From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.philosophy.meta
Subject: Re: Excluded Middle
Date: 6 Nov 92 17:18:11 GMT
Organization: University of Helsinki

In article <TORKEL.92Nov6160230@isis.sics.se> torkel@sics.se 
(Torkel Franzen) writes:

>  I'm not asking for any opinion on the status of tautologies. I am
>just asking if you can explain what you yourself are saying. In
>ordinary discourse, a logically equivalent formulation of a statement
>conveys the same information. In your discourse this is not the case,
>since apparently "the conditions obtain or do not obtain" cannot be
>replaced by e.g. the logically equivalent "if the conditions obtain then
>they obtain".  Hence you are making some special use of the tautology.
>I am asking you if it is at all possible to explain what you mean by
>this special use.

OK.  What I have in mind is the relativist argument that there is no
such thing as "the way things really" are.  Margolis makes this attack
on realism.  If there is no objective, mind-independent reality, then
it makes no sense to ask of a certain proposition, is it a fact or is
it not.  This is the point where the Law of Excluded Middle comes in.
It says that every possible proposition is true or false.  It follows
that in every instance, it _does_ make sense to ask whether a certain
proposition describes some fact in the real world, or does not.

The picture is very simple now.  If there is a collection of all true
propositions, then what the Law of Excluded Middle does is say that
every possible proposition either belongs to this collection or it 
does not.  If it does, it is true, if it does not, it is false.
While if there is no collection of all truths, the relativist can say
that it makes no sense to ask whether a certain proposition is
really true ie. does it belong to the collection of all truth?
All we say that it is "true from us" or "true for them".

So, what I'm setting up here is a parallel between the realism/relativism
issue and the Truth exists/Truth does not exist issue.  It would be very
nice if we could get these two issues to stand or fall together, if we
could just build a logic in which we can rigorously show that there is
a collection of all truths, and if this achievement then meant that we
had refuted the skeptic or relativist.  You see what I mean?  It is an
appealing picture.  Maybe it's not a good picture.  But there it is.

What I'm trying to do in a sense is to reduce metaphysics to logic.
To reduce the problem of existence to the problem of whether there is
a collection of all truths about the world or not.  There are a vast
number of different arguments for the conclusion that one can never
have a collection of all truths in a logic.  Tarski's Theorem says that
no nice logic can pick out all its true sentences.  If we change the
principles of the construction of logic, the truth-set can be defined.

But, given the correctness of this metaphilosophical picture, and
assuming that Hintikka's IF logic, eg. can actually define its own
truth-set, and assuming that he is right in claiming that IF 
corresponds in some way to the logic of our actual every-day language,
then we must ask ourselves:  Is IF an acceptable logic?  One of the
things that happen in this logic is that the Law of Excluded Middle
Fails.  This follows in a natural way from the altered nature of the
quantifiers in IF (the scope rules are changed).

But, he has achieved his results by violating the Law of Excluded Middle.
Someone who holds that this tautology ought to apply to all logics can
reject his results.  But how does he justify this rejection?  How does
anyone justify the rules of logic, rules of deduction?  We set up a
formal system and hope that it does some things for us.  It is another
mattter entirely to say that building such a logic shows something
about the real world.  But this is what Hintikka suggests.

To suggest this is to call for some kind of non-circular justification
of logical laws.  Can this be done?  Dummmett tries to do just that.
I can't give his analysis off the top of my head, although I've read
his book twice.  What it boils down to in his opinion is what basic
form our meaning-theory is to take, and in particular, what is the
meaning of the logical constants.  In classical logic their meaning
differs from that in intuitionism or quantum logic.  And this
difference in meaning results in different logical laws.

So, the question of the validity of Hintikka's system would,
according to Dummett, turn on the meaning of the symbols involved,
the logical constants and in particular the quantifiers.  There is
an underhanded trick in the way Hintikka introduces branchin
quantifiers in his paper.  He merely states that it is a prejudice of
logicians that quantifiers should have the scope rules they have.
But this is absurd.  The quantifiers have the scope rules they have
because the resulting logic is well-understood.  To change them is
to change the whole ball-game.  IF is so different from classical
logic that it is fair to ask is Hintikka talking about truth in
the classical sense at all.

We are faced with the question of the applicability of formal logic
to philosophical problems.  Assume certain axioms or principles and
certain results follow.  Assume others and things look totally
different.  But what does any of this formal manipulation have to
do with reality and truth in the real world any more.  One can make
the same kind of criticim about epistemic logic.  If the agent knows
the consequences of all his beliefs, what does this formal system
have to say about the actual beliefs of actual people?  If the
nature of the `beliefs' is so different, why even call it belief 
any more.

But, it is just too easy to say that formal logic has _nothing_
to do with real thinking.  I thought this for a long time, while
I studied mathematics and when I started to study philosophy.
Prof. Rota of MIT argues in an article "The Pernicious Influence
of Mathematics on Philosophy" that all of 20th century Anglo-Saxon
philosophy is the result of one big mistake.  The mistake of
assuming that formal logic describes how people really think.
But, can it be that simple?  Have we learned nothing about thought
by studying formal logic?  If not, why the stange similarity 
between implication in logic and the way we infer about facts in
the real world?

If mathematical logic is just another branch of mathematics and
has nothing to say about real thinking, a lot of philosophers
have been kidding themselves in a dumb way.  If we deny this,
how are we to show that formal logic has something so say?
What I have been describing, my Master's thesis topic, is one
exploration of the extent to which a question in logic, can
the collection of all truths be defined has bearing on a
problems in philosophy.  I really have no idea how valid all this
is.


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