Here are two objections that will have occurred to many of you.  I'm sure they've been put to Dretske many times since 1970, but I can't now recall how he's replied.

    First, although I myself am sympathetic to denying closure, the case for closure can be sharpened.  Suppose Fred is at the zoo and claims to know that the animal in front of him is a zebra.  Of course he knows that being a zebra entails not being a painted mule.  We ask, then, if he knows that the animal is not a painted mule.  He replies, no, because he has done nothing to rule out the painted-mule hypothesis.  But, we remind him, that the animal is not a painted mule is trivially entailed by its being a zebra.  Of course, he says.  Well, then, if he does know that the animal is a zebra how could he fail to know that it is not a painted mule, when the entailment is vividly present to his attention??  Surely he could not help performing the inference.
    Two possible replies:  (1) Dretske could agree that he would and does perform the inference, but deny that the resulting belief counts as knowledge.  He would have to reject the principle that what is consciously inferred from knowledge by a simple deduction known to be valid is itself knowledge, but that shouldn't be hard for someone who already rejects closure.
    Yet surely knowledge is transmissible by such deduction.  We often do come to know things by deducing them from things we already knew.  That is part of what deduction is for.  So if he makes reply (1), Dretske will have put himself in the business of giving criteria for when such deduction does yield knowledge and when it doesn't.  The obvious criterion is: It does when and only when no antecedently irrelevant alternative is introduced.  (But what's "introducing" exactly?)
    Or, (2) he might appeal to his unusual doctrine that objects of knowledge are more finely-grained than propositional sentence meanings.  (As I said in class, the parallel doctrine concerning explananda is almost universally accepted, but this one is more contentious.)  He knows that the animal is a zebra(-as-opposed-to-a-giraffe-or-an-elephant-or-an-unpainted mule), but he does not know that it is a zebra(-as-opposed-to-a-painted-mule-or-a-hologram-or-a-hallucination).
    I am not sure how that idea should be applied back into my modified zoo example.  The obvious application is not Dretske's:  It is to deny that the fine-grained object of knowledge expressed by "the animal is a zebra" in the original knowledge-claim does entail that the animal is not a painted mule, on the grounds that the animal's being a painted mule is not in the original claim's contextual set of relevant alternatives.  If that were Dretske's idea, there would be no need to deny closure.  So even the fine-grained object of original knowledge does entail that the animal is not a painted mule.  Therefore, I don't yet see how the unusual doctrine would help against the first objection.

     The second objection does not seem to me to damage Dretske's main contentions.  I'm not sure what hangs on it, but it's worth pointing out: The idea of a semi-penetrating operator is not obviously tenable.  What distinguishes a semi-penetrating operator from a nonpenetrating operator is that it is closed under single elementary deductive implications.  (Or at least that it penetrates some of them; for "knows," Dretske gives the examples of Conjunction Elimination and vel-Introduction.)  But now consider a deductive calculus that is semantically complete in the sense that every semantically valid sequent is derivable in the system.  And choose such a calculus each of whose primitive rules is transparently valid, known to be valid by anyone who understands and uses the system.  So, for every semantic consequence of a given premise, no matter how remote, there will be a derivation that proceeds by a series of individually elementary deductive inferences.  Thus, if "knows" is closed under single elementary deductive implications, it is also closed under semantic consequence generally; anyone who knows a given proposition knows every one of that proposition's infinitely many semantic consequences, including ones that would take fifty years to write down, much less ever understand.
     It does not follow that "knows" is a fully penetrating operator, because (a) one might suppose that not all entailments can be codified in a semantically complete formal system such as the predicate calculus or the various complete modal logics, and (b) Dretske may not grant that "knows" penetrates all single elementary deductive inferences.  So it is fair for him to continue to call the epistemic operators semi-penetrating.  But, contrary to what he suggests, they are much closer to being fully penetrating than they are to being nonpenetrating.
     And, much more to the point, even if the epistemic operators do fall short of being fully penetrating, my argument shows that they penetrate far enough to wreck Dretske's rejection of skepticism based on denial of closure.  The animal's being a zebra semantically entails its not being a painted mule rather than a zebra, in a way easily represented in the propositional calculus:

         (1) Za                    Premise
         (2) Ma & ~Za        Assumption for Reductio
         (3) ~Za                  2, Conjunction Elimination
         (4) Za & ~Za         1,3, Conjunction Introduction
         (5) ~(Ma & ~Za)   2,4 Reductio
 

     If that's so (which it is), why did I opine that this second objection does not damage Dretske's main contentions?  Because he is perfectly free to give up the claim that the epistemic operators are semi-penetrating, and claim instead that they are flatly nonpenetrating.  That's what I'd urge him to do.