I Kripke's fixed point for truth predicate: justification?

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nomadreid

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What justifies saying that a fixed point for Kripke's process exists?
If I understand correctly (dubious), given a consistent theory C (collection of sentences), Kripke proposes to add a predicate T so that, if K = the collection of all sentences T("S") for every sentence S in C, ("." being some appropriate coding) then the closure of K∪C forms a new theory C*; one reiterates this until a fixed point is reached, Cn* =C(n+1)*. Then T is suitable as a truth predicate for this final theory.
Two questions:
(a) is this a proper statement of Kripke's truth predicate? If not, please correct.
(b) If so, what justifies the statement that such a fixed point exists?
 

andrewkirk

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I don't think he is asserting that a fixed point exists. I think he is saying that IF a fixed point exists THEN T is suitable as a truth predicate. It may be that for many theories there is no fixed point, in which case it follows that there is no suitable truth predicate for that theory.

I assume the closure referred to is closure under deduction, so that any sentence that can be deduced from K union C is in the closure. Under that operation, the closure of a consistent theory should be consistent.

Although a fixed point may not be reached by any finite number of iterations, we can define a theory C* that is the union of all finite iterates. That theory will be a fixed point, but we lose the guarantee of consistency. Based on Godel's work on incompleteness, my guess is that, for any theory C that can express Peano arithmetic, no finite iterate will be a fixed point, and C* will be an inconsistent theory.
 
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nomadreid

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Thanks, andrewkirk, but my impression is the following: by using a three-value logic, his resulting fixed point exists even for a consistent theory containing PA, but there will be some sentences for which the truth predicate will give the third value: that is, the truth predicate will identify all the statements that are capable of being identified. So, if G is Gödel's sentence, and T is Kripke's predicate T with codomain {t,f,n}, then T(G) = n, not contradicting Tarski's theorem on the indefinability of truth. This doesn't seem to solve the Liar, but what I am looking for here is a rough explanation of the justification for his fixed point theorem.... say, how Zorn's Lemma or something might be applied.
 

andrewkirk

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OK, you didn't say you were using three-valued logic. In that setting there's no obstacle. You can find a formal presentation of the necessary definitions and a proof of the existence of a fixed point here. Theorem 4.5 is the Fixed Point Theorem.
 

nomadreid

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Thanks, andrewkirk. That answers the question. :smile:
 

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