Interpreting the third truth value in Kripke’s theory of truth

March 28, 2010

Notoriously, there are many different theories of untyped truth which use Kripke’s fixed point construction in one way or another as their mathematical basis. The core result is that one can assign every sentence of a semantically closed language one of three truth values in a way that $\phi$ and $Tr(\ulcorner\phi\urcorner)$ receive the same value.

However, how one interprets these values, how they relate to valid reasoning and how they relate to assertability is left open. There are classical interpretations in which assertability goes by truth in the classical model which assigns Tr the positive extension of the fixed point, and consequence is classical (Feferman’s theory KF.) There are paraconsistent interpretations in which the middle value is thought of as “true and false”, and assertability and validity go by truth and preservation of truth. There’s also the paracomplete theory where the middle value is understood as neither true nor false and assertability and validity defined as in the paraconsistent case. Finally, you can mix these views as Tim Maudlin does – for Maudlin assertability is classical but validity is the same as the paracomplete interpretation.

In this post I want to think a bit more about the paracomplete interpretations of the third truth value. A popular view, which originated from Kripke himself, is that the third truth value is not really a truth value at all. For a sentenc to have that value is simply for the sentence to be ‘undefined’ (I’ll use ‘truth status’ instead of ‘truth value’ from now on.) Undefined sentences don’t even express a proposition – something bad happens before we can even get to the stage of assigning a truth value. It simply doesn’t make sense to ask what the world would have to be like for a sentence to ‘halfly’ hold.

This view seems to a have a number of problems. The most damning, I think, is the theory’s inability to state this explanation of the third truth status. For example, we can state what it is to fail to express a proposition in the language containing the truth predicate: a sentence has truth value 1 if it’s true, has truth value 0 if it’s negation is true, and it has truth status 1/2, i.e. doesn’t express a proposition, if neither it nor its negation is true.

In particular, we have the resources to say that the liar sentence does not express a proposition: $\neg Tr(\ulcorner\phi\urcorner)\wedge\neg Tr(\ulcorner\neg\phi\urcorner)$. However, since both conjuncts of this sentence don’t express propositions, the whole sentence,  the sentence ‘the liar does not express a proposition’, does not itself express a proposition either! Furthermore, the sentence immediately before this one doesn’t express a proposition either (and neither does this one.) It is never possible to say a sentence doesn’t express a proposition unless you’ve either failed to express a proposition, or you’ve expressed a false proposition. What’s more, we can’t state the fixed point property: we can’t say that the liar sentence has the same truth status as the sentence that says the liar is true since that won’t express a proposition either: the instance of the T-schema for the liar sentence fails to express a proposition.

The ‘no proposition’ interpretation of the third truth value is inexpressible: if you try to describe the view you fail to express anything.

Another interpretation rejects the third value altogether. This interpretation is described in Fields book, but I think it originates with Parsons. The model for assertion and denial is this: assert just the things that get value 1 in the fixed point construction and reject the rest. Thus the sentences  “some sentences are neither true nor false”, “some sentences do not express a proposition” should be rejected as they come out with value 1/2 in the minimal fixed point. As Field points out, though, this view is also expressively limited – you don’t have the resources to say what’s wrong with the liar sentence. Unlike in the previous case where you did have those resources, but you always failed to express anything with them, in this case being neither true nor false is not what’s wrong with the liar since we reject that the liar is neither true nor false. (Although Field points out that while you can classify problematic sentences in terms of rejection, you can’t classify contingent liars where you’d need to say things like ‘if such and such were the case, then s would be problematic’ since this requires an embeddable operator of some sort.)

I want to suggest a third interpretation. The basic idea is that, unlike the second interpretation, there is a sense in which we can communicate that there is a third truth status, and unlike the first, 1/2 is a truth value, in the sense that sentences with that status express propositions and those propositions “1/2-obtain” – if the world is in this state I’ll say the proposition obtails.

In particular, there are three ways the world can be with respect to a proposition: things can be such that the proposition obtains, such it fails, and such that it obtails.

What happens if you find out a sentence has truth status 1/2 (i.e. you find out it expresses a proposition that obtails)? Should you refrain from adopting any doxastic attitude, say, by remaining agnostic? I claim not – agnosticism comes about when you’re unsure about the truthvalue of a sentence, but in this case you know the truth value. However it is clear you should neither accept nor reject it either – these are reserved for propositions that obtain and fail respectively. It seems most natural on this view to introduce a third doxastic attitude: I’ll call it receptance. When you find out a sentence has truth value 1 you accept, when you find out is has value 0 you reject and when you find out it has value 1/2 you recept. If haven’t found out the truth value yet you should withold all three doxastic attitudes and remain agnostic.

How do you communicate to someone that that the liar has value 1/2? Given that the sentences which says the liar has value 1/2 also has value 1/2, you should not assert that the liar has value 1/2. You assert things in the hopes that your audience will accept them, and this clearly not what you want if the thing you want to communicate has value 1/2. Similarly you deny things in the hope that your audience will reject them. Thus this view calls for a completely new kind of speech act, which I’ll call “absertion”, that is distinct from the speech acts of assertion and denial. In a bivalent setting the goal of communication is to make your audience accept true things and reject false things, and once you’ve achieved that your job is done. However, in the trivalent setting there is more to the picture: you also want your audience to recept things that have value 1/2, which can’t be achieved by asserting them or denying them. The purpose of communication is to induce *correct* doxastic state in your audience, where a doxastic state of acceptance, rejection or receptance in s is correct iff s has value 1, 0 or 1/2 respectively. If you instead absert sentences like the liar, and your audience believes you’re being cooperative, they will adopt the correct doxastic attitude of reception.

This, I claim, all follows quite naturally from our reading of 1/2 as a third truth value. The important question is: how does this help us with the expressive problems encountered earlier? The idea is that in this setting we can *correctly* communicate our theory of truth using the speech acts of assertion, denial and absertion, and we can have correct beliefs about the world by also recepting some sentences as well as accepting and rejecting others. The problem with the earlier interpretations was that we could not correctly communicate the idea that the liar has value 1/2 because it was taken for granted that to correctly communicate this to someone involved making them accept it. On this interpretation, however, to correctly express the view requires only that you absert the sentences which have value 1/2. Of course any sentence that says of another sentence that it has value 1/2 has value 1/2 itself, so you must absert, not assert, those too. But this is all to be expected when the obective of expressing your theory is to communicate it correctly, and that communicating correctly involves more that just asserting truthfully.

Assertion in this theory behaves much like it does in the paracomplete theory that Field describes, however some of the things Field suggests we should reject we should absert instead (such as the liar.) To get the idea, let me absert some rules concerning absertion:

• You can absert the liar, and you can absert that the liar has value 1/2.
• You can absert that every sentence has value 1, 0 or 1/2.
• You ought to absert any instance of a classical law.
• Permissable absertion is not closed under modus ponens.
• If you can permissibly absert p, you can permissibly absert that you can permissibly absert p.
• If you can absert p, then you can’t assert or deny p.
• None of these rules are assertable or deniable.

(One other contrast between this view and the no-proposition view is that it sits naturally with a more truth functionally expressive logic. The no-proposition view is often motivated by the motivation for the Kleene truth functions: a three valued function that behaves like a particular two valued truth function on two valued inputs, and has value 1/2 when the corresponding two valued function could have had both 1 or 0 depending on how one replaced 1/2 in the three valued input with 1 or 0. $\neg, \vee$ is expressively adequate with respect to Kleene truth functions defined as before. However, Kripke’s construction works with any monotonic truth function (monotonic in the ordering that puts 1/2 and the bottom and 1 and 0 above it but incomparable to each other) and $\neg, \vee$ are not expressively complete w.r.t the monotonic truth functions. There are monotonic truth functions that aren’t Kleene truth functions, such as “squadge”, that puts 1/2 everywhere that Kleene conjunction and disjunction disagree, and puts the value they agree on elsewhere. Squadge, negation and disjunction are expressively complete w.r.t monotonic truth functions.)

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.