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.]