# FOM: reverse logic; classical logic in the natural sciences

Stephen G Simpson simpson at math.psu.edu
Sun Oct 18 13:33:19 EDT 1998

```This is a reply to Neil Tennant's posting of 15 Oct 1998 17:30:40.

I had suggested that the correct logic for natural science is
classical rather than intuitionistic.  Neil argued against this.  I am
now recording a few off-the-cuff reactions to Neil's comments.

Neil says:
> Wouldn't this require an argument to the effect that only classical
> (but not intuitionistic) logic can handle certain inferences that
> are needed by the natural scientist?

Why?  I am saying that classical logic is true and appropriate for the
natural sciences.  If this is so, then the fact that it is needed for
certain particular natural science inferences would seem to be merely
icing on the cake.

However, your remark is interesting to me, because it suggests an idea
of "reverse logic", i.e. a program of justifying some of the axioms of
classical logic by showing that they are needed for certain
inferences.  This is perhaps parallel to reverse mathematics, where we
show that certain set-existence axioms are needed in order to prove
certain mathematical theorems, and this can be interpreted as in a
sense justifying those axioms.

"Reverse logic" strikes me as a novel idea, one that I had never
considered before.  But at first glance it seems much less compelling
than reverse mathematics, simply because the axioms of classical logic
is much less questionable than the set-existence axioms which are the

It seems to me that classical logic is deeply embedded in most if not
all fields of science.  If someone proposes to throw it away, we must
first examine carefully what the effects of that move would be across
the board.  I think that such an examination has only begun, and even
that bare start is based on dubious assumptions.

> There's no compelling reason to use classical logic to develop
> natural science, no matter how much of a `realist' you happen to be
> in your philosophical outlook.  Whatever you do (refutationally)
> with classical logic, you can do with intuitionistic relevant
> logic.

That may be, but I don't subscribe to the Popperian idea that science
consists of refutations.  It seems to me that universal statements
(i.e. laws) are an essential part of science.

> The classical mathematical theorems sort of "melt away" and vanish
> from the proof, in IR, of the main refutational result.  ...
> As a tough-minded fellow, shouldn't you too throw away your
> crutches (i.e., the law of excluded middle)?

I may be tough-minded, but not to *that* degree.  :-)

Seriously, I don't view classical logic as a crutch but rather as a
basic part of science, essential for the unity of human knowledge.

Speaking of the unity of human knowledge, I would like to point out
that intuitionistic logic was invented by mathematicians for
mathematicians and is still of interest mainly to mathematicians.
What are the prospects for getting other scientists interested in it?

-- Steve

```