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In article <26505@cca.CCA.COM>, g-rh@cca (Richard Harter) writes:

[but first, a word from our sponsor:]

>>_The Liar: An Essay in Truth and Circularity_, by Jon Barwise and
>>John Etchemendy, Oxford University Press, 1987.  It only costs $20;
>>its ISBN is 0-19-505072-X.  Buy this book.

>Re the example of claire and the three of clubs.  It sounds at first
>hearing that the difference [is merely adding contextual indentifying
>clauses, which isn't interesting; it can't be that simple.]

You are correct, it isn't that simple.  For example, you can't "add" con-
text to a *self-referential* sentence.

>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
>or false.

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.

Kripke first made concrete part of this feeling with the following example:

 (A)	Most of Nixon's assertions about Watergate are false.
 (B)	Everything Jones says about Watergate is true.

Pretty harmless looking, no?  But what if A is Jones' *only* assertion
about Watergate, and B was asserted by Nixon, who also said, coinciden-
tally, 2k other things about Watergate, k of them clearly true, and k
of them just as clearly false?  As Kripke said, "there can be no syn-
tactic or semantic `sieve' that will winnow out the `bad' cases while
preserving the `good' ones."

>	    However we can classify statements as reducible (founded)
>or irreducible (unfounded).  Reducible statements can be resolved in
>terms of true and false.  True and false do not directly apply to
>irreducible statements; instead the relevant question is "Can truth
>or falsity be consistently assigned to the statement".  There are
>four possibilities (both T and F, T only, F only, and neither.)

What about

 (L*)	 "This sentence is either false or irreducible."  ?

If true, then it isn't false, and it isn't irreducible.  Nope.  If false,
then true.  Nope.  So it's irreducible, hence true.  Whoops.

>An immediate corollary is that it is not permissable to quantify over
>irreducible statements in the usual form of truth and falsity, because
>true and false are not correct categories for irreducible statements.

So how do you *identify* if a statement is reducible or not?  You need
to know *before* you attempt to permissibly analyze a statement, accord-
ing to your above comments, and yet the reducibility may hinge on the
internals of the sentence.

Unless you come up with an *internal* categorization of reducible/irredu-
cible statements--and Kripke's example or variants thereof makes this very
unlikely--any such theory will always be vulnerable to just strengthened
liars.  You as might as well stick to identifying truth.

>Whether this in fact works, is quite another matter.  But, if it does,
>it seems satisfactory to me.

Ignoring the L* problem, it is unsatisfactory in that it doesn't *explain*
anything.  Just saying, "paradox go home" doesn't mean anything.  Indeed,
rejecting "vicious circles" on the grounds that they are "vicious circles"
is--you guessed it--a viciously circular argument.  In contrast, and this
was the point of view that bugged so many people when I raised it so many
months ago, *justifying* vicious circles in a circular manner is self-con-
sistent.

And that is the beauty of B&E's book.  They give an intuitive explanation
for what happens with these self-referential sentences.  No hokey rules
about what is allowed or isn't.  As an example of a non-hokey rule, con-
sider the following pseudoproposition p:

	p := Max has the 3 of clubs OR p.

Its negation is q:

	q := Max doesn't have the 3 of clubs AND q.

It works out that *both* are true under their natural truth predicates.
But the following self-reference *is* acceptable:

	p := Max has the 3 of clubs OR True(p).

Its negation is q:

	q := Max doesn't have the 3 of clubs AND False(p).

B&E give a simple *internal* grounding rule that eliminates the former
pair but not the latter.  And they point out how natural this rule is by
comparing the situation with English:  "Max has the 3 of clubs or this
sentence is true", is something we try to analyze, but "Max has the 3 of
clubs or this sentence", is ungrammatical.

In contrast, a reducible/irreducible criterion is an *external* distinc-
tion, and cannot be justified before the fact.

ucbvax!garnet!weemba	Matthew P Wiener/Brahms Gang/Berkeley CA 94720