🚀 go-pugleaf

In article <1992Apr13.194859.5426@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk writes:
> In article <1992Apr5.213320.27618@morrow.stanford.edu>,
> BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
> 
[...]

> In particular, if D is a statement of classical set theory, then ZFC(S5) /- D
> if and only if ZFC(S5) /- D.
[...]
> --
> R Loader, Oxford.

If this is all you wanted to say (modulo a typo in the above statement, I
preassume) then it is an easy excercice, indeed, to figure out that:


(*) For any modal-free theory T and modal-free sentence p,
    T |- p  iff  T + S5 |- p.

This fact (*), however, is irrelevant to whether S5 (or its variants) are
usefull in answering the consistency questions like "Is ZFC + \neg HC con-
sistent?". The real problem is somwhere else. Namely, it has been the
treatment of negation that makes S5 too weak to prove non-trivial (as far
as my taste is concerned) consistency results. In particular, if T is a
modal-free theory and p is a modal-free sentence with neither T |- p nor
T |- \neg p, then neither T + S5 |- Kp nor T + S5 |- \neg Kp. Actually,
what one may need here is some kind of "nonmonotonic" rule of inference
which allows (under certain conditions) to infer \neg Kp from non-provabili-
ty of p.

Currently, there is a lot of research in progress (specially on West Coast)
with people trying to find (a reasonable approximation of) such nonmonotonic
variant of S5. Nonmonotonic S5 seems strong enough to prove e.g. certain 
consistency results (this is not the major concern, though). Please note, 
that nonmonotonic S5 still satisfies (*). Therefore, as I said, (*) is
rather irrelevant in this context.

Greetings,

Marek A. Suchenek
Computer Science
Cal State Dominguez Hills
1000 E Victoria St
Carson CA 90747