Friday, October 05, 2007

ACA0, #5: An aside on PA + Th versus T(PA)

Suppose we bolt onto to first-order PA the axioms of Th, the arithmetization of a natural theory of a new truth-predicate T, in a way that – prima facie – shouldn’t upset even a deflationist/minimalist about truth. As we'd expect, the composite theory PA + Th proves each biconditional T("φ") ↔ φ for sentences φ. And as we’d also expect -- though it isn’t entirely trivial to demonstrate it -- PA + Th is conservative over PA: it proves no new arithmetical truths.

But now let’s reflect. Bolting the arithmetical truth-theory onto PA ‘externally’ leaves us, in particular, with just the same induction axioms as we started off with. However, you might say, why stop there? It would seem that the line of argument that I sketched a few posts ago for being generous with induction will apply again, and will motivate extending the induction axioms from instances involving just the original L-predicates (L is the language of PA) to instances involving the new predicate T too. So let T(PA) be the theory we get taking PA plus Th plus the closures of all instances of the first-order induction schema for new predicates constructed from T as well. Then, the argument seems to be, T(PA) should be as compelling a theory as PA + Th. But as is well known, T(PA) is not conservative over PA (it proves Con(PA) for a start).

However, the inflationary argument for generosity with induction can be resisted. But how exactly? A deflationist might be tempted to say: ‘As a deflationist, I don’t accept that truth is a genuine property: more precisely, ‘T’ in the theory Th doesn’t express a genuine property, so we can’t use it in inductive arguments.’ But this isn’t in fact terribly helpful, unless augmented by an independent account of the initially murky notion of ‘not expressing a genuine property’. So let’s proceed more carefully.

Suppose someone with a taste for formalizing his knowledge – call him Kurt – accepts PA (this is the apparatus he uses in fixing his arithmetical beliefs). Suppose we now offer him the axioms Th as a partial characterization of the uninterpreted new predicate T. If we give Kurt any particular number which happens to be the Gödel number of a sentence φ then he will in principle be able to prove the corresponding theorem T("φ") ↔ φ. What he can’t do, since he doesn’t yet have induction axioms for T, is prove anything general about T to the effect that, for every n, if n = "φ" for some φ, then T(n) ↔ φ, or else T(n) ↔ ⊥. So, while still working from inside PA + Th, Kurt has no way of knowing whether T(n) has been defined for all numbers n. In this sense, then, Kurt doesn’t yet know whether T expresses a determinate property of numbers. So, for a start, he isn’t entitled to employ universal quantifier introduction applied to complex expressions involving T. Hence, he won’t be in a position to establish the quantified antedecent needed to make use of an instance of the induction scheme involving a predicate embedding T. In other words, Kurt is not entitled to make any use of the extended instances of induction allowed in T(PA). In sum, a suitably cautious Kurt – so far – is in no position to inductively inflate PA + Th.

And now there’s an added wrinkle. For we can see in retrospect that talk of the theory PA + Th in fact glosses over an issue that matters. In bolting the axioms Th onto the theory PA, were we intending these new axioms to interact merely with sufficient logical rules governing the elimination of quantifiers and the use of conditionals to enable the extraction of the information packaged in those axioms? Or were we intending that the whole weight of first-order logic can be brought to bear on axioms from either pool – so that we can trivially prove, for example, ∀x(T(x) ∨ ¬T(x))? The issue doesn’t arise for practical purposes. But now the question has been raised, we see that the second alternative overgenerates in enabling us to deduce more than we are entitled to in just being given the axioms Th as (partially) characterizing the new predicate T. A cautious Kurt should only use quantifier elimination and the conditional rules on Th.

Note, it is not being suggested that Kurt reject the instances of the induction schema that embed the predicate T as false. How can he? As the argument for inductive generosity reminds us, if the antecedents of such an instance are true, the consequent has to be true too. Rather, as we said, the point is that Kurt so far doesn’t find himself entitled to get to the starting line for using such an instance.

Of course, Kurt can now start to ‘think outside the box’. He can stand back from PA, think about his practice, commit himself explicitly to the thought that every sentence of L is either true or false, reflect this this thought using an arithmetized truth-predicate which he takes to be fully defined, so induction must apply to it -- and so he comes to endorse T(PA). We certainly don’t want to ban Kurt from such reflections or suppose that he must make some mistake if he takes on these further thoughts, and so comes to be able to demonstrate Con(PA), at least to his satisfaction. The point to emphasize is only that they are further thoughts, not commitments already implicity accepted in rationally endorsing PA in the first place. (Cf. Isaacson's Thesis.)

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