[FOM] Defined truth

[Sandy Hodges]

I wonder if anyone can direct me to references concerning defined truth
predicates.   I have a couple of references (Taisuke Sato, e.g.) on the
logic programming side, but none on the philosophical side.  The sort of
thing I'm thinking about is given in this example, in which I have made
"Tarski" and "Skolem" characters:

Skolem speaks a sentence; Tarski does not know what Skolem said, but
thinks he said something true.    But the language Skolem and Tarski are
using does not contain a truth predicate which applies to its own
sentences.    What can Tarski say?    Suppose the language does have a
"Spoke" relation, which holds between speakers and the Gödel numbers of
the sentences they have spoken, and a "Denotes" relation which holds
between the Gödel numbers of terms (noun phrases) and the items the
terms designate.

With these, Tarski can express the truth of Skolem's unknown sentence,
provided he can place an upper bound on the logical complexity of the
sentence.    Suppose for example he knows that Skolem's sentence, in
prenex normal form, has only existential quantifiers.   Then Tarski
says:

There exists a number, S, a list of numbers, T, a list of things, D, and
a set of ordered triples of numbers, M, such that:

Skolem Spoke S, and

for each k, either T(k) is the Gödel number of a variable, or T(k)
Denotes D(k), and

for any ordered triple <d, i, j> in M, [
    if d=2, then ( D(i) = D(j) )  &
    if d=3, then ~ ( D(i) = D(j) ) &
    if d=4, then ( D(i) e D(j) )  &
    if d=5, then ~ ( D(i) e D(j) ) &
    if d=6, then ( D(i) Spoke D(j) )  &
    if d=7, then ~ ( D(i) Spoke D(j) ) &
    if d=8, then ( D(i) Denotes D(j) )  &
    if d=9, then ~ ( D(i) Denotes D(j) )
], and

the number S is related to the list T and set M as follows: put S into
prenex normal form.   There is some disjunct such that every term that
occurs in that disjunct is in T, and for each atomic sentence Q(A R B)
in that disjunct, <d+o, i, j> is in M, where i and j are such that A is
T(i) and B is T(j), and
   d is 2 if R is "=",
   d is 4 if R is "e",
   d is 6 if R is "Spoke",
   d is 8 if R is "Denotes",
and o is 1 if Q is "~", 0 if Q is absent.
----------
By saying this, Tarski has said something which implies and is implied
by the truth of what Skolem said, whatever it was, provided it was of no
more than the indicated complexity.

This defined truth predicate (predicates, actually, one for each
complexity level) does not produce a paradox, because whatever upper
bound Tarski puts on the complexity of Skolem's sentence, the sentence
Tarski then speaks to call Skolem's sentence true, is of higher
complexity.   Thus if Skolem had tried to say that what Tarski says is
not true, then one of the two sentences refers to a sentence of
complexity no lower than itself, and thus fails to imply what it seeks
to imply.

Thus the paradoxes are blocked, as they always are, by a hierarchy, but
this hierarchy is of complexity.

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