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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
S4 and S5 (and most other common modal logics), are subsystems of classical
(non-modal) logic extended with the axiom: A iff \box A.
Call this latter theory Soo. Obviously a theorem of set theory whose logic
is Soo is just a theorem of usual set theory with a few \box inserted.
Any modal logic which gives a set theory which is not extensional over usual
set theory thus cannot be a subsystem of Soo.
IMHO, extending the underlying logic of set theory is unlikely to give
interesting set theorys. On the other hand, perhaps one could introduce new set
theoretic principles involving modal operators?
--
Ralph Loader,
Mathematical Institute, Oxford.