🚀 go-pugleaf

In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
> In article <1992Apr5.213320.27618@morrow.stanford.edu>,
> BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
>   (Edited Highlights:)

[irrelevant stuff deleted]

> S4 and S5 (and most other common modal logics), are subsystems of classical
> (non-modal) logic extended with the axiom: A iff \box A.

Not true. S5 contains only \box A => A, and the rule of inference:

                      |- A
                      ---------
                      |- \box A

which is strictly weaker then an axiom A => \box A (i.e. it may happen that A
is true while \box A is not).


> 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.

This is true under the above false assumption.

> Any modal logic which gives a set theory which is not extensional over usual
> set theory thus cannot be a subsystem of Soo.

Any prove of this?

> IMHO, extending the underlying logic of set theory is unlikely to give
> interesting set theorys. 

There is a lot of interesting results in the so called autoepistemic logic and
its relationships to set theoretic forcing by Marek and Truszczynski (see
Journal of the ACM vol 38 No 3 (June 1991) for details.)

Marek Suchenek
Cal State Dominguez Hills