🚀 go-pugleaf

	This is one of several articles in response to Matthew's comments.
In this article I clarify (?) my remarks on the liars paradox and reducibility
and irreducibility.

In article <8307@agate.BERKELEY.EDU> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes:

>>Re the liars paradox.  The following (in loose form) seems satisfactory
>>to me:  We make a distinction between statements and data.  Statments
>>about data are either true or false.  We can also make statements about
>>statements.  Statements about statements do not have to be either true

>That this is unsatisfactory has been felt for a long time by many.  Our
>intuition very strongly says that statements about statements "refer",
>and it is only after some analysis that we discover limits on this.  We
>can say, so much for intuition, but no one has ever made good, convinc-
>ing *models* for these self-referential sentences in the first place be-
>fore, with or without funny logics.


	Well, my intuition doesn't say this at all, but that's probably
the fault of my intuition.  Let me see if I can make this clear.  Suppose
I have a rather long list of statements.  Suppose I want to go through
the list and determine which ones are true and which are false.  What
happens?  I find that there are statements about data which are true or
false and can be marked accordingly. I winnow out these.  Call this set
of statements S_0.  Then I notice that there are statements that refer
to statements in S_0.  I can unambiguously mark these as true or false.
And so on.  After I have marked S_0, S_1, and so on I may have some left.
These I call irreducible.

	Paranthetically, I'm using the term irreducible, because they 
don't have to form a closed referential loop, e.g.

(1)	Statement 2 is true.
(2)	Statement 3 is true.
	.....

Now, it's possible that I can tease out the truth of these statements in
the manner of a logic puzzle.  I.e. I would mark statement x as true and
then see what other statements would then be true and vice versa, mark it
as false and see what happens.  Now if it happened to be the case that
all statements could be assigned truth values in this manner, I might be
satisfied.

In fact, I can't.  There is no unique assignment.  Even if I could, however,
I am actually doing two different things.  The truth value of the reducible
statements follows directly from the data.  In the case of the irreducible
statements I am really saying "The only consistent assignment is to mark X
as true, etc."  I am not really saying that X is true;  I am only saying that
it will be consistent the rest if it is treated as being true.  If this be
truth it is a different kind of truth.

Now it simply doesn't bother me that there is no unique list of assignments,
that there is no unique model that fits all cases.  What I can actually DO
is to say and determine things such as

"If I mark statement X as being true then it is consistent with all of the
reducible statements, but it is not if I mark it false."

and

"I cannot consistently mark statement X as being either true or false."

Now these are statements of fact -- I can either do the markings as claimed
or not.  These statements have truth value.  In short, irreducible statements
have no intrinsic truth value at all.  This is not the same as saying that
they are meaningless or that they have no referents; it is simply saying that
they don't (despite appearances) fall into the category of things with truth
values.  It is the statements about how they can be marked that have truth
value.
-- 

In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.
	Richard Harter, SMDS  Inc.