Knowability and Competent Deduction Closure

Ryan Wasserman was here this week–what a great guy…and superb philosopher!–and we talked about knowability. He had an interesting idea. For standard knowability, you need to get from K(p&q) to Kp&Kq. Since nobody sophisticated about closure thinks that knowledge is closed under entailment, or that knowledge is closed under known entailment, he wondered how we’d get the argument to work. The most plausible closure principles are versions of competent deduction closure principles, and Ryan’s idea was that this might block the paradox. The idea is that some truths might be knowable only to incompetent deducers, and knowledge that a given claim is an unknown truth might be just such a knowable claim–only the logically challenged could have such knowledge.

Very interesting idea, and surely an advance beyond anything in the literature to this point. But conjunction seems special to me, so I’m not sure it works. To possess the concept of conjunction seems to be about as close to the Dummettian ideal as anything can be: to possess the concept is just to wield properly the intro- and elim-rules for &. If so, one wouldn’t be able to have any conjunctive beliefs without seeing that the belief either resulted from &-intro or was susceptible to &-elim. So conjunctive beliefs couldn’t be the kind that could only be known by incompetent deducers.

There’s also Williamson’s conjunctive knowledge bypass argument, but I won’t go into that here. Still, there are so few really interesting and new ideas about the knowability paradox that it is worth noting them when one finds them.


Knowability and Competent Deduction Closure — 13 Comments

  1. I was thinking about the Williamson argument as well as I was reading this. Can’t you imagine a person that psychologically constituted in such a way that they won’t believe q, for some q, unless it occurs as a component of some associative operator? Granted if this is systemic, they probably lack the concept; but all you need for a failure of closure is one case.

    By the way, does the paradox require some such K-distribution principle? I recall Williamson is rather gifted at finding new ways to generate the paradox without a questioned principle.

  2. Ted, you’re right that Williamson as an argument that bypasses the need for closure–that’s the reference to conjunctive knowledge in the last paragraph. It’s from his AJP piece; very neat argument!

    I also think you may be right about your thought experiment, but I’m not sure it helps here unless you think it can be run when the connective is &. I also worry about the claim in question, since it is so schematic. Lots of things look possible, or at least don’t look impossible, when presented in such a schematic way, but a defense of the possibility in question needs to fill in the details.

  3. “… to possess the concept is just to wield properly the intro- and elim-rules for &.”

    Hi Jon-

    Why think that grasping the concept of conjunction must involve these proof theoretic notions (which also happen to sinks Wasserman’s thesis)? Wouldn’t this view of the concept of conjunction place it uncomfortably far away from general meet operations (i.e., any commutative, associative, idempotent binary operator)? Isn’t this the concept of conjunction? Or have I missed your line entirely?

  4. Hi Greg, I don’t see any particular reason to think the semantic meaning of terms in ordinary language are tied to the capacity to form beliefs involving that particular logical operation, as a general rule or guiding principle. In some cases, there may be such a connection, but it isn’t needed and in general isn’t there, so far as I can tell. That, in spite of the motivation behind natural deduction systems to try to encoded such a connection in the intro and elim rules for each constant. But maybe ‘and’ is special, and the intro and elim rules are connected here in the right way, so that nobody could have a conjunctive belief and incompetently deduce each conjunct. If so, difficulties arising from a general closure principle might not make any difference to the paradox.

  5. Hi Jon and Greg:

    The logical connections between (M) K(p&q) -> K(p)&K(q) and closure under know entailment (K) [K(p) & K(p -> q)] -> K(q) are not usually made explicit. If one accepts:

    (E) p q / K(p) K(q)

    K entails M as long as one assumes (N) K(True). But the converse entailment depends on the acceptance of the more controversial:

    (C) K(p)&K(q) -> K(p&q)

    In fact, it is not difficult to show that if one accepts E, M entails K in the presence of C.

    If one is a probabilist about knowledge and accepts a high probability threshold rule for knowledge then C should be abandoned. But if one accepts an argument like Jon’s in terms of rules of elimination and introduction, then C seems as plausible as M. Nevertheless, as I explained above, M and C entail the problematic K (in the presence of E). Perhaps one can abandon E but this is tantamount to abandon the idea that the carriers of knowledge are propositions, something that many would like to preserve (unless they are worried about some form of logical omniscience — Graham Priest offers an interesting way of abandoning E in a paper that we will soon publish in the Review of Symbolic Logic).

    In a nutshell commitment to M requires commitment to the more problematic principle of closure under known entailment as long as one accepts C and E. While C and M can perhaps be justified in terms of the proof theoretic argument advanced by Jon, this justification is not available in the case of the stronger K.



  6. Hi Horacio,

    WordPress has a one-track mind, (html), so it mangled your post. For readers, the modal inference rule (E) should read:

    (E) p # q / K(p) # K(q), where # is the biconditional.

    Thanks for the tip on Priest’s upcoming paper! I’m looking forward to seeing that argument.

  7. Hi Horacio, you say,
    “But if one accepts an argument like Jon’s in terms of rules of elimination and introduction, then C seems as plausible as M.”

    I think I almost agree, but not quite. The proof-theoretic argument I used is likely to be as good for the competent deduction closure version of either C or M, but it might provide a stronger claim as well. Nobody should think that believing p and believing q entails believing the conjunction, so that’s a good reason to reject C. But it is a least a bit more promising to claim that believing the conjunction involves believing each of the conjuncts. If so, the argument might provide some basis for endorsing M, rejecting C, and at the same time endorsing a competent deduction closure analogue of C.

  8. Dear Greg: thanks for correcting (E).

    Dear Jon: Perhaps I am missing the content of he proof-theoretical argument you have in mind. I am not particularly inclined to accept it in the first place, but it seems that if one accepts M in view of proof-theoretical reasons, then C is its dual (they would be rules for introduction and elimination respectively). But perhaps you see some proof-theoretical asymmetry separating them. Do you? Of course, as I explained in the previous posts there are ways of distinguishing between M and C, by, for example, by being a probabilist about knowledge. In this case C fails and M holds, but this is another story.

    It seems that the proof theoretical argument requires to accept M and C as a package as epistemic rules of introduction and elimination of `and’. The point that is perhaps less clear is that this package is rather powerful. In the presence of E it entails closure under known entailment (K). It is not strong enough to guarantee that all tautologies are known, and it does not guarantee necessitation for the knowledge operator. But it gives us the axiom K. So, if one does not like K something has to go. But I do not see a proof theoretical way of blocking C and at the same time accepting M.

    The probabilistic argument though seems to give you most of l the elements you want. It accepts M, N and E. Rejects C and, most importantly it also rejects the axiom K. The resulting epistemic system is (as Kyburg and Teng made clear in a recent article) the system EMN (see Chellas for details) which is clearly weaker than the smallest normal (Kripkean) system K (which requires both the axiom K and necessitation for the epistemic operator).

  9. Horacio, the proof-theoretic argument is about concept possession–that to have the concept in question is to be competent with the rules in question. That doesn’t require that one have applied the rules in any context of use. That would require some further psychological hypothesis. What I was pointing out was that the psychological process that would be needed to make M come out correct is considerably more plausible than the process that would be needed to make C correct.

  10. Jon, I’d offer that proof-theoretic rules are not necessary conditions for (logical) concept position, since they sneak in assumptions about the arguments that a function, in this case &, take. And those assumptions need not apply to all logics in which & occurs yet is perfectly well understood.

  11. Hi Greg, so, first, I didn’t make any general claims about the connection between proof-theoretic rules and concept possession. But your last remark raises the really hard question of what limitations there are between the two, so that a substantive philosophy of logic is possible. That’s a big topic and I won’t go into it here, but I will note that endorsing the claims I made about & don’t threaten it.

  12. Dear Jon: Perhaps then there is no proof theoretic argument for M (C). An agent can have the right &-concept because he is a competent user of the rules for introducing and eliminating &, but this might not be enough to guarantee neither M or C, which require the deployment of the notion of belief over and above the basic logical notions.

    >What I was pointing out was that the psychological process that would be >needed to make M come out correct is considerably more plausible than the >process that would be needed to make C correct.

    This seems a different argument based on a psychological asymmetry between the principles. The psychological argument does not need to be based on proof theoretic considerations. There are as well normative reasons for accepting M while rejecting C as I explained above (reasons based on a probabilistic account of knowledge, for example). Deploying a probabilistic argument (or any other normative argument) for M presupposes that the agent in question is a competent user of & (the calculus of probability presupposes classical logic).

    By the same token some well known accounts of closure, for example Dretske (and Nozick’s ?) abandon M as well as K. Of course Dretske presupposes competence in the use of &. He just thinks that the principle should be abandoned for normative reasons anyhow. Dretske has been criticized for abandoning M as well as K but this criticism has not altered his view about these matters as far as I know.

  13. Hey Jon

    Let me just clarify one point about the “incompetent” deducer: He doesn’t need to be incompetent. He just needs to know (p&~Kp) without being able to deduce each of the conjuncts. ONE way to be such a person is to be incompetent. Another way is to be unlucky.

    Here’s an analogy: It might be possible to travel back in time and meet one’s grandfather. And it might be possible to travel back in time AND try to kill one’s grandfather. But it’s not possible to travel back in time and SUCCESSFULLY kill one’s grandfather, since it’s not possible to kill your own grandfather (or to change the past at all). In some cases, you might fail because you’re an incompetent assassin. But in other cases you might have a change of heart or slip on a banana peel or whatever.

    In the same way, it may be possible for one to know (p & ~Kp). And it may be possible to know (p&~KP) AND try to deduce the conjuncts. But it’s not possible to know (p&~Kp) and SUCCESSFULLY deduce the conjuncts since it’s not possible to know both p and ~Kp. In some cases, you might be too incompetent to do the deduction. In other cases you might have a change of heart before you carry out the deduction. In other cases you might slip on a banana peel and get distracted from doing the disjunction. But the point is just that, one way or another, you’re going to fail!

    In any case, this is all moot since it doesn’t help with Williamson’s version of the argument and, thus, doesn’t get to the heart of the issue. But it was fun talking about this stuff with you anyway.

    Thanks again.

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