Failures of Single-Premise Closure? The “Just Barely” Problem

Multi-Premise Closure (MPC) can seem a less secure principle than is Single-Premise Closure (SPC), because MPC is subject to what I’ve for a long time called the problem of the “accumulation of doubt,” which is, or is at least very close to, what John Hawthorne, perhaps a bit more appropriately, would call the aggregation of risk. (I’ve always had to quickly add that the problem needn’t be one of actual doubts accumulating, but may be one where, whether doubt accumulates in the believer or not, it should so accumulate.) Here is Hawthorne on this potential problem:

Deductive inference from multiple premises aggregates risks. The risk accruing to one’s belief in each premise may be small enough to be consistent with the belief having the status of knowledge. But the risks may add up, so that the deduced belief may be in too great a danger of being false to count as knowledge. (Knowledge and Lotteries, p. 47)

Now I should quickly point out that John does not abandon MPC on the basis of these thoughts. On the pages that immediately follow, he rallies to MPC’s defense, closing the first chapter of his book a few pages later with a fairly open mind about the issue, but seemingly leaning toward the pro-MPC position. I, on the other hand, think MPC fails because of the problem of the accumulation of risk. (I agree with John the relevant notion of risk is far from unproblematic. But I think there is some good notion there on which this problem is real.)

But what I want to discuss here is conditional in nature: If you think MPC fails because of this problem, what should you think about SPC? Since, as its name so clearly indicates, there’s only a single premise involved in cases of SPC, there seems to be no room for risk to accumulate. And what immediately follows the quotation above is this:

Granted, deductive inference from a single premise does not seem like a candidate for risky inference. (p. 47)

But wait! Though there’s only one premise involved in cases to which SPC would apply, there are two pieces of knowledge involved. For SPC – in order to be at all plausible – is not formulated as the principle that if S knows that p, and p entails q, then S knows that q. Rather, it’s the principle that if S knows that p, and S knows that p entails q, then S knows that q. (Other complications, which I’m ignoring here, have to added as well.) And this gives rise to the possibility that risk can accumulate even in cases of deduction from a single premise. Given how I think risk (or, perhaps, “dis-warrant”) aggregates, I think the failures of SPC that would arise through the aggregation of risk would have to be cases in which S’s knowledge of p counts as knowledge, but just barely so counts, and S’s knowledge of the entailment is also just barely a case of knowledge, while S’s belief that q falls just barely short of being knowledge. And what that means is that you shouldn’t expect to find clear counter-examples to SPC, even though we have reason to think not-so-clear ones exist. Which is good, because clear counter-examples to SPC of this type seem impossible to find!

Though I myself don’t take things this direction, I should note the possibility of agreeing with me in my conditional thought that if MPC fails for the reason in question, so does SPC, but, because SPC can seem so compelling, go the modus tollens route and take all this as reason to think we shouldn’t abandon even MPC, at least for this reason. For my part, I think both SPC and MPC need to be modified to handle this problem – as of course, they have to be anyway to handle several other problems.

[I should note that, though it was buried in an obscure footnote, I raised this problem to SPC several years ago: See note 14 to my Editor’s Introduction (draft available on-line here) to DeRose & Warfield, ed., Skepticism: A Contemporary Reader (Oxford UP, 1999). However, I think given current interests in lotteries and closure principles, it’s worth re-raising the problem at this time.]


Failures of Single-Premise Closure? The “Just Barely” Problem — 5 Comments

  1. For reasons that Lewis Carroll brought out in “What the Tortoise Said to Achilles”, we really don’t want to make the “knowledge” that p entails q an extra premise of the inference. Anyway, can’t children, and probably even some non-human animals, engage in deductive reasoning without possessing the *concept* of entailment?

    What’s really needed for a good formulation of SPC, I think, is not the *knowledge* that p entails q, but the notion of competence with the relevant type of inference. So the following formulation of SPC seems better (due to Oxford graduate student Nico Silins):

    If S knows p, and competently deduces q from p, then S knows q.

    However, Keith’s point is still correct. If the deduction of q from p is fantastically long and complicated, then it could be rational for S to remain in some doubt about q, even though she knows p and has competently deduced q from p.

    Suppose that we appealed to something like Harman’s idea of “immediate implication”. (So, p immediately implies ‘p or q’, but does not immediately imply ‘Not both: not-p and not-q’.) Then we might try formulating SPC as follows: If S knows p, p immediately implies q, and S competently immediately infers q from p, S knows q. Would that get round the problem?

  2. Let the “Uncomplicated Closure Principle” (UCP) be the very simple (and blatently incorrect) principle that if S knows that P, and P entails Q, then S knows that Q. UCP has to be modified in several directions to have a hope of being right.

    (A) The problems with UCP that motivate many to beef up the second conjunct of the antecedent to “S knows that P entails Q” do call for a beefing up that’s at least approximately that beefy. If we don’t require knowledge of the entailment, we do have to require something at least close to that to take care of the problems in question. Thus, if we go the route Ralph suggests, in terms of “competent deduction,” the notion of “competence” involved will have to be a quite substantial one – one that can come close to the requirement that S know that the entailment holds.

    (B) But Ralph is right, it seems, that it’s too much to require propositional knowledge of the fact that the entailment holds.

    The way to put A and B together, I think, is to require that the deduction be sufficiently warranted for S. The notion of a sufficiently warranted deduction will be such that it can be applied to both subjects who possess the notion of entailment (and can easily entertain the proposition that P entails Q) and those who do not, but will be substantial enough that, with respect to the former group, the subject will have sufficient warrant for the deduction at least roughly when & only when, s/he has warrant sufficient for knowledge for the proposition that the entailment holds. (S/he can have this warrant sufficient for knowledge even if s/he doesn’t in fact hold any propositional attitude toward the fact that the entailment holds.)

    If so, the “just barely” problem will arise, even where we don’t require knowledge of the fact that the entailment holds in the antecedent of our closure principle (but instead require that the deduction be sufficiently warranted for S, in the way described above) – and can arise in cases where the deduction is not “fantastically long and complicated” – it just has to be a little bit insecure for S. Even if the deduction is extremely short & uncomplicated, if it’s a bit insecure for S, the problem can arise.


    If the notion of “immediate implication” is a psychological one, I don’t think it can provide the needed patch for the closure principle in the way Ralph points. If it’s an evaluative notion, then there’s hope.


    One way to patch UCP in response to the problem of the aggregation of risk is to require something along the lines of absolute certainty of/in the entailment/deduction – then there’s no dis-warrant to aggregate with the dis-warrant that the premises contribute. But that seems too strong a restriction. We first-and-foremost want a true closure principle, but we also want one that’s as strong as it can be while still being true. And putting something *that* beefy into the antecedent makes the principle *extremely* weak, I fear. Also, I, for one, hope for a patch for the problem of the aggregation of risk that will save MPC as well as SPC, and the patch under discussion can only save SPC. (B/c in MPC, there can be a knowledge-killing aggregation of risk even where the entailment/deduction adds no dis-warrant to that contributed by the premises.) So I’m working on & trying to find patches a) of the right size and b) that can save both MPC & SPC…

  3. Keith, you say that we should not expect to find clear counterexamples to SPC, but should be able to find not-so-clear ones. I’m not sure this follows from the “just barely” problem, but if we don’t find clear counterexamples, that’s compatible with there being a “just barely” problem.

    Second, do you have any not-so-clear counterexamples? Especially, I’d like to see counterexamples where the person competently deduces q from p without having any propositional attitude toward the entailment claim between the two. I’m not convinced that there aren’t such counterexamples, but it would help to see what they might look like.

  4. Hey, everyone has to come up with their own cases! Not only is that fair, but here it’s important because at points the cases have to be adjusted to taste.

    But I can briefly describe how to come up with a case.

    So start with a situation where S knows that p, but just barely knows it. For such purposes, I find cases where the subject knows via someone else’s say-so useful. Since this is an epistemology blog, might as well have it be a case involving the notion of knowledge. So, Uncle Alonzo tells Bernice that Chip knows that the Bulls won the 1996 NBA championship. Where Uncle Alonzo is a horribly unreliable source on such matters, and has been unreliable in Bernice’s experience, Bernice won’t know that Chip knows this on Alonzo’s say-so. Where Uncle Alonzo is perfectly impeccable on such matters, generally and in Bernice’s experience, Bernice will know that Chip knows. You should be able to find intermediate cases where it’s quite unclear whether Bernice knows on Uncle Alonzo’s say-so. Here’s where everyone should devise their own case: Adjust Alonzo’s reliability, generally and in Bernice’s experience, so that Bernice does know, but just barely knows, that Chip knows that the Bulls won.

    Now also devise the case so that Bernice has no other grounds for believing that the Bulls won the ’96 championship, knows that Chip’s knowing that the Bulls won entails that the Bulls did in fact win, but that she just barely knows that the entailment holds. Here, I’m supposing that Bernice has the notion of entailment, and so can have, and just barely have, knowledge of the proposition that the entailment holds. That makes things easier to describe. If you want the other kind of case, imagine is so that, though Bernice doesn’t have the concept of entailment and can’t know that the entailment holds, the deduction “Carl knows that the Bulls won; so, the Bulls won” is about as secure for your Bernice as it is for mine. You can make the case more concrete by imagining such things as this. Generally, Bernice will not describe anyone as knowing something that is not true — except in cases of sarcasm or the like (“Oh yeah, Rumsfeld just knew that the Sunni triangle was swarming with weapons of mass destruction.”) And she generally is inclined to deduce *p* from *S knows that p* — as she’s inclined to in the case at hand. But just somehow make this deduction just a bit insecure. How insecure? That’s to taste, but in any case, about as insecure as it is for you in the case of my Bernice, who considers the proposition that the entailment holds, and knows, but just barely knows, that it does hold.

    Now, from her just-barely knowledge of the fact that Chip knows that the Bulls won, have Bernice ever-so-shakily deduce that the Bulls won. Here, her belief that the Bulls won should be a bit more shaky than is the corresponding belief of Diane, who is as shaky as Bernice on the fact that Chip knows the Bulls won (Diane, too, knows this, but just barely knows it), but is *far* less shaky than is Bernice on the entailment. If all has gone well, as opposed to Diane who just barely knows that the Bulls won, Bernice should fall just short of knowing that the Bulls won the ’96 championship (though it shouldn’t be at all clear to you that she doesn’t know), and you should now be the proud owner of a very unclear counter-example to single-premise closure.

  5. Keith–yes, you’re right to say I should construct my own case. I see now what you have in mind, but the cases put us in strange territory for evaluating SPC. It looks like this: as long as we can describe a case that isn’t obviously a case of knowledge where SPC was used, then we have grounds for rejecting SPC. That’s too strong, though–nearly any account will admit of some penumbral region cases. So I wonder what weaker methodological principle could be used to get such cases to count against SPC.

    Here’s a nice baseball quote for you. When Texas lost to Cal State-Full., the Texas coach remarked to the press, “I know the quality of this team. We’re champions, even though we lost. We don’t have to be champions to know we’re champions.” I laughed, but when it was reported on ESPN, the announcer said about the last claim, “Sorry coach, yes you do.” A blossoming epistemologist!

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