From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Excluded Middle
Organization: Boise State University
Date: Fri, 6 Nov 1992 18:26:38 GMT

In article <1992Nov5.153919.27994@klaava.Helsinki.FI> amnell@klaava.Helsinki.FI (Marko Amnell) writes:
>Reading Hintikka's article "Defining Truth, the Whole Truth and
>Nothing but the Truth" today I was struck by the surprising role
>of _tertium non datur_ in the problem of defining truth inside
>a logic.  It seems that if one drops the Law of Excluded Middle
>as a principle in the construction of logic, it becomes _possible_
>to build a logic in which the set of all true sentences can be
>defined.  Gabriel Sandu claims to show this rigorously in a new
>paper "IF first-order logic and truth-definition".

How about the sentence "The truth value of this sentence is not
"true""?  However many other truth values there may be, the truth
value of this sentence will be "true" precisely if its truth value is
anything other than "true".  The thing which this paradox shows cannot
be defined is the function which takes a sentence to its truth value;
of course, the set of true sentences gives us this function if there
are two truth values and does not if there are more.  But the paradox
has not actually been evaded.


>
>What I wish to ask you all is your general opinion about the 
>_metaphysical_ status of the Law of Excluded Middle.  Reading
>Hintikka's article, I was struck by the dissimilarity of the
>role of this principle in the problem of Truth, when compared
>with its role in the metaphysical problem of Reality or
>Existence.
>
>In _The Logical Basis of Metaphysics_ Michael Dummett attempts to
>show that if one accepts _tertium non datur_ then the realist
>approach to physial objects and the existence of mathematical
>objects is supported. (This is an over-simplification, but bear
>with me, I have a point)  This suggested to me that if the Law
>of Excluded Middle supported Realism, then it would also be
>on the side of the existence of a set of all truths.

Not necessarily.  Not all collections are sets, as Russell's paradox
showed us; similarly, not all properties can be reified.

>
>But Hintikka shows that this is manifestly false.  Klaus Grue's
>work in CS also supports the same conclusion.  What we see is
>that if we maintain Excluded Middle, we are led to realism in
>metaphysics and philosophy of mathematics (what one might call
>`global realism') but when it comes to examining the problem
>of the definition of truth in logic, and the metaphysical
>problem of whether there is or is not a set of class of any other
>kind of collection of all truths _in the real world_ (and distinguished
>from within a formal system) maintaining Excluded Middle has
>a conclusion that seems antithetical to the aims or beliefs of the
>realist.

But observe that you cannot in any case have the function which sends
a sentence to its truth value, which is just as compelling an object
to the realist, no matter how many truth values there are.

>
>I find this to be a deep result.  As Von Wright would put it, the
>status of this one principle is a clue about more profound things
>in philosophy at large.  I am quite surprised at the starkness of
>the contrast in the consequences of dropping or maintaining 
>Excluded Middle in these two groups of philosophical problems:
>those surrouding truth and the whole truth, and those surrounding
>existence and the nature of the being of various kinds of
>objects -- sets, chairs, persons.

Again, it is NOT profound; what happens when you add more truth values
is that the set of all truths becomes an innocuous object, but there
is still an interesting object which cannot be defined for the same
reasons that truth could not be defined originally.  The profound
result remains the non-existence of Truth (or of the full truth-value
assignment scheme).

>
>I'm probably rambling a bit here.  But, in any case, what do you
>think about the Law of Excluded Middle?  Is it valid in all cases?
>Why?  How would one justify dropping it?  How does one decide
>in cases like this which philosophical principle to maintain and
>which to drop?

I think that Excluded Middle is true and that the existence of Truth
cannot be supported (as opposed to the existence of truths).  The
class of all classes that are not members of themselves is a pretty
appealing object, too.

>
>-- 
>Marko Amnell
>amnell@klaava.helsinki.fi
>Graduate Student in Philosophy

For Mikhail's benefit (since he has been waiting for me to say
something about truth), I think that truth is a property of sentences
in a language (and so Truth[L] is a set of sentences of the language
L).  The result I have in mind when I say that Truth does not exist is
the following: let L be the language of first-order logic with
equality, membership, and whatever other predicates you favor, and let
L be adequate to talk about arithmetic and "enough" set theory and
contain a constant for every object in the universe (names for
everything).  Whatever the "set of all truths" should be, it should
enable us to construct Truth[L], the set of all true sentences of L.
But Truth[L] can be neither a set nor even a definable proper class,
by an application of Tarski's paradox.  This works in NFU, in ZFC, and
in any other reasonable theory.  If you switch to multi-valued logic,
I invite you to consider the function which takes sentences to their
truth values, rather than the set of truths (which can then exist).

Now Truth[L] can exist for languages L which do not have names for all
objects (they will not, in particular, have names for Truth[L], if it
happens to be a set).  It can exist for languages L which do not
purport to provide all predicates (Truth[L] would be such a
predicate).  Truth can be saved by limiting the scope of reference.
But then one is no longer able to talk about all "truths".

Just as Russell's paradox does not show that non-self-membership does
not occur, this argument does not show that there are no truths.  But
there is no abstract object which captures Truth as a whole.  I
continue to maintain that statements are either true or false, if they
are well-defined.  "Not well-defined" is not a truth value; it calls
upon one to withdraw the question.

Notice that the language L with names for all objects in the universe
_can_ exist as a definable proper class (in ZFC) or as a full-fledged
set (in NFU); this object can be thought of as the collection of all
"propositions".  The collection of all propositions appears not to be
paradoxical, but the set of all true propositions, or, more generally,
the function which assigns truth-values to propositions, is
paradoxical.

Truth lurks in the Absolute Infinite.

-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Xref: sparky sci.philosophy.meta:2519 sci.logic:1993