From: sasghm@theseus.unx.sas.com (Gary Merrill)
Newsgroups: sci.philosophy.meta
Subject: Re: Excluded Middle
Date: 6 Nov 92 14:11:52 GMT
Organization: SAS Institute Inc.
Originator: sasghm@theseus.unx.sas.com


In article <Bx9JuL.CJq@unx.sas.com>, sasghm@theseus.unx.sas.com (Gary Merrill) writes:

|> own oddities.  I have the vague recollection (the only kind I
|> have any more) of analyzing and criticizing approaches of this
|> sort in my dissertation.  If I can remember this weekend, I'll
|> wipe the mold off it and see if there is anything useful in it.

Let me begin with a kind of "vote of confidence" for Torkel Franzen.
I confess that my feelings about the discussion and issues raised
here pretty much mirror his, and I think that his remarks and
criticisms (aside from the name-calling, which is somewhat pardonable)
should be taken seriously.

I did unearth my dissertation and scrape the mold off it (this is not
an exaggeration).  If you are seriously into alternative (non-Tarski)
approaches to theories of truth (or truth predicates) you might
benefit from looking at a copy (if you could get one).  The title
is _A Semantically Closed Theory of Truth_.  In skimming it I was
reminded that the multi-valued logic approach to all this is even
older than I recalled:  Chrysippus (I forget the dates).  My own
approach bears certain similarities to that of Jean Buridan (14th
century;  see _Sophisms on Meaning and Truth_).

An analysis of several more modern attempts (most based upon denial
of excluded middle; van Fraassen, Skyrms, Robert Martin) is performed
and weaknesses in these approaches are described.  If you are inclined
toward such an approach, you need to beware of certain unsavory
features it may exhibit.  For example, you will discover that the
deduction theorem frequently fails in systems of the sort under discussion.
And it often turns out that you cannot meaningfully say things that
you seemingly should be able to.  For example, if you use a three-
valued approach in which each sentence is either true or false or
neither (sometimes "meaningless" or "neuter" is used here), you
may likely discover that the statement

	Every sentence is either true or false or neuter.

is neither true nor false.  You may discover that your truth
prediate is non-extensional.  Other problems can arise as well.
My point is that there will be *some* price to pay for solving
the paradoxes, and that price may be regarded as too high.

While I'm at it, I should describe the approach I took.  This
involved constructing a formal theory T* which embodies S5 as its
sentential logic and is otherwise a "free" logic (in that
denotationless terms are allowed).  The language contains a
quotation operator and a concatenation operator (suitably
axiomatized).  The quantifiers receive a substitution inter-
pretation.  Given this, the expressive power of T* is somewhat
greater than that of the standard first-order predicate
calculus.  A truth predicate for T* can be defined within T*
itself, and theorems are proved to show that this predicate
saatisfies criteria of adequacy for such predicates.  The
theory is a classically two-valued semantically closed theory
capable of defining its own notion of truth.

A full axiomatization and semantics for T* is provided and
Tarsk's Convention T is valid relative to that semantics.
Several familiar semantic paradoxes (including the Liar and
the Grelling-Nelson pardoxes) are formulated in the language
of T* and it is shown that no contradictions are
generated within the theory.  Soundness and completeness
proofs for T* are provided.

Late in the game Pollock discovered an error in the proof of
a fundamental lemma in the soundness proof.  The problem was
quite complex and involved the key definition of "representing
function" in the semantics.  It was clear to me how to avoid
the problem, but the truth predicate in the resulting theory
became non-extensional.

None of this was ever published.  I felt that I had failed in
achieving some of my original goals -- which included having an
extensional truth predicate and having all the semantic
paradoxes fail in a uniform and principled way.  As it turned
out (in order to repair the problem discovered by Pollock)
the truth predicate was not extensional.  And while the pardoxes
could be formulated and seen to fail (i.e., they weren't
pardoxes -- didn't result in contradictions), different
paradoxes tended to fail for different reasons.  While the
resulting theory certainly wasn't any goofier than what others
had done, it did not seem to me to offer any genuine advantages
either.  You win some; you lose some.












-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
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