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.)



  1. Hi Andrew,

    Just a quick question. I can see that in the theory you cannot assert that the liar is true, not true neither true nor false etc. The claim that it gets value 1/2 however would be in the meta-language and unless you take that to be non-classical I can’t see the motivation for thinking that it should not be asserted. Any help would be great.


  2. Presumably the object language is a proper fragment of the metalanguage, so you can say things like “s has value 1 iff s is true”, “s has value 1/2 iff s is neither true nor false” and so on in the metalanguage. The metalinguistic statement that the liar doesn’t have value 1, I think, is is equivalent in the metalanguage to the statement that the liar isn’t true. Which just is the liar: so even in the metalanguage you’re asserting untrue sentences.

    But I was imagining talk about the three values as being part of the object language already. I could simply define “s has value 1/2” in the OL as “s is neither true nor false”. Or alternatively I could have added “has value 1/2” into the language right at the beginning and supplemented the fixed point construction appropriately.

  3. Although I should of course add that people like Tim Maudlin don’t see anything wrong with asserting a sentence which has value 1/2, or is untrue. But his isn’t a view in which you’re asserting sentences which don’t express a proposition.

    Thanks for your question, BTW, I wasn’t being particularly clear on that point.

  4. That’s interesting. Presumably, though, you can’t assert that s gets value 1/2 iff it is neither true nor false. So when you carry out the fixed point construction you don’t assert that every sentence not assigned 1 or 0 at some level gets value 1/2, but absert it?

    I think that’s a reason in favour of Field’s theory (we don’t given necessary and sufficient conditions for truth on the basis of our semantic values (model theory and truth come apart)).


  5. Ah, I see.

    I think there are two uses of the word “meta-theory”/”metalanguage” that are in use here.

    One is a language according to which we are describing the intended model of the object language. Thus, if the metalanguage contains the object language, we should want to communicate things like s has value 1/2 in the intended model iff s is neither true nor false (but as you rightly point out, we have to absert these things.) If these claims weren’t correct (in my technical sense) it seems you wouldn’t be talking about the intended model. The metalanguage in this sense must be non-classical (as it contains the non-classical object language.) Furthermore, the use of a distinct metalanguage isn’t necessary, as the object language contains all the resources it needs to describe the intended model already, as you can define semantic value 1, 1/2 and 0 already.

    Another use of the “metalanguage” might be the language in which we carry out a consistency proof for the theory of truth (to describe a “theory” now you must specify not just the set of assertable sentences, but also the absertable and rejectable sentences.) In this metalanguage I see no reason to have say that having value 1 and being true amount to the same thing, as the sole purpose of this model is to show contradictions don’t follow from what you absert/assert. Also it seems that this metalanguage is essentially stronger than the object language – as to be expected when doing consistency proofs.

  6. I wonder, though, if you really can carry out Kripke’s fixed point construction in set theory with a background Kleene logic. It seems unlikely. It seems even more unlikely that you can carry out such a fixed point construction with set theory in the ground language (although I don’t know if the second incompleteness theorem extends to Kleene logic.)

    At any rate, I think the actual construction Kripke gives can’t be providing a picture of the intended model. The thought that value 1/2 is due to ungroundedness goes more naturally with the no-proposition view. On the view I’m envisaging a sentence having semantic value 1/2 corresponds to a way the world could be. You might think there are possible worlds where the truth teller has value 1 or 0, as these are at least consistent, even though the truth teller is ungrounded. (Although, necessarily, the liar has value 1/2.)

  7. Thanks Andrew, that’s really helpful!


  8. Standardly, both the paracomplete and the paraconsistent versions of the “no proposition” view will assert that “you ought not assert and deny the same thing” and “everything (that you’re not agnostic about) ought to be either asserted or denied”. This, if nothing else,

    In this framework though, I’m not sure what sense to make of the claims (i) “You ought not assert and absert the same thing” and (ii) “You ought not absert and deny the same thing.” On this view, you can absert (i) and (ii), but presumably you won’t assert (i) or (ii) or deny them. Is that right?

    If so, I wonder what role absertion is supposed to play. You might think that absertion is meant to update people’s commitments by forcing them to recept the things they absert (which is supposed to somehow clash with accepting or rejecting them). But of course the theorist still can’t *assert* this. It seems they can’t assert what role absertion is supposed to play in updating people’s commitments.

    Another thought: I’m not sure I agree that “agnosticism comes about when you’re unsure about the truth value of a sentence”. It seems that agnosticism is a perfectly legit response to the Liar in this case. I’m not sure whether I should assert the liar or deny it simply because I use negation to deny and truth to assert. But this sentence is it’s own negation, and so I’m left with conflicting evidence about whether to assert or deny. Moreover, unlike recepting, agnosticism is an already recognized propositional attitude. We already understand what it means, whereas I’m still not sure what it would mean to recept something.

    Finally, is there any real difference between this view and the “dual” paraconsistent view? In other words, suppose I absert a premise p and then infer conclusion q, and proceed to deny the conclusion. Have I done something wrong, or not? If yes, then one presumably would want to take the paraconsistent account of validity, that validity is preservation of obtaining and obtailing. Whereas, if the answer is no, presumably validity is merely truth preservation.

  9. Hi Aaron,

    Thanks for the comments.

    Can I ask for a bit more clarification on the first question? The idea is, roughly, you’re doing well epistemically when you accept only true things, reject only false things and recept only value 1/2 things. Of course, there’s the twist that some of the relevant norms themselves have value 1/2, so one should not be asserting those. Are you suggesting we should actually be *asserting* them even though they have value 1/2? I claim that asserting such norms would communicate them incorrectly (listeners might end up accepting them instead of recepting them.)

    Regarding the point about agnosticism: I can well understand the view that you should be agnostic regarding the liar. However, I don’t think you can have this view and take the third truth status seriously as a *truth value*, i.e., as a way for a proposition to obtain. To describe the world on the no proposition view undefined sentences don’t even come into the picture, whereas if you’re taking the third truth value seriously you haven’t completely described a world until you’ve specified the truth value of every sentence including sentences like the liar and the truth teller. On the latter view, if you know the distribution of truth values, you know exactly which possible world you’re in and there’s nothing left for you to be ignorant or agnostic about. For example, once you’ve found out the value of the truth teller there will be no worlds (thought of in the three valued way) compatible with your epistemic/doxtastic state which differ on the truth value of the truth teller.

    Regarding the very last question – I think this view is very different from the paracomplete and paraconsistent versions. After all, the latter two are only distinguished by which sentences they assert and which they reject (and which are conditionally accepted, etc.) I don’t really know what you mean by ‘infer’ q in this setting. Inference, in the sense of saying when it’s good to accept the conclusion if you accept the premisses, is paracomplete, but there’s a paraconsistent version for receptence too I guess. And these don’t exhaust the norms of reasoning: if you accept p, and recept p->q you should recept q is a norm of good reasoning that isn’t captured by either.

  10. Suppose you absert a norm. What should I do? Well, “recept it,” I guess would be the answer. Sure, but now what? How does this inform me on what I should do with that norm now that I’ve got the right propositional attitude toward it? Should I act accordingly, or no?

    I suppose (wrt the last point) I’m assuming a kind of picture where it’s essential to valid inference that one ought not assert the premises and deny the conclusion (thinking of a view like Greg’s). But that underdescribes the consequence relation in this case. Is there any clash in asserting the premises and abserting the conclusion? Also, is there any clash in abserting the premises and rejecting the conclusion? Which ever way you go has an effect on what argument forms are good to use. But if the answers to those questions are only abserted, then I’m not sure how I should reason.

    Wrt to the second thing about agnosticism, you’re right. It probably doesn’t square all that well with taking it as a truth-value in the sense you mean.

  11. Ok, I think I misunderstood your first question then.

    It might be helpful to consider a more familiar case: suppose it’s vague whether you ought to X in a certain situation. (I claim such cases of vagueness are commonplace given the vagueness of many important morally relevant concepts such as personhood, et cetera.)

    It’s quite tempting to carry on asking what you should do in these situations. Finding out that it’s vague whether you ought to X is all well and good but how does that piece of information help you decide whether to X or not?

    Even though it’s tempting to continue asking what you should do, I think it’s ultimately misguided. If you know it’s vague whether p it’s fundamentally misguided to continue to ask whether p, to wonder whether p, and so on and so forth.

    Regarding your second question, I guess I was suggesting that we just replace familiar talk about consequence with a bunch of norms governing what you should accept, recept and reject. Ordinary consequence can be thought of just encoding norms of the form “if you accept P_1, …, P_n you should accept Q”, which provides us with the whole story in the classical setting, but in the non-classical setting not all the relevant norms of reasoning are of this form. (I think this is so even for the standard paracomplete theorist, where rejection and acceptance of the negation come apart. I’m guessing this is what the Restall paper you mentioned covers – although I haven’t read it yet.)

  12. Thanks! The “vague whether to do X” is really helpful.

    And you should have a look at the Restall stuff. His notion of consequence is right along these lines. In the typical non-classical settings (standard paracomplete/paraconsistent) you can tell the whole story of consequence with assertion and denial (which in the end behave pretty classically). But on your view, you’ll need additional norms about reception (and that doesn’t behave classically at all).

    You should also talk to Dave Ripley who has a paper about “paracoherentism” or something. It’s basically the standard paraconsistent story about the liar, and yet you deny by asserting negations (negation *does* express denial). So, one ends up both asserting and denying the liar. But why is that so bad? 🙂

    • Oops, I meant to say that I have included this post in the latest Philosophers’ Carnival. Hope that’s ok!

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