[FOM] Re: Combining classical and intuitionistic features in one logic

[Joao Marcos]

Dear Alex:

I haven't checked the result of your investigation yet, but...

> I has been curious about combining the 'best' of classical and
> intuitionistic logics in one intermediate logic. On one hand,
> it would be nice to have classical models based on truth tables
> for the propositional fragment. On the other hand, it would be
> nice to have the disjunction and existence properties holding
> for at least some other syntactically-defined fragment.
> Unfortunately, I was not able (even after checking with few
> colleagues) to find any work aiming at blending classical and
> intuitionistic features in one logic.

Obviously, this depends on what you mean by "blending" those features and
having the best of both worlds.  Now, if your Median Logic is to be obtained
by *constrained fibring*, sharing the classical and intuitionistic implication
symbols, then you are bound to lose the intuitionistic implication, and some
of the constructive character of the blend altogether.  Now, even if you
combine classical and intuitionistic logic sharing no symbols at all, still
you can end up with a "collapse" of intuitionistic into classical logic, if
you combine them through the standard fibring mechanism.  A fix to this
situation appeared recently in a paper called:

C. Sernadas, J. Rasga, and W. A. Carnielli. Modulated fibring and the
collapsing problem. Journal of Symbolic Logic, 67(4):1541--1569, 2002.
(preprint available at
http://www.cs.math.ist.utl.pt/ftp/pub/SernadasC/01-SRC-fiblog7.pdf)

In case I might be misconstruing something of the above, I would be happy to
have anyone correcting or complementing my comments.

> I ended up with my own investigation:
> http://www.mathpreprints.com/math/Preprint/asakharov/20040206/3/
> but the question remains.
>
> Perhaps, somebody can point to some works of the kind, or have
> an argument why this combination should not be attempted, or
> have an argument that some other features should be considered
> instead.

Well, the above-mentioned *collapse* argument against this combination,
depending on the way you intend to do it, appeared at least as early as in the
paper:

Luis Fariñas del Cerro and Andreas Herzig. Combining classical and
intuitionistic logic, or: intuitionistic implication as a conditional. In
Franz Baader and Klaus U. Schulz, editors, Frontiers in Combining Systems
(Proc. Int. Workshop FroCoS'96), number 3 in Applied Logic Series, pages
93-102. Kluwer Academic Publishers, 1996.
(preprint available at
http://www.irit.fr/ACTIVITES/EQ_ALG/Herzig/P/frocos96.ps.gz)


Yours, Joao Marcos