$ and $S5_{(m)}$

• $S5$ - 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 $B$.

Equivalence frames

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.

Theorem

(Soundness of $S5_{(m)}$)

$S5_{(m)}\vdash \phi \Rightarrow \mathcal{S}5(m)\models \phi.$

Proof

(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

1. $\mathfrak{W},w \models K M p$ if and only if $\forall u$ with $R w u$, $\mathfrak{W},u \models M p$, but

2. 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.

References

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:

• J.- J. Ch. Meyer and W. Van der Hoek, Epistemic logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995.