Here is an argument for Tony's conjecture that his (4) (closure of knowledge under known entailment) falls with (2) (closure of belief under believed entailment).  It is not a rigorous argument, but only suggestive.
     I shall assume that knowledge is some species of justified true belief, and let's read "justified" pretty strongly, as containing an anti-Gettier qualification such as "undefeatedly."  Now, let's unpack (4) accordingly, using the obvious notation:

            {(Jp & p & Bp) & [J(p=>q) & p=>q & B(p=>q)]} => (Jq & q & Bq)

And now suppose that (2) is false, so  Bp & B(p=>q)  does not logically guarantee Bq.
    Bq occurs as a conjunct in the consequent of unpacked (4), so (4)'s complex antecedent is supposed to guarantee Bq.  That antecedent is, again, (Jp & p & Bp) & [J(p=>q) & p=>q & B(p=>q)], and it contains Bp and B(p=>q) as conjuncts.  By hypothesis, Bp and B(p=>q) do not together guarantee Bq.  So if (4)'s antecedent does guarantee Bq, it must be in virtue of what has been added to Bp and B(p=>q) in order to make (4)'s antecedent, viz., (Jp & p) & [J(p=>q) & p=>q].  That addition does obviously guarantee q, and arguably it guarantees Jq.  But what in  (Jp & p) & [J(p=>q) & p=>q]  is supposed to guarantee Bq, even in conjunction with Bp and B(p=>q)?  If I were setting out to guarantee Bq, I don't see how adding  (Jp & p) & [J(p=>q) & p=>q]  would help.  So, if (2) is indeed false, why should we think (4) is true?
    (The reason that is not a rigorous argument is that even though none of the individual added conjuncts entails Bq, of course some conceptual interaction between them might have the effect of guaranteeing Bq.  But I don't offhand see what sort of interaction that would be.  Possibly one might be caused by the hidden anti-Gettier element.)
    Notice that my argument is not weakened if we suppose, contrary to custom but with D.M. Armstrong, that there cannot be an epistemically justified false belief, i.e., that Jp => p.
    Are there theories of epistemic justification that themselves sustain (4)?  I don't think so.  Take the simple nomic-reliability theory I sketched in class.  If my belief that p nomically implies p, and my belief that p=>q nomically implies p=>q (which it does trivially since p=>q is itself a necessary truth), does it follow that I believe q?  Not that I can see.  Or try an even more demanding theory, a Cartesian one according to which one knows that p only if one has deduced p by self-evident steps from beliefs which are incorrigible or infallible for one.  If I know that p in that strong sense and (what is easier) know that p=>q ditto, does it follow that I believe q?  Not that I can see.
    Notice that all this present discussion of (4) is independent of skepticism.  I'm not urging that we should give up (4) because (4) leads to skepticism and skepticism is implausible or un-credible or otherwise bad.  Thus, (4) is an antecedently controversial philosophical claim.