## Truth as an operator and as a predicate

November 5, 2009

Suppose we add to the propositional calculus a new unary operator, T, whose truth table is just the trivial one that leaves the truth value of its operand untouched. By adding

• $(Tp \leftrightarrow p)$

to a standard axiomatization of the propositional calculus we completely fix the meaning of T. Moreover this is a consistent classical account of truth that gives us a kind of unrestricted “T-schema” for the truth operator.

On the face of it, then, it seems that if we treat truth as an operator operating on sentences rather than a predicate applying to names of sentences we somehow avoid the semantic paradoxes. But this seems almost like magic: both ways of talking about truth supposed to be expressing the same property – how could a grammatical difference in their formulation be the true source of the paradox?

My gut feeling is that there isn’t anything particularly deep about the consistency of the operator theory of truth: it just boils down to an accidental grammatical fact about the kinds of languages we usually speak. The grammatical fact is this. One can have syntactically simple expressions of type e but not of type t. Without the type theory jargon this just means we can have names that can be the argument of a predicate but not “names” that can be the argument of an operator. Call these latter kind of expressions “name*s”. If $p$ is a name* then $\neg p$ is grammatically well formed and is evaluated as the same as $\neg \phi$ where $\phi$ is whatever sentence p refers* to. If pick $p$ so that it refers* to “$\neg p$” then we are in just the same predicament we were in the case where we were considering names and treating truth like a predicate. One could simply pick a constant and stipulate that it refers to the sentence “~Tr(c)”.

We could make this a little more precise. By restricting our attention to languages without name*s we’re remaining silent about propositions that we could have expressed if we removed the restriction. Indeed, there is a natural translation between operator talk (in the propositional language with truth described at the beginning) and predicate talk. So, on the looks of it, it seems we could make exactly the same move in the predicate case: accept only sentences that are translations of sentences we accept. The natural translation I’m referring to is this:

• $p^* \mapsto p$
• $(\phi \wedge \psi)^* \mapsto (\phi^*\wedge\psi^*)$
• $(\neg \phi)^* \mapsto \neg \phi^*$
• $(T\phi)^* \mapsto Tr(\ulcorner\phi^*\urcorner)$

Here’s a neat little fact which is quite easy to prove. Let M be a model of the propositional calculus (a truth value assignment.)

Theorem. $\phi$ is the translation a true formula in M if and only if $\phi$ appears in Kripke’s minimal fixedpoint construction using the weak Kleene valuation with ground model M.

Note that, because we don’t have quantifiers, the construction tapers out at $\omega$ so we can prove the right-left direction by induction over the finite initial stages of the construction. Left-right is an induction over formula complexity.

If the rule is to simply reject all sentences which aren’t translations of an operator sentence then it appears that the neat classical operator view is really just the well known non-classical view based on the weak Kleene valuation scheme. It is well known that the latter only appears to be classical when we restrict attention to grounded formulae; it seems the appearance is just as shallow for the former view.

Incidentally, note that there’s no natural way to extend this result to languages with quantifiers. This is because there’s no “natural” translation between the propositional calculus with propositional quantifiers and a quantified language with the truth predicate capable of talking about its own syntax.

1. Cool! It’s well known that if one has a naive truth predicate, then operators are available for free. But I’m not sure anyone’s ever shown anything like the converse. If you have a truth operator theory (within certain assumptions) it is similar to a certain kind of predicate theory.

Part of the problem of with the operator theory of truth is that it doesn’t yield the generalization capabilities of unrestricted predicate type theories. So it’s not really a good candidate if you’re a deflationist. That fact is born out here in an interesting way: the weak Kleene theory doesn’t allow generalization over the ungrounded sentences at all.

2. Hi Aaron,

I’m glad you commented actually, as I’m coming to this with basically no knowledge of the literature at all. What is the well known result you mentioned in your second sentence? Is there anything written on the operator versus predicate theories of truth? (After writing this I realised that I’m still not crystal clear why one turns out to be consistent.)

Incidentally, I’m not sure I agree with your point about the generalization capabilities of the operator theory. At least, if you’re willing to accept that name*s are illegitimate then can’t you just formulate the generalizations using propositional quantification?

3. The result is just this: if you have a truth predicate s.t. $T(\ulcorner A \urcorner)$ and $A$ can be substituted for each other in all (extensional) contexts w/o change in semantic value, then for any operator $\odot$ one can form the predicate $\odot T(\ulcorner A \urcorner)$. You can also take any predicate P and turn it into an ‘operator’ on sentences using $T(P(\ulcorner A \urcorner))$. I suppose that’s not as obviously relevant to your idea as I initially thought.

I don’t know much about the operator theories of truth, either. Probably the most worked out theory in this vein is the prosentential theory (Grover, et al). I guess the generalization issue is this: the truth predicate allows us to generalize over sentences using out regular old first-order quantifiers (the truth predicate allows for semantic ascent). On an operator theory of truth there’s no “semantic ascent” going on; we’d still need propositional or substitutional quantification in order to generalize over the true sentences. So we’re introducing a new kind of quantification to play the generalizing role that the truth predicate was intended to play.

4. Cool paper here:

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bsl/1255526080

Seemed like it’d be right up your alley.

5. Thanks!

Actually I saw him give a talk on this. I should definitely work through it properly though.