Deflationism, Conservativity, Truth.

Posted on October 19, 2014 by

Over the summer I spent a lot of time learning about truth, and particularly deflationary theories of truth. In this post I will outline some of my thoughts about one particular criterion that is often argued for (and against) by various players in the recent literature on deflationism about truth: conservativity. The plan is to write a paper about these issues sometime in the (hopefully not to distant) future, and I would be happy to hear peoples thoughts.

It is notoriously difficult to say exactly what deflationism about truth amounts to, to the point where there is controversy about whether particular theories of truth count as deflationist or not. Deflationism is often just characterized via slogans like “there is no more to truth than its role as a devise of disquotation/generalization/inference etc,” or “truth is metaphysically insubstantial.” What I am concerned with here is whether (formal) theories of truth need to be conservative over whatever base theory they are formulated over to count as deflationist.

To be a conservative extension of a theory means roughly that adding a theory (of truth) to a base (mathematical theory) does not allow you to prove any new theorems of the base theory. One familiar example from outside of the truth debate is von Neumann-Goedel-Bernays class theory (NGB), which is conservative over ZFC. The problem in the case of truth is that it doesn’t seem right to say that truth is insubstantial if it has non-trivial mathematical consequences.

With preliminaries out of the way, here are my thoughts on the issue. First of all, it seems to me that one important goal of the philosophical debates about the nature and behaviour of truth is to work towards a theory that will work for the truth predicate as it appears in natural language. In my opinion, one reason we tend to study (especially axiomatic) theories of truth formulated over mathematical theories, and in particular Peano arithmetic (PA), is because such theories are well understood and well behaved. The question then is, should we expect (non-) conservativity to carry over from PA to English (say)? I think the answer to this is probably no. After all, what would it mean for a theory of truth to be non-conservative over a natural language like English? For that matter, how could we even tell? –natural languages, unlike many mathematical theories, do not have straightforward deductive closures or completeness properties, in fact we have good reason to believe that English is not first-orderizable, which suggests that English probably behaves more like a second-order theory.

Furthermore, there are some more formal considerations that suggest that as we move to stronger theories, non-conservativity decreases, in other words, more theories of truth are conservative over stronger base theories. In one direction, theories of truth that are conservative over PA will still not be conservative over classical quantified first-order predicate logic, because any theory of truth will prove that there are at least two (non-syntactic) things.

In the other direction, recent work by Kentaro Fujimoto on theories of truth formulated over set theory, suggests that (given the conservativity of NGB over ZFC) some theories of truth that are not conservative over PA will be conservative, or at least more conservative, over set theory.

All of this is to say that deflationists shouldn’t worry about conservativity over PA, or even ZFC, when working out their theories of truth, as it will likely turn out that non-conservativity results will not carry over to natural language. Of course, if we are only worried about mathematical truth, then we might want our deflationary theories of truth to be conservative, though only if we don’t expect mathematical truth to correspond to truth simpliciter.

Disclaimer: Although I know a fair amount about this subject, this is all off the cuff so to speak, and so should be taken with a grain of salt. That said, I stand by the general ideas expressed above (at least until someone convinces me otherwise).