The KK principle and empirical data

So, today I went up to Groningen to attend a lecture by T. Williamson, with the title "Very improbable knowledge". As it turns out, I didn't make it on time for the lecture, as there were massive train delays due to adventurous goats crossing the train-tracks, who then got hit by a train (I do feel sorry for the goats!). I arrived just in time for Q&A, and while I should probably not have had the nerve to ask a question, there was this question that just *had* to be asked.

At some point during Q&A, the KK principle came up, namely the principle that to know p entails to know that you know p (Kp --> KKp). Williamson was arguing against it (as he must have done during the lecture) on conceptual grounds, suggesting that there are situations where it simply does not seem to hold (some convoluted thought-experiment). His interlocutor at that point, Allard Tamminga, insisted that the KK principle is fundamentally correct, and it wasn't clear how the debate could be carried on any further.

So I asked Williamson whether he thought that a debate on the KK principle might benefit from attention to empirical data. He first thought that I meant carrying out surveys and asking people around whether they thought that the KK principle holds -- which sort of beats the point, as they may know the principle and yet not know that they know it! (duh...) I then clarified that I actually had psychological experiments in mind, in particular the kind of thing that has been investigated under the heading of meta-cognition in recent years (very cool stuff!). He was still not very enthusiastic about my idea, arguing that, just as psychological phenomena are not reliable guides for the truth of mathematical statements, they cannot be reliable guides for the truth of logical principles, and according to him KK is a logical principle.

But is it really? I admit that I tend to think that an awful lot of questions are ultimately empirical questions, but I take KK to be essentially about cognition, and thus not a logical principle as such (unless one wants to take the Kantian transcendental road to cognition and infer everything a priori). In fact, many of the results in the meta-cognition tradition seem to suggest that we often 'know' without knowing that we know (and similarly, that we often think we know when we actually do not know). In other words, the meta-cognition literature investigates, among other things, the accuracy with which we judge our own epistemic capacities. My hunch is that the results basically support the view that KK does not hold, but a more serious investigation would have to be carried out for more definitive claims. In any case, I think it is pretty obvious that it could be very interesting to look into the meta-cognition material from the point of view of KK.

I must say that I was surprised to see Williamson so dismissive of the possible relevance of empirical data for this issue. After all, in The Philosophy of Philosophy, he does enlist the mental models theory of reasoning to argue against the inferentialist program (something that Ole still owes us a paper on!). (Basically, his argument doesn't work, but at least it departs from the premise that empirical data could be relevant for such philosophical debates.) But today, he was not in any way sympathetic to the idea -- a shame, if you ask me.


  1. I get the impression here (link below) that the KK Principle was supposed to be about exploring the use of /know/ in modal logic. That might explain the disconnect with Williamson.


  3. Catarina: "I take KK to be essentially about cognition, and thus not a logical principle as such."

    Right - if "Kp" means "A knows that p", then

    (KK) Kp -> KKp

    is an empirical claim about cognition, and an implausible one. I'd guess that Williamson wishes to argue, using counterfactual thought experiments, that it is not analytically true, whereas empirical evidence - non-counterfactual experiments about thoughts! - might show it's merely false.

    Relatedly, there is a verificationist claim advocated by intuitionists, that

    (VER) p -> <>Kp

    (sometimes given the weird name "epistemic constraint"), saying that if p, then one can know that p. That seems to me to be an empirical claim also, and one that presumably only applies to creatures with superhuman powers of cognition.
    (Of course, there's also the famous argument - Fitch's paradox - against (VER): namely that it implies "p -> Kp".)

  4. Hi Ronald,

    If you read my previous post on logic and target phenomena, it will become clear that my claim here is that *cognition* is the target phenomenon of, say, epistemic logic (although one might still want to differentiate the descriptive from the prescriptive enterprise). So while the general formal apparatus was borrowed from modal logic, the target phenomenon is something external to the phenomenon. But of course, this might well be something that Williamson would disagree with, so the debate is still on :)

  5. Jeff, I'm very pleased to see that I am not the only nut-case who thinks that so many of these so-called logical principles have at least some degree of empirical content! :)

    But indeed, perhaps what Williamson wanted to argue for is that KK is analytically false. However, at least judging from people's comments during Q&A, he hadn't been sufficiently convincing; my suggestion then was that, given such an impasse, i.e. a 'clash of intuitions', paying attention to empirical data may shed new light on the debate. (But between you and me, KK just strikes me as implausible on all accounts.)

  6. Also, let me mention that Pete Wolfendale wrote a post prompted by this post, where he bashes my anti-Kantianism :)

  7. Hmm... this is probably simple-minded, but why isn't the following just a knock-down argument against the KK-principle?

    First off, for any logical operator, there ought to be a corresponding predicate (I'm just not willing to negotiate on this, so if you disagree, you might as well stop reading here).

    Then the KK-principle (formulated in terms of the predicate) is inconsistent. The only other principles needed are: proofs are knowledge generating, knowledge is closed under modus ponens, and that K(~F) entails ~K(F) (note that the latter is substantially weaker than factivity).

    Now, by diagnalization, obtain a sentence G such that (I am suppressing annoying brackets for Godel coding):

    G <-> ~K(G)

    Since this is a theorem of arithmetic, assume we find the proof and read it. Then we get:

    K(G <-> ~K(G))

    which by closure under modus ponens gives us:

    K(G) <-> K~K(G)


    K(G) -> K~K(G)

    So by the principle about negation:

    K(G) -> ~KK(G)

    but the KK-principle gives us:

    K(G) -> KK(G)

    so we now have a proof of:


    But, by diagonalization, this is just:


    But since we have now proven this, this means that:



    Of course, this is just a variant on Montague paradox stuff, so if this rules out the KK-principle then presumably the Montague paradox itself rules out factivity. It is kind of an interesting variant since it doesn't depend on the facticity of knowledge!

  8. I agree with the comments so far on the empirical status of the KK-principle, and its non-logical status when K is read as something idealized version of "is known". It's more controversial when we consider the analogous principle where K is read as "is in principle knowable". This version is, in my opinion, not logical (since the knowability operator just isn't logical), buts its empirical status is I think very much up for debate.

  9. But Roy, weren't you the guy who thought that there should be limits to diagonalization procedures? :)

    But yes, it looks like KK can be put under fire for different reasons and from different angles. The nice thing about the proof you give above is that you show that very minimal assumptions need to be made for the argument to go through.

    And I would say that "is in principle knowable" will still have an empirical component, if the 'in principle' part is indexed to cognitive agents with certain characteristics (unless you want to assume ideal, boundless agents).

  10. "Ideal, boundless agents"...

    Actually, I have a move here. One doesn't need to assume that there are these weird, ideal boundless agents to nevertheless want an idealized notion of knowability (or truth, or whatever). Instead, one just has to want an account of the concept that is not provincial - i.e. an account of the concept that will apply to all possible knowers, and not just knowers who are relevantly like us. In that case, presumably a lot of the particular empirical aspects of the notion should be idealized away.


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