Jack has been posting on epistemic closure principles. I know that I have hands. I also know that, if I have hands, then I'm not a (handless) brain in a vat (BIV). But, it's generally supposed, I can't know that I'm not a BIV.
Something weird's going on here. For an intuitive complaint, look at the abominable conjunctions:
# I know that I have hands, but I don't know that I'm not a handless BIV.
# (Expressed in assertion:) "I have hands, but I can't say whether I'm a handless BIV."
For a principled complaint, it makes no sense to think that I could have less epistemic warrant or evidence for the logically weaker claim. The other entails it. The worlds where I have hands are a proper subset of the worlds where I'm not a handless BIV. It's as silly as thinking that Linda is more likely to be a feminist bank teller than a bank teller.
My favoured solution is due to Keith DeRose. S's belief that P is epistemically safe precisely to the extent that certain possible worlds -- namely, those where S believes P falsely -- are distant. Degree of safety is the fundamental epistemic property, and it satisfies closure principles perfectly, as we should expect. I think I have hands, and I think I'm not a BIV, and the latter belief is at least as safe as the former. (You'd have to go out to at least as distant a possible world in order to find one where I hold the belief falsely.)
Why do closure principles seem to fail for knowledge, then? Simply because the standards for knowledge vary. Knowledge is belief that is sufficiently safe for our purposes. Whether a belief that's safe to degree N so qualifies is an open question, and one that will receive different answers in different contexts. Raising certain possibilities to salience will tend to raise the bar, requiring that the safety level of the belief extend to the possible worlds under consideration. Once the bar is raised so high, even our ordinarily safe beliefs will not qualify as "knowledge". Abominable conjunctions are thus avoided.
So we find that the apparent failure of closure is an artifact of our shifting standards. At the fundamental level, epistemic qualification (i.e., safety) transmits across entailments just fine.