1 KNOWING AGAINST THE ODDS Cian Dorr, Jeremy Goodman, and John Hawthorne §1 Here is a compelling principle concerning our knowledge of coin flips: FAIR COINS: If you know that a coin is fair, and for all you know it is going to be flipped, then for all you know it will land tails. The idea is that the only way to be in a position to know that a fair coin won’t land a certain way is to be in a position to know that it won’t be flipped at all. 1 One class of putative counterexamples to FAIR COINS which we want to set aside involves knowledge delivered by oracles, clairvoyance, and so forth. A second, and more interesting, class of counterexamples involves knowledge under unusual modes of presentation. For example, if you introduce the name ‘Headsy’ for the first fair coin that will be flipped but will never land tails, then Headsy is arguably a counterexample to FAIR COINS: you know Headsy is fair, you know it will be flipped, and you know it will not land tails. Cheesy modes of presentation pose a challenge to a wide range of intuitive epistemological principles about objective chance (see Hawthorne and Laasonen‐Aarnio 2009, §3): nevertheless, let us set them aside for the remainder of this paper. 2 1 We will treat ‘For all you know, Φ’ as equivalent to ‘You are not in a position to know not‐Φ’. If we instead treated ‘For all you know, Φ’ as equivalent to ‘You don’t know that not‐Φ’ or ‘What you know is consistent with Φ’, we would have to consider putative counterexamples to FAIR COINS in which you know that a coin will not land tails but fail to know that it will not be flipped simply because you have failed to consider whether it will be flipped. 2 It is important to distinguish FAIR COINS from the stronger principle that when a coin is in fact fair and will be flipped for all you know, it will land tails for all you know. This principle is clearly false: suppose that you’re not sure whether a coin is fair or double‐headed, but know it will be flipped only if it is double‐headed.