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From: jer@peora.UUCP (J. Eric Roskos)
Newsgroups: net.singles,net.math
Subject: Re: Logic (for logicians)
Message-ID: <1252@peora.UUCP>
Date: Tue, 9-Jul-85 14:16:11 EDT
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Posted: Tue Jul  9 14:16:11 1985
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Gee, I didn't know there were any intuitionists out there.  I remember
references to intuitionistic logic in Mendelson's text, though they were
just cursory.

>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.... |-).

I have, though!  In fact, that is close to how I met my most recent SO; one
day I was working on the University computer, and saw this person who had
"Logical" in her login name.  So I asked her, "What kind of logic does a
Logical  use?"  And this was how we met.

Now... to get down to what you were talking about here...

I have some reservations about claiming that there is any problem with my
stating the "classical" notion that A(x)y <=> ~E(x)~y; I mean, you can make
complete & consistent logics that contain the above, can't you?  It seems
to me merely a philosophical question; the fact that the logics exist and
are "ok" in this formal sense should be enough as far as mathematics is
concerned, I would think.

> Mr. Roskos has (un?)knowingly raised a philosophical issue
  ^^

>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....

I am not much of an expert on this constructive logic, but don't there exist
proofs for which no effective procedure exists?  Consider any theorem
containing an existential quantifier which can only be proved by choosing
elements out of the universe at random and trying them to see if they make
the quantified predicate "true"; e.g., theorems that say something like
"There exists a number with property x," where it is not possible to compute
the number given the property effectively.

Or does this relate to the question of whether there exist any languages
recognizable by a nondeterministic finite-state automaton that aren't
recognizable by a deterministic one in polynomial time?  I am not sure,
though there seem to be vague analogies there, since the epsilon-
transitions in an NFA are roughly analogous to "guesses," as are (I think)
the non-polynomial equvalent DFAs (which just try all the guesses by
enumeration rather than by nondeterministic transitions). But I don't know
for sure... it has been several years since I studied that stuff, and now
I am left with mostly just the intuitive recollections...

If you post a reply to this (which will go to net.math), please mail me
a copy too, since I don't read net.math, since I'm not a mathematician really.
-- 
Shyy-Anzr:  J. Eric Roskos
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