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From: rance@cornell.UUCP (W. Rance Cleaveland)
Newsgroups: net.math
Subject: Re: Logic (for logicians)
Message-ID: <3033@cornell.UUCP>
Date: Wed, 10-Jul-85 10:46:15 EDT
Article-I.D.: cornell.3033
Posted: Wed Jul 10 10:46:15 1985
Date-Received: Sat, 13-Jul-85 09:16:21 EDT
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> 
Classical logic is indeed consistent and complete (as, indeed, are many logics)
and you are right that the "problem" I allude to is a philosophical one.  On the
other hand, I believe that one should approach logic first from a philosophical
perspective and decide first what sorts of intuitive underpinings one wants
one's notion of logic to have.  Classical logic takes truth as a given, which
is to say that truth is defined in terms of truth valuations ("truth tables").
In this setting, then, a proposition must either be true or false, and it is 
this requirement which gives some of us pause.  Constructive logic, on the 
other hand, defines truth as provability; we know only that which we can prove,
and we prove things by giving constructions.  What I'm trying to say, I guess,
is that while completeness and consistency are certainly germane to the dis-
cussion of logics, other issues are important as well.  Incidentally,
Ax(y) <=> -Ex(-y)  is not true constructively because it cannot be proven
constructively, that is, without recourse to the law of the excluded middle.
Intuitively, the constructions which establish the truth of either side of
the bi-implication are inherently different, and while one for Ax(y) can
be translated into one for -Ex(-y), thereby establishing Ax(y)=>-E(-y),
the converse does not hold.

Historically, constructive logic was de rigeur until the time of Cantor
(late 1800's).  For the Greeks, for instance, truth reduced to constructibility
with a compass and straight-edge, and until Cantor began talking about point
sets and sets of sets mathematicians were under tremendous peer pressure to 
give constructive accounts of their work.  (Cantor in fact was sharply crit-
cized by many of his contemporaries for his unorthodox work, and several 
authors have speculated that the professional ostracism he underwent con-
tributed to his ultimate mental breakdown.)

> >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.
> 
Constructively proof is equivalent to procedure, so this point is moot in a
constructive setting.  However, in the situation you describe to prove an
existentially quantified statement in the manner you propose you must give
some assurance that eventually you will pick a number satisfying property
x.  If you can demonstrate that this process terminates then you have given
an effective procedure for computing x, and constructively you are fine.

> 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...
> 
> -- 
Ah, P=NP.  There's not really any connection between complexity theory and
logic, except that some logics yield inherently more complex proofs than
others....  (For someone who hasn't studied the stuff in a while, thought,
your recollections are admirable :-)).

Regards,
Rance Cleaveland