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From: jmc12 on 20 Jul 2010 07:32
I've been thinking about the modal logic S5 and a famous result proved
by Scroggs (paper easily found on google) that every normal extension
of S5 is a finitely many valued logic (S5 itself isn't finitely
valued). Thus, every such extension will have amongst its theorems a
formula which (intuitively speaking) imposes a finite bound on the
number of distinct (i.e. not necessarily equivalent) propositions. A
simple example of such a formula is

NEC(p1 <--> p2) v NEC(p1 <---> p3) v NEC(p2 <---> p3)

Intuitively, for any three propositions at least two of them will be
necessarily equivalent----there are at most 2 propositions. Of course,
not every extension of S5 will impose this bound, but it imposes a
bound for some n.

Scroggs' argument is quite complicated and although I can pretty much
follow it, I've no idea what's going on intuitively here. That is, I
can't really see what it is about S5 (and the assumptions about
modality that it encodes) which makes it the strongest logic which
does not impose a bound on the number of propositions.

Of course, one philosophically interesting fact about S5 is that
(according to it), the modal facts are themselves non-contingent, i.e.
for any modally closed formula P, (P <--> NEC(P)) is a theorem of S5.

Is this fact about S5 connected with Scroggs' result?

Could one give an informal argument for Scroggs' result that helps us
see what's going on here modally?

Thanks in advance for any help.


From: Ken Pledger on 20 Jul 2010 16:44
In article
<b92b2d94-e8e6-44e6-a83c-2f4a69b87535(a)d16g2000yqb.googlegroups.com>,
jmc12 <ajm133(a)gmail.com> wrote:

> I've been thinking about the modal logic S5 and a famous result proved
> by Scroggs (paper easily found on google) that every normal extension
> of S5 is a finitely many valued logic ....
>
> Scroggs' argument is quite complicated and although I can pretty much
> follow it, I've no idea what's going on intuitively here....


I've never read Scroggs's proof; but according to Hughes &
Cresswell [1996] p.157, there's a proof also in Segerberg, "An Essay in
Classical Modal Logic" [1971].

Ken Pledger.
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