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1. As John Hawthorne and Maria Lasonen-Aarnio appreciate, some of the central issues raised in their ‘Knowledge and Objective Chance’ arise for all but the most extreme theories of knowledge. In a wide range of cases, according to very plausible everyday judgments, we know something about the future, even though, according to quantum mechanics, our belief has a small but nonzero chance (objective probability) of being untrue. In easily constructed examples, we are in that position simultaneously with respect to many different propositions about the future that are equiprobable and probabilistically independent of each other, at least to a reasonable approximation. Taking the contents of all these pieces of knowledge as premises, we can competently deduce their conjunction, and believe it on that basis. By a very plausible multi-premise closure principle for knowledge, we thereby come to know the conjunction. Since the chance that our belief in the conjunction is true is the product of the chances that the separate conjuncts are true, given independence, it can be made arbitrarily close to zero by choosing an example with enough conjuncts. But this contradicts the very plausible-