What Theories of Validity Should be Like (but Cannot be)

Hannes Leitgeb has a nice paper called ‘What Theories of Truth Should be Like (but Cannot be)’. Our MCMP reading group on truth just read the paper, and it struck me that it might be worth thinking about whether Hannes’s conditions on formal theories of truth also apply to formal theories of validity.

Let me explain what I have in mind. In recent literature – for example Shapiro (2010) and Beall & Murzi (ms) – there has been some discussion about the nature of a validity predicate. (See also Field’s book, ch 19, and Graham Priest’s reply for their takes on the validity predicate.) For simplicity, let us think about Val(<A>,<B>) as a two-place predicate between (codes for) formulae, and forget about multiple-premise and -conclusion for now. A formal theory for validity, then, is like a formal theory of truth in that it adds the predicate to the language and introduces some inference rules governing the predicate. In particular, both Priest and Beall & Murzi consider the following pair of rules:

The rules look suspeciously like intro- and elim-rules for a conditional, and, of course, the validity-predicate is a lot like a conditional. In fact, Shapiro’s rules explicitly gives an interaction between the validity predicate and an entailment connective. Unfortunately, as Beall & Murzi show, the fact that the validity predicate has conditional-like rule makes it Curry-susceptible, i.e. a Curry-like paradox follows when we let Y be equivalent to Val(\langle Y\rangle, \langle \bot\rangle).

So-called validity-paradoxes are nothing new, but they have still been given relatively little attention in a formal setting. With the analogy to a truth predicate emphasised, we should investigate what sort of desiderata a theory of validity should satisfy. This is precisely what Hannes’s paper does for theories of truth, so let’s list the conditions and see how they apply to a theory of validity. For truth, it goes without saying that we cannot fulfil all the conditions in one theory. If we could, it wouldn’t be much fun would it. So let us paraphrase some of the desiderata:

(a) Validity should be expressed by a predicate. Hannes’s first desideratum is that truth be expressed by a predicate. I think the reasons for having a truth predicate carries over to validity, so I won’t pause on this. Shapiro’s paper gives a neat statement of why (a) should be accepted for validity.

(c) Validity should not be subject to type restrictions. Again, I think that the reasons for having a monolithic truth predicate applies to a validity predicate as well. (Although, for an alternative, see Myhill’s ‘Levels of Implications’, 1975.)

(d) V-biconditional should be derivable unrestrictedly. Hannes’s condition (d) is that the T-biconditionals are derivable unrestrictedly. Beall & Murzi suggest that from the two rules for the validity predicate there is a corresponding V-schema:

\vdash Val(\langle A\rangle, \langle B \rangle) iff A \vdash B

Granted, the analogy with truth is far from perfect. Yet, if one accepts the suggested rules for the validity predicate as playing a role analogously to the naive truth rules, then the V-schema is a plausible desideratum. (That’s not to say that it can’t be the thing that has to go on pain of paradox.) The schema can be thought of as expressing a sort of soundness and completeness for the validity predicate. It captures precisely the arguments valid in the logic of the theory.

(f) The theory should allow for standard interpretations. When a truth predicate is added to PA over some logic, it happens that although the theory does well on other desiderata, it does not have a standard model (the theory might be \omega-inconsistent). Some people are a bit skeptical about (f) for formal theories of truth, but presumably mostly because something has to give. I suppose the same condition can be added to a theory of validity.

(g) The outer and inner logic should coincide. For the truth predicate, if the theory has, say, A \vee \neg A as a logical law, then it should also include T\langle A \vee \neg A\rangle. I.e. logical laws are the same inside and outside the scope of the truth predicate. If all the T-biconditionals are derivable, then we get (g) for the truth predicate as well. For validity, however, the issue is a bit more complicated. Obviously, we’re not only interested in the special case of theorems, but of what follows from what in the logic of the theory. In other words, the V-schema might be what we’re after for (g) as well. Alternatively, one might insist that there is a corresponding entailment connective \rightarrow which expresses validity (again, see Shapiro’s paper). In that case one simply replaces the right hand side of the V-schema with \vdash A \rightarrow B.

(h) The outer logic should be classical logic. Following the reasoning of (g), this should be interpreted as the desideratum that the theory’s logic ought to be classical – not only with respect to theoremhood, but with respect to consequence in general. Clearly, with (g) in place, (h) entails that the inner logic is classical as well.

This leaves the conditions (b) and (e). The former states that the when a theory of truth is added to a mathematical or empirical theory it should be possible to prove the latter true. I can’t see any obvious way of paraphrasing this desideratum in terms of validity. Similarly, condition (e), that truth should be compositional, also doesn’t seem obviously applicable.

Are there further desiderata that could be added for a theory of validity? One candidate that strikes me as fairly uncontentious is that one might expect there to be a connection between validity and truth. How to express this connection, however, is not an entirely straightforward issue.