[FOM] Another easy solution does not work

[Sandy Hodges]

Harvey Friedman:
> In particular, there is a closed term t such that notT(t) has godel
number |t|.


Some example sentences:
1.  Sentence 2 is true.
2.  Sentence 1 is false or is paradoxical.
3.  Sentence 1 is false or is paradoxical.
4.  Sentence 1 is false.
5.  Sentence 1 is paradoxical.

I am a token-relativist, so I think sentences 2 and 3 can have different
truth values (2 is paradoxical and 3 is true).   Sentences 2 and 3 must
have the same Gödel number, however.    Any project to define a
predicate True(x) as having as its extension some set of Gödel numbers,
is not a token-relative approach.    Approaches that are not
token-relative, sometimes have difficulty calling their own conclusions,
true.    But there is one non-token-relative approach that can call its
conclusions true, a proposal by Gupta and Martin in 1984.    This
proposal requires that the semantic predicates "True" and "False" be
treated as 'weak'.   This means that if x is the name of a paradoxical
proposition, then "True(x)" and "False(x)" are both paradoxical.   This
contrasts with 'strong' treatment of the semantic predicates, which
calls both "True(x)" and "False(x)" false, when x names a paradoxical
proposition.

Since "True(x)" and "False(x)" can be paradoxical, 3-valued truth tables
are needed to find the truth value of compound formulas.    Gupta and
Belnap proposed using weak Kleene tables.  With weak Kleene, (A and B)
and (A or B) are both paradoxical, if either A or B is paradoxical.
Their proposal would call sentences 1 and 2 paradoxical.    Sentence 5,
expressing this conclusion, is true.    Sentence 4, however, is
paradoxical.    And sentence 3 is paradoxical, since its right disjunct
is paradoxical.   Thus sentences 2 and 3 have the same truth value.
Any non-token-relative approach that wants to call its own conclusions
true must use both weak predicates and weak Kleene, I think, otherwise
it will be unable to call sentence 3 paradoxical, when it calls sentence
5 true.

So any approach that looks for sets of Gödel numbers as the extensions
of "True" and "False", if it wants to call its conclusions true, must
use weak truth tables, I think.   And it must have a predicate
"Paradoxical," since even though each sentence is either true, false, or
paradoxical, "~True(x) & ~False(x)" can't take the place of
"Paradoxical(x)" when True and False are treated as weak.

( None of this is to say that Gödel numbers play no role in
token-relative approaches. )
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Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.