[FOM] Re: Defined truth (Joao Marcos)

[Sandy Hodges]

JM: By your usage of the expression "truth predicate" it seems to me you

are actually looking for "truth operators", or "truth connectives", is
that right?  If that is the case, you might be interested in having a
look at the following paper:

von Wright, Georg Henrik
Truth-logics.
Logique et Analyse, Nouv. Sér. 30, No.120, 311-334 (1987).
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SH:

thanks for the reference.
I'll get the paper, the more so since I remember something interesting
by von Wright.

What I'm looking for at the moment is ways to call a sentence true, in a
language that does not have a truth predicate at all, but does have a
"Denotes" relation.   Suppose you knew some unknown sentence A was of
the form "(X e Y)".    Sentence A might be, for example, the sentence on
a card named "CardOne."    Suppose "WrittenOn(x,c)" means that x is the
Gödel number of a sentence written on c.   Let "X /. Y" be the Gödel
number of the formula that is the concatenation of the formulas of which
X and Y are the gns.   Let E be the gn. of "e" (the set membership
relation), LP the gn. of "(", and RP the gn. of ")".  Then you could
say:

(E x,y,z,p,q)  ( WrittenOn(x,CardOne) &  x = ( LP /. y /. E /. z  /. RP
) &
Denotes(y,p) & Denotes(z,q) & (p e q) )

I can assert this formula even though I don't know what A is, and this
formula implies and is implied by A, whatever A is, provided A is indeed
of the form "(X e Y)".    I wondered if anything had been done to
explore the limits of what could be done using the Denotes relation
only, or if there was a known paradox based on such a Denotes relation.
(Or if on the other hand there was a consistency proof).

I have in mind a Denotes relation that is purely extensional, so that if
69 is the gn. of the null set sign, 69 also Denotes { x e N | x > 5 & x
< 5 }. Also,
(V x,y,z) ( ( Denotes(x,y) & Denotes(x,z) ) -> y = z ).

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Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.