Desiderata for theories of truth

As is well known (but sometimes forgotten), in his seminal work on truth, Tarski proposed his condition T, now known as the T-schema, as a condition of material adequacy for any formal theory of truth. By this he meant that the T-schema was intended to capture the conceptual core of the antecedent, informal notion of truth as correspondence.

Since Tarski, and especially in recent decades (roughly since the 1970s), there has been an explosion of formal, axiomatic theories of truth proposed in the literature. Now, whenever there are multiple contestants, the question arises as to how one could determine which of the candidates is the correct theory of truth. (If truth is a uniquely determined concept, arguably there can only be one true theory of truth!) These theories can be compared with respect to their formal properties; but most of them were proposed by very able logicians, and are thus all formally/technically adequate and sophisticated. So the debates have to revolve around the extent to which the different theories capture the antecedent conceptual core of the notion of truth, which in turn requires the informal discussion of which properties a formal theory of truth should display in order to be ‘materially adequate’.

In an influential 2007 article, ‘What theories of truth should be like (but cannot be)’, H. Leitgeb discusses eight plausible desiderata for theories of truth, but notes that they cannot be jointly satisfied (taken together, they lead to inconsistency and triviality, and this happens even with some subsets of these eight conditions). Now, if the ideal of satisfying all these conditions at once cannot be realized, what can we do?

Just as in the case of moral dilemmas, if a set of prima facie norms is not satisfiable simultaneously, the next best option is to search for maximal subsets that can be satisfied. (Leitgeb 2007, 284)

He then identifies different subsets of these eight desiderata that can or cannot be satisfied by specific theories. However, there does not seem to be a unique 'peak of maximization' among the different candidate subsets, so the discussion will have to revolve around how to weight and compare the different informal desiderata – once again, by and large a conceptual affair.

Last Friday, I attended some of the talks of the second Amsterdam workshop on truth (here is my blog post prompted by the first installment, two years ago), and again many of these issues resurfaced, reminding me of Leitgeb’s article and of the whole debate on desiderata for theories of truth. For example, Philip Welch argued that revision theories of truth cannot be said to be about truth, properly speaking, in particular because the phenomenon they identify – ‘jump operators’ that are built up by quantifying over fixed points – is not unique to truth as such (it underpins other widely dissimilar hierarchies such as infinite-time Turing machines). Welch was relying on the idea that it is not sufficient for a theory of truth to satisfy some conceptual desiderata; it should also not count as a plausible account of something other than truth.

In the last talk of the workshop, Leon Horsten explicitly raised the question of what counts as a ‘good’ theory of truth, and discussed some desiderata other than those discussed in Leitgeb’s paper. Horsten noticed that it depends on what the proposed theory of truth is a theory of (of the truth predicate in natural language? Of the philosophical concept of truth?), and adopted the perspective of theories of truth specifically for meta-mathematics. He then went on to propose the following set of desiderata:

• Non-interpretability
• Conservativeness
• Speed-up 

This was all stage-setting to argue that there is at least one theory of truth in the market, namely a theory proposed by Martin Fischer (not coincidentally, his co-author in the paper!) in 2009, which satisfies all these constraints. Now, here again a discussion needs to be had on why these are indeed adequate desiderata for a formal theory of truth, and at Q&A I suggested to Horsten that this is essentially a conceptual matter. He replied that he and his co-author Fisher had debated the exact status of these desiderata, and that they were not sure that the issue belonged to a conceptual level.

So here follows a modest attempt to argue that the items on Horsten’s list are indeed conceptual in the sense that they are in the ballpark of Tarski’s notion of material conditions of adequacy. Naturally, whether a given formal theory does or does not satisfy these desiderata will be a purely technical, formal matter, but the criteria as such must be plausible at an informal, conceptual level. They represent bridges between the prior, informal realm and the formal realm of the theory.

As Horsten presented it, non-interpretability is a matter of expressiveness. The idea is that a truth theory T for meta-mathematics must be non-interpretable in the sense that there is not another theory which is not about truth, but on which T can be interpreted (in the technical sense investigated by e.g. Albert Visser and others). If that were the case, then T would not be specifically about truth, and this seems to me to be in the spirit of Welch’s objection to revision theories of truth. Although it is based on a highly technical notion of interpretability, the demand here seems to be that a theory of truth should be about truth and not something else.

Conservativeness is a property of axiomatic theories of truth which is usually formulated in proof-theoretical terms: a theory T which is obtained by adding a truth predicate to a theory S is conservative over S iff T cannot prove anything with the vocabulary of S (i.e. statements not involving the truth predicate) that could not be proved in S already. But Horsten proposes to think of conservativeness in model-theoretic terms: T is a (semantically) conservative extension of S if and only if all the models of S can be expanded to models of T. In other words, no models are ‘lost’ in the transition from S to T, since there are no models of S which are not models of T. The conceptual rationale for this desideratum is that a theory of truth for S should still be about the exact same ‘things’ that S itself is about – no more,  no less.

Initially, the discovery that some respectable axiomatic truth theories were conservative over their base theories was viewed as an argument in favor of deflationist conceptions of truth, in the sense that adding a truth predicate did not add anything of ‘substance’ to a theory. But it seems that the demand that a theory of truth be conservative goes well beyond deflationist concerns, and is sufficiently neutral so as to count as a condition of material adequacy for any theory of truth about a given topic/domain S, on a par with Tarski’s T-schema.

But conservativeness is not the whole story. It has also been known for a while that some theories of truth also display the phenomenon of speed-up, which is best understood as the idea that theorems which could be proved in S can then receive much shorter proofs in T. So while conservativeness seems to suggest that the truth predicate does not add anything of substance to the base theory, speed-up goes in the opposite direction: a truth predicate can make proofs significantly shorter.

Now, in what sense can we say that speed-up is a desideratum for a theory of truth? Again, it seems to me that there is a plausible conceptual story here: a truth predicate functions very much like a second-order quantifier, and so it seems that an increase in expressive power should be expected when a truth predicate is added to a theory S. In turn, increase in expressive power should allow for derivational ‘shortcuts’ in the new theory T with respect to S. So arguably, the more speed-up one observes, the more successful the theory has been in capturing the expressive role of the truth predicate (again, something that deflationists and non-deflationists alike agree on).

To conclude, let me just reiterate my initial suggestion that Horsten’s proposed desiderata for theories of truth (for meta-mathematics) are very much in the spirit of Tarski’s notion of material conditions of adequacy, and thus must be (and indeed are) backed by compelling conceptual arguments.