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From: bwk@mbunix.mitre.org (Barry W. Kort)
Newsgroups: comp.ai,talk.philosophy.misc
Subject: Re: Fun with the semantics of paradox
Summary: Reinventing the excluded middle.
Keywords: Multivalent Logic, Ambiguity, Undecidability
Message-ID: <43519@linus.UUCP>
Date: 11 Jan 89 00:40:57 GMT
References: <551@soleil.UUCP> <1975@cadre.dsl.PITTSBURGH.EDU>
Sender: news@linus.UUCP
Reply-To: bwk@mbunix (Barry Kort)
Organization: Semantic Antics, Wordsworth, MA
Lines: 50

In article <1975@cadre.dsl.PITTSBURGH.EDU> geb@cadre.dsl.pittsburgh.edu
(Gordon E. Banks) enters the fray on Dave's paradoxical sentence:

>In article <551@soleil.UUCP> peru@soleil.UUCP (Dave Peru) writes:

>>When you view the meaning of a paradox, your brain is on a razor's edge.
>>Depending on what side you fall, the paradox is decidedly true or false.
>>Example:  This statement is false.

>On the contrary, when presented with a paradox, one's mind tends to first
>call it true, then false, then true, then false as one considers it
>over and over.  It is not resolvable.

I disagree.  I suggest that we consider the law of logic that trips
us up here:  Aristotle's Law of the Excluded Middle.  This law says
that a sentence must be either True or False.  There are no other
possibilities.  We now know better.  A sentence may be formally
undecidable.  A sentence may be ambiguous, admitting multiple meanings.
A sentence may be a meaningless sequence of words, admitting no meaning
whatsoever.

Now, let us consider the pathological locution, "This sentence is false."
If we abandon the Law of the Excluded Middle, we are left with the
problem of categorizing the locution in question as one of: 1) True,
2) False, 3) Undecidable, 4) Ambiguous, or 5) Meaningless.

We have already tried to categorize the sentence as either True
or False, and come to a contradiction in either case.  So we discard
those two possibilities.  The locution is apparently neither
Meaningless nor Ambiguous.  This leaves only Undecidable, which seems
consistent with everthing else we know about the pathological sentence.

This example illustrates the main idea of paradoxes:  they reveal
the incompleteness of our thinking.  The paradox is not unresolvable.
We resolve it by inventing the previously excluded middle.

Interestingly enough, Saul Kripke has developed several new branches
of logic (Modal Logic and Intuitionist Logic being the two that seem
most interesting here).  Like non-Euclidean geometry, new logics
arise by throwing away unnecessary restrictions in the Axioms of
the formal system.  

Not all thinking is deductive.  Some of the most fascinating thinking
is creative.  Cantor and Conway created transfinite numbers in
different ways.  What new possibilities can you imagine if you
throw off the yoke of unnecessarily restrictive rules?

--Barry Kort

and unambiguous.