Can you have knowledge that is based on a false belief?

Suppose you have two kids: A and B, and you believe that A is left-handed, and that B is left-handed. Being a deft reasoner, you conclude:

At least one of my kids is left-handed.

Now suppose that, really, B is not left-handed (though you're right, and knowledgeably so, about A). It seems to me that it is still right to count you as knowing that at least one of your kids is left-handed, and it also seems plausible that your belief is in some good sense "based on" a false belief (that B is left-handed). Someone might challenge the latter claim, by observing that your belief about B is inessential to your justification for the "at least one" belief---if you hadn't believed that B was left-handed, after all, you still would have been justified in concluding that at least one of your kids was left-handed. But on that way of viewing things, your "at least one" belief is based on neither of your two beliefs about the left-handedness of A and B (for each of them individually is such that, without it, you'd still have concluded that at least one of your kids was left-handed). And that seems crazy. So, yes, you can have knowledge based on a false belief. It's quite another question whether you can have knowledge for which a false belief is essential to its justification. If you can think of a solid example of that, please send it to the journal Analysis.

I'm not planning to send this to Analysis, but it seems to me there might be such a case, at least under certain conceptions of knowledge (where I am not proposing the case as a counterexample to the conception). Suppose we think (as I actually do) that knowledge requires the application of appropriate epistemic virtues. Being fallible (at least for the most part, as far as we can tell), one such epistemic virtue is epistemic humility. The one with epistemic humility is a fallibilist, and accepts that she could be mistaken (but actually is not). But suppose she actually has a kind of knowledge about which she is, in fact, infallible. Even so, she thinks she could be mistaken about what she knows. But that is false, she cannot be mistaken. However, were she not to think she could be, she would be epistemically immodest, thus not virtuous, and thus wouldn't know. Might that be such a case?

It is not difficult to come up with a case where you have knowledge that you would not have had if you did not have a false belief. Suppose you are a medical researcher, doing the experiments to get the medical data is very difficult, and you never would have been able to make the effort if you didn't believe it would enable you to cure the common cold. That belief was alas false; but the data you gathered enabled you to gain new knowledge (though no c0ld cure). This is a case where you would not have had that knowledge were it not for your false belief; but I would not describe this as a case where the justification itself includes a false belief. The new evidence is true, it is just that you would not have found it if it were not for your false belief.

One can slightly simplify Mark's case as follows. Suppose one believes that p and also believes that q. One therefore believes that p and q, but also that p or q. The disjunctive belief surely must be "based upon" one's beliefs in the disjuncts, but neither of them is essential: The belief in the disjunction would survive failure to believe either disjunct. Should we say that your knowledge is "based upon" a false belief?

It seems to me that, in this case, one has two independent reasons for the disjunctive belief, namely, the beliefs in its disjuncts. The reasons are independent in several senses: (i) one could have each reason even if one did not have the other; (ii) as I'm imagining the case, anyway, one's justification for the beliefs in the disjuncts is independent of one's justification for the othe belief; (iii) each reason is sufficient, on its own, to underwrite knowledge in the disjunction. As it happens, however, only one of the reasons actually underwrites knowledge, and it is in that sense that this example doesn't give one what one would really want: a case where knowledge is essentially based upon a false belief.

I, anyway, am inclined to think there are no such cases.

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