Re: Kripke on truth predicate for Sigma_1 sentences

On Apr 4, 6:55 am, LauLuna <laureanol...@xxxxxxxx> wrote:
In his renowned 'Outline of a Theory of Truth' (The Journal of
Philosophy, 72, 1975, p. 697) Kripke writes:

"One surprise to me was the fact that the orthodox approach by no
means guarantees groundedness in the intuitive sense mentioned above.
The concept of truth for Sigma_1 arithmetical statements is itself
Sigma_1, and this fact can be used to construct statements of the form
of (3)."

Where '(3)' is a name for '(3) is true', an ungrounded sentence, and
the orthodox approach is the Tarskian hierarchical approach to the
Liar.

As I see it, Kripke says there is a Sigma_1 arithmetical sentence TT
saying (via arithmetization and diagonalization, I presume) that TT is
true.

Yes.

TT, like (3), seems to have no definite truth conditions,

It does have truth conditions. You go about checking whether it's true
in the same way as any other Sigma-1 sentence. The fact that a certain
Sigma-1 predicate is a truth predicate for Sigma-1 sentences gives you
a way of showing that the truth of a certain Sigma-1 sentence is
equivalent to the truth of another Sigma-1 sentence. In this case we
don't get any useful information this way because we just get the
triviality that the sentence is true if and only if it is true.
However, the sentence's truth-conditions are just as well-defined as
those of any other Sigma-1 sentence.

but
this would imply the concept of truth for Sigma_1 arithmetical
sentences being not well-defined. And this seems absurd since those
sentences can be interpreted as propositions about numbers.

So, please, what's going wrong here?

.

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