From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta
Subject: Re: Excluded Middle (Long)
Date: 5 Nov 92 16:53:09 GMT
Organization: University of Helsinki

A bit of elaboration.  The reason why a realist might want to
believe that there is a set of all truths is simple.  He believes
that there is such a thing as the Whole Truth about the world.
This is an objective, mind-independent entity, and the aim of
science is to approach this object, to have our theories match
the structure of this metaphysical object.  This is a gross
caricature of the various realist positions, but it serves my
purpose.

So, if Dummett is right that Excluded Middle is necessary for
metaphysical realism, and my caricature above holds in broad
outlines, and one cannot define a set of all truths in a logic
without dropping Excluded Middle (such as in Hintikka's IF system,
which he claims matches the deep structure of natural language and
hence the way we think) then what follows is a very interesting dilemma.

If one holds Excluded Middle, one can maintain Popper-style 
realism about the world (and Platonism about mathematical
reality) but this position is itself undermined by sticking
to Excluded Middle, for the only way that one could really
define a set of all truths explicitly is to drop the Law of
Excluded Middle.

On the other horn of dilemma we have the metalogical fact that
if one drops Excluded Middle as a basic principle in the 
construction of logic, one can define a set of all truths,
eg. Hintikka and Sandu's IF system, or Grue's Map Theory.
But, if one drops Excluded Middle, and Dummett's meta-philosophical
analysis is right, then one cannot at the same time be a
realist about physical objects or mathematical reality.
One is led to eg. Berkeleyan idealism and intuitionism.

What is one to make of this, if my analysis is indeed correct?
Well, one cannot help being struct by the _irony_ of the
dilemma (and by the fact that a dilemma is only a dilemma if
Excluded Middle holds!!).  Apart from this, it all looks pretty
confusing.  Are we to say that there is a set of all truths in
a metaphysical sense, but since we are idealists and intuitionists,
there is no ultimate or objective reality, but only physical
objects constructed from sense data, and mathematical theorems
and axioms built from intuitions?  Or, are we to say that there
is an ultimate mathematical and metaphysical reality, but that
there is no set or class or other collection of all true
propositions correctly ascribed of it?

What would these two alternatives mean?  Neither seem to be very
appealing.  The first is simply a surrendering of the realist
ground and an embracing of Idealism.  The second is even more
problematical.  If there is an ultimate reality, why is there not
a collection of all propositions that are true of it?  Is this
because the very act of making an assertion can be repeated
indefinitely, and hence one could always `step back' from any
completed totality and assert something of the whole, which
would create a truth not included in the original totality?
This is essentially Peter Unger's argument against a Whole
Truth about the world.

Unger's third way is even more unappealing.  There is an objective
reality but we cannot know it because our way of thinking, the
logic of our language, prevents us from defining a set of all
truth about the world.  But, if we drop Excluded Middle and try
to build a logic that avoids this kind of radical skepticism,
then we run up against Dummett's claims that Excluded Middle
is necessary to have realism at all.  Popperian philosphy
of science might survive, since there is no directly appeal
to Excluded Middle in theories of verisimilitude or notions
of truthlikeness.

One possibility does suggest itself.  One could say that there is
a set of all truths about the world and Unger is right that our
limited understanding cannot grasp it, but it exists nonetheless.
No realist logic that maintains Excluded Middle can define its
own truth-set since a realist logic that could do this would
grasp the Whole Truth about the world in its totality.  But, no
logic can do this as Tarski showed.  If one relaxes the implicit
critaria that Tarski's Truth Theorem imposes on the features of
the logic, then one can try to define the truth-set within a
logic (one can do this in other logics in stronger systems).

Now, since such a logic drops Exluced Middle in order to define
Truth, and since dropping Excluded Middle leads to anti-realism,
then one could say that Hintikka's or Grue's systems no longer
talk about the real world (or ultimate, noumenal reality)
but only about the world of experience (the phenomenal world,
the world described by our best scientific theories, and the
world as it appears to our minds) not the objective reality
itself.  This is an explanation of the apparently contradictory
status of Excluded Middle in the problems or reality and truth.

But, is it a satisfactory explanation?  Unger would say that if
our current categorial framework does not match the real world,
but only somehow approximates it, then we have absolutely no
guarantee that the way we think the world is is even close to
the way it really is (in-itself, as Kant would say).  This 
position sounds crazy, but it does point out the fact that no
notion of verisimilitude has even really been shown to be more
than a nice picture (and few are still trying to carry out
this application of intensional logic).  Nevertheless, one can
argue on pragmatist grounds that the relative or approximate
truth of our world-view can be verified by experience.

One can say that if our world-view was totally false, and reality
were completely different (as Unger claims is possible) then
how could we successfully interact with the world?  The fact that
we are able to avoid walking into walls, and that we can build
a rocket and fly to the moon, these feats show that the way we
see the world is close in some sense to the way things really are.
We could be wrong here and there, just as the Ptolemaic astronomy
was shown to be incorrect, but again, the fact that we could
_see_ that it was incorrect defeats the radical sceptic.  We
have a grasp on reality, but the picture of the world is always
incomplete, only an approximation of the objective and complete
reality, which may indeed be too complex for a human being to
ever grasp in all its detail.
-- 
Marko Amnell
amnell@klaava.helsinki.fi
Graduate Student in Philosophy
Newsgroups: sci.philosophy.meta