[ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
To add a bit in support to Uday's remark about "presumption of
supremacy" of classical logic:
In fact, it is well known that classical logic can be embedded in
intuitionistic logic
using, for example, negative interpretation (classical "exists" becomes
\not\forall\not etc.)
From the point of view of provability, the interpretation of a
classical theorem is provable
intuitionistically if and only if the original theorem was provable
classically.
So, what is bad rationally, if instead of respectable "exists" we shall
say "it is
not true that for all x does not exist..."? It is clear, that it is a
"bad publicity",
a less "convincing" way to say - bad for "supremacy", but it has nothing
to do
with scientific rationality itself. (Constructive mathematics is just
more subtle
concerning existence.)
Best regards
Sergei Soloviev
Uday S Reddy wrote:
[ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
Tadeusz Litak writes:
It is time to leave behind the classical logic. In fact, we should have
done it a long time ago.
(even if it wasn't intended, it does indeed sound "like a total and
unconditional rejection"... such things happen in the fervor of a
discussion :-)
Having thought about why it sounds like a "total and unconditional
rejection", I believe the difference is in the perspective of what logic is
about.
Logic consists of the principles of "reasoned discourse", as per Aristotle.
Our reasoned discourse happens in natural language, which is a humongous
ocean. We may never be able to understand fully all the principles of logic
that are there. But it is clear that the logic that we do understand (all
the known logics put together) represents only a miniscule proprotion of the
vast ocean of "logic" that is employed in reasoned discourse. So, it seems
to me that a great deal of humility is warranted in talking about "logic" in
general.
In contrast, people that vax about classical logic seem to have the
presumption that classical logic has it all cased. They seem to think that
it represents the sum total of all reasonable principles of reasoned
discourse (even if they are willing to admit modal logics of one kind or
another as being reasonable *extensions* of classical logic). Hence,
anybody that talks about alternative logics is seen to be mounting an attack
on the classical logic, denying the supreme position of classical logic as
the one true logic.
We, the non-believers, of course deny that classical logic is supreme in any
sense. However, that is not an attack on classical logic itself. It is
just an attack on the *presumption* that classical logic is supreme.
All that we can say about classical logic is that it seems to be the
canonical logic for the present-day mathematics. Given that mathematics is
a very conservative discipline, with the bar of entry for new ideas set very
high, it has an abundance of depth but not so much in breadth. Thus, a
canonical logic for mathematics in no way represents a canonical logic for
all of human thought.
In particular, in a young and dynamic discipline like Computer Science,
which has none of the mathematical conservatism, we should be free to
explore all possible logics and invent new ones. In fact, devising logics
is our very main business. We should be very wary of any presumptions about
"the canonical logic" of any kind.
Cheers,
Uday
