Gaps in the truth about truth according to Leon Horsten?

I very much like Leon Horsten’s The Tarskian Turn as an attempt to make more widely available some recent work on formal axiomatic theories of truth,  without scaring readers off by excessive technicality or by unnecessarily spelling out tricky proofs. Students doing an advanced course on the notion of truth really ought to know the headlines about this  formal stuff, for a reason Tim Williamson has stressed:

One clear lesson [of these logical investigations] is that claims about truth need to be formulated with extreme precision, not out of kneejerk pedantry but because in practice correct general claims about truth often turn out to differ so subtly from provably incorrect claims that arguing in impressionistic terms is a hopelessly unreliable method.

So three cheers for Horsten’s efforts at lucid and lively explanation here. But his book is more than a merely expository essay. It is philosophically opinionated, takes sides, and even is prepared to endorse one particular axiomatic theory of truth as philosophically in good shape. This makes The Tarskian Turn engagingly provocative.

As is common in these sorts of investigations, Horsten takes as a test case the enterprise of adding a theory of truth to first-order Peano Arithmetc, PA. And the theory he recommends is PKF (that’s Feferman’s axiomatization of Kripke’s three-valued model, but redone in a Partial, gappy, logic). Roughly the idea is that you augment the language of PA with a truth-predicate T, take a rule version of Kleene’s strong logic, add the non-induction axioms of PA and the rule form of induction. Then we have the T-biconditionals of atomic sentences of PA, plus two-way rules that allow us to commute T with the logical operators: so, in Horsten’s sloppy but readable shorthand, we can infer $latex \neg T(\varphi)$ from $latex T(\neg\varphi)$ and vice versa, and similarly for other operators. Finally we have a two-way rule that allows us to infer $latex T(T(\varphi))$ from $latex T(\varphi)$ and vice versa.

The T-rules here are entirely natural, so PKF has all kinds of nice features. In particular, it is easily seen that everything remains classical for T-free sentences, and so classical PA can proceed undisturbed. So it is only some sentences involving T for which PKF’s non-classicality really matters and where e.g. excluded middle fails (where indeed we might want it to fail). Horsten thinks PKF is a best buy. Is it?

In his much more expansive, much more technically detailed book Axiomatic Theories of Truth Volker Halbach also investigates  PKF but is much less enthusiastic about it. Partly this is because of a general resistance to the idea of departing from classical logic if that can be avoided. And partly this is because of technical observations about the mathematical limitations of PKF (it can’t do much transfinite induction on open wffs involving the T-predicate). However, I don’t see why Horsten need be moved by these considerations. After all, logical regimentation always involves trade-offs of costs and benefits: and Horsten will say that given pure arithmetic remains classical, the cost of having to go non-classical in ones background logic (in ways that only really matter in cases that involve troublesome uses of the T-predicate) is a price worth paying for the benefit of skirting round paradox. And if that leads us to restrict transfinite induction in cases of no ordinary mathematical interest, why care?

But there are other worries about PKF. As Horsten notes, it isn’t that PKF rejects excluded middle in the troublesome cases, but rather it is silent. But is silence really what we want from a theory here? Horsten cheerfully says

The system PKF … is not vulnerable to a strengthened liar attack because it makes no claim concerning the truth value of the liar sentence. PKF simply does not assert the liar sentence, nor its negation, nor that it is true, nor that it is not true.

Indeed. But now the theory is out there, on the table for all to see, can’t we as philosophers stand back and reflect on it, and forcefully raise the question of the truth or otherwise of claims on which PKF fails to give a verdict? And off we go again … (To be sure there are philosophical positions where refusal to affirm or deny a putative proposition P is backed up with therapy that is supposed to massage away our temptation to suppose that P does express a contentful proposition. But Horsten is the business of theory not therapy.)

Horsten himself, then, is remarkably silent on what seems to be an obvious question. Instead he concludes his book by considering whether PKF is at least consonant with a broadly deflationist or minimalist stance. He thinks it is. For the theory treats truth as an insubstantial property without ‘a fixed nature or essence’ in the sense that there is no more to truth than is grasped in grasping some inference rules (though Horsten in fact  doesn’t rule out there being inference rules beyond those codified in PKF). But what about the fact that the compositional theory is non-conservative over arithmetic? Indeed PKF is arithmetically as strong as a transfinitely ramified system of predicative analysis that goes by the label $latex ACA_{\omega^\omega}$, whose first-order arithmetical consequences go far beyond those of PA.

Horsten is remarkably unworried. I think he is too swayed by a (quoted) claim of Feferman’s that suggests that systems of predicative analysis only elaborate commitments that are already implicit in accepting PA, a suggestion that runs clean against Isaacson’s well-known thesis (which I’ve defended elsewhere) that PA marks the natural boundary of those truths that can be reached by purely arithmetical reasoning. We can’t examine who is right about that here. But we might complain that neither does Horsten: he fails to acknowledge just how contentious it is to suppose that the progression through systems of predicative analysis stronger than the arithmetically conservative system $latex ACA_0$ can somehow be regarded as insubstantial rather than as involving new infinitary ideas (and so it remains equally contentious to suppose that a theory of truth arithmetically equivalent to a strong system of analysis can still count as deflationary). We might well dissent at the end of the book, then, about Horsten’s philosophical assessment of the merits of PKF.

Still, I think we should still be very grateful for a beautifully structured guided tour, with thought-provoking commentary, making some recent formal work on truth accessible to a wide student audience interested in the truth about truth (and accessible as well as to non-expert colleagues who want to know what the logicians down the corridor have been up to).