[FOM] Foundational Challenge

[Michael Lee Finney]

>  I'd like to comment on just one of Michael Finney's assertions:
>
>  "Not everyone accepts classical propositional logic as providing valid reasoning under all circumstances.
>  After all, if it did, would there have been paradoxes in the first place?" 
>
>  Do not blame the strictly classical part of logic for any of the paradoxes. 
>  They all arise for the constructivist.

I, perhaps, was not really clear. I apologize for that. The formalization of
classical propositional logic arose as a result of the paradoxes in an attempt
to both explain and avoid them, so you are right classical logic cannot be
directly blamed for the paradoxes.

However, that formalization does not avoid the paradoxes, since paradox still
follows with the addition of naive set theory. And the paradoxes arose where
the implicit logic was essentially the same as classical logic. So, if those
reasoning principles had been sufficient, then there would not have been
paradoxes in the first place. So, in that sense the informal logic that
preceded formalized logic is to blame. Given that the only real difference
was formalization, it can be argued that classical logic is just as much to
blame, because the only clothes have changed.

>  They can all be derived using only Core Logic (the relevant fragment of Intuitionistic Logic).
>  And Core Logic suffices for constructive mathematics.
>
>  Neil Tennant

Yes, they can. They can be derived using surprisingly little. Nor am I
suggesting that I or anyone else has solved the problem. At least not in a
manner that is consistent with normal mathematical practice.

However, my viewpoint is that is not naive set theory that is at fault, but
rather the logic that is at fault. Just because restricted versions of set
theory have been developed that appear to avoid the problems (and, we all
think that they do) does not mean that the problem has been solved. It is
essentially the situation that we found a nice, bright corner to hide in,
where everything seems to be safe. But the rest of the room is still in the
dark and we have no idea of the dangers there.

Michael Lee Finney