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In article <1992Apr5.213320.27618@morrow.stanford.edu>,
BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
  (Edited Highlights:)
> I am not aware of any attempt to construct a set theory with modal
> logic as its underlying logic.  I'd appreciate hearing about any
> research in this area.
> 
> Following are some speculations as to why this line of research
> might prove interesting.  Readers not interested in speculation are
> invited to hit <next> or <delete> now.
> 
> So closure is not definable absolutely.  To remedy this, I adopt as
> my underlying logic S4 or S5.  In these systems, as is well known
> (e.g. Sikorski, "Boolean Algebras", p.199), the possibility
> operator is a Kuratowski closure operator.  So in my set theory
> whose logic is S4 or S5 I may define: *the* closure of x is the set
> of all y such that it's *possible* that y is a member of x.
>
> So it would seem that using a modal logic of a certain sort as our
> underlying logic might permit us to follow Shoenfield's program of
> "introducing new operations on sets" in order to deal with
> "problems that cannot be settled from the axioms of ZFC."
> 
> No?
> --
> Michael Carroll                br.mjc@forsythe.stanford.edu
> Research Libraries Group
> 4/5/92
My first posting on this was terribly ambiguous - I was trying to be just too
concise - and seems to have caused lots of confusion, so I'll say it all
properly this time.
Define a "modal" logic Soo by adding the axiom A iff \box A to classical logic.
It's now really easy [exercise folks :-)] to see that:
(i) If B is a theorem of S5 (or S4, or any common modal logic), then B is
   a theorem of Soo.
(ii) Let C be any formula, and C' be C with all \box removed. Then:
   C is a theorem of Soo if and only if C' is a theorem of classical logic

Now, define theories ZFC(Sn) in the obvious way: ie the axioms of Sn plus
the axioms of ZFC (with comprehension / separation allowed with formulae
containing \box). Then easily (i), (ii) hold for these systems also:
(i) If  ZFC(S5) /- B  then  ZFC(Soo) /- B
(ii) ZFC(Soo) /- C  iff  ZFC /- C'

In particular, if D is a statement of classical set theory, then ZFC(S5) /- D
if and only if ZFC(S5) /- D.

Thus it seems that simply formulating set theory in a modal logic doesn't
introduce any new theorems that don't involve modal operators: Eg we can't
solve the continuum hypothesis in this way.

On the other hand, it may be that adding new axioms stating how modal operators
interact with sets might give us theories that prove things like the continuum
hypothesis. But I'll leave that to real set theorists.
--
R Loader, Oxford.