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From: rance@cornell.UUCP (W. Rance Cleaveland)
Newsgroups: net.singles,net.math
Subject: Logic (for logicians)
Message-ID: <2958@cornell.UUCP>
Date: Mon, 8-Jul-85 13:14:32 EDT
Article-I.D.: cornell.2958
Posted: Mon Jul  8 13:14:32 1985
Date-Received: Tue, 9-Jul-85 06:36:56 EDT
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Organization: Cornell Univ. CS Dept.
Lines: 43
Xref: watmath net.singles:7685 net.math:2126

> EVERYONE should know how to do proofs with quantifiers!  For one thing, it's
> taught in the introductory CS math courses most places, and for another,
> it'll probably be on that test you're going to take, if it's any good...
> I mean, everyone should know that
> 
> 	"For all x, y"
> 
> is the same as
> 
> 	"It is not the case that there exists an x such that not(y)".

Sorry to be pedantic, but when one's thesis topic comes up, can one help it?
Mr. Roskos has (un?)knowingly raised a philosophical issue which has caused a
great deal of acrimony among logicians and foundational mathematicians for at
least a century, the issue being "should logic be constructive?"  In layman's
terms, the debate turns on whether, as so-called classical logicians say,
establishing the truth of something is the same as establishing that the
negated statement is false, or whether, as constructive (or "intuitionistic")
logicians say, establishing the truth of something requires that you, well,
establish the truth of something.  To use Mr. Roskos' example, proving

	"For all x, y"

classically reduces to proving

	"There does not exist an x such that not(y)"

while constructively proving the former statement requires that one pick an
arbitrary x and prove y (since x is arbitrary, the proof that y holds will
work for any x, hence all x.).

The difference?  In the above example, the constructive philosophy requires
that we give a procedure for proving y for any x, while the classical frame-
work has no such requirement.  In this sense then, constructive logic has an
inherent computational flavor, while constructive logic does not....

Yeah, this probably doesn't belong on net.singles; at least, I've never had
any success starting a romantic conversation with "Whose your favorite logi-
cian?"  Still, maybe some of you out there.... |-).

Regards,
Rance Cleaveland