Probability and Danger

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge (MPC). Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths are known. Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. The Amherst Lecture in Philosophy (ISSN: 1559-7199) is a free on-line journal, published by the Department of Philosophy, Amherst College, Amherst, MA 01002. Phone: (413) 542-5805. E-mail: alp@amherst.edu. Website: http://www.amherstlecture. org/. Copyright Timothy Williamson. This article may be copied without the copyright owner’s permission only if the copy is used for educational, not-for-profit purposes. For all other purposes, the copyright owner’s permission is required. In all cases, both the author and The Amherst Lecture in Philosophy must be acknowledged in the copy. P the amherst lecture in philosophy Lecture 4, 2009 Probability and Danger Timothy Williamson 1 Probability and Danger Timothy Williamson University of Oxford 1. Much of recent and not-so-recent philosophy is driven by tensions, or at least apparent tensions, between common sense and natural science – in the terminology of Wilfrid Sellars, between the manifest image and the scientific image of the world. These tensions arise most saliently in metaphysics and the philosophy of mind, but are far from confined to those branches of philosophy. In this lecture, I will discuss one specific form they take in contemporary epistemology. Central to common sense epistemology is the distinction between knowledge and ignorance. Knowledge is not usually conceived as coming in quantifiable degrees: we do not ask and could not answer ‘To what degree does she know where the station is? ’. 1 By contrast, a continuum of numerical degrees of probability is central to contemporary natural science. The point is not merely that a framework of probabilities has to some extent displaced a framework of knowledge and ignorance in the scientific image of cognition. Worse, probabilistic reasoning seems to destabilize common sense conceptions of knowledge. As so often, we cannot just blandly assert that the manifest image and the scientific image are both fine in their own way, but useful for different purposes. We face prima facie conflicts between them which seem to imply that if the scientific image is accurate, then the manifest image is radically misleading. We have to do the hard work of analysing the apparent conflicts in detail, to determine what their upshot really is. 1 For discussion of the (un)gradability of knowledge ascriptions see Stanley 2005: 35–46. P the amherst lecture in philosophy Lecture 4, 2009 Probability and Danger Timothy Williamson 2 Elsewhere, I have argued that the sort of probability most relevant to the epistemology of science is probability on the evidence, and that the evidence is simply what is known; thus knowledge is a precondition, not an outdated rival, of probability in science (Williamson 2000). I have also shown that much of the supporting argument for that conclusion is robust even when recast in probabilistic form (Williamson 2008). But those arguments do not entitle us to ignore specific probabilistic considerations that seem to undermine common sense epistemology. This lecture concerns one such threat to a non-probabilistic conception of knowledge. 2. Why is deduction useful? The obvious answer is that it is a way of extending our knowledge. It is integral to that answer that extending one’s knowledge in this way depends on the temporal process of carrying out the deduction, for one knows more after doing so than one did before. Moreover, one cannot expect to obtain knowledge thereby unless the deductive process involves forming a belief in the conclusion. This suggests a principle along the following lines, now often known as ‘Multi-Premise Closure’ (Williamson 2000: 117): MPC If one believes a conclusion by competent deduction from some premises one knows, one knows the conclusion. Here competence is intended to stand to inference roughly as knowledge stands to belief. One can no more hope to attain knowledge of the conclusion by less than competent deduction than one can hope to attain it by deduction from premises of which one has less than knowledge. But competence does not require knowledge that the deduction is valid, otherwise the attempt to use MPC to explain how deduction extends knowledge would involve an infinite regress of knowledge of the validity of more and more complex deductions (Carroll 1895). MPC is closer to the dynamics of cognition than is the static principle that one knows a conclusion if one believes it, knows it to follow deductively from some premises, and knows the premises. At first sight, there is no tension between MPC and a scientific account of cognition. Mathematics is essential to science, and its main role is to extend our knowledge by deduction. P the amherst lecture in philosophy Lecture 4, 2009 Probability and Danger Timothy Williamson 3 Perhaps some fine-tuning is needed to capture exactly the intended spirit of MPC. Nevertheless, some such principle seems to articulate the compelling idea that deduction is a way of extending our knowledge. I will not discuss any fine-tuning of MPC here. Nor will I discuss challenges to MPC that are closely related to traditional sceptical puzzles, for instance where the premise is ‘That is a zebra’ and the conclusion is ‘That is not just a mule cleverly painted to look like a zebra’. It is generally, although not universally, agreed that such examples do not refute a properly formulated closure principle for knowledge.2 Even if we start by answering the question ‘Do the spectators know that that is a zebra? ’ in the affirmative and then the question ‘Do they know that it is not just a mule cleverly painted to look like a zebra? ’ in the negative, once that has happened and we are asked again ‘Do they know that it is a zebra? ’ we are now inclined to answer in the negative. Thus the supposed counter-example to closure is not stable under reflection. The probabilistic threat to MPC starts from the truism that many acceptably small risks of error can add up to an unacceptably large one. The most obvious illustration is a version of the Lottery Paradox (Kyburg 1961). Suppose that for some positive real number δ a risk of error less than δ is acceptable. Then for any suitably large natural number n, in a fair lottery with n tickets of which only one wins, for each losing ticket the statement that it will lose has an acceptably small risk of error, but all those statements together logically entail their conjunction, which has a probability of only 1⁄n – the structure of the lottery being given – and a fortiori an unacceptably large risk of error. This does not constitute a clear counter-example to MPC, since one can deny that the premises are known: even if a ticket will in fact lose, we do not know in advance that it will; we only know that it is almost certain to. But can we legitimately treat knowledge of lotteries as a special case? 3 For example, does not a scientific study of human perception and memory show that even in the best cases they too involve non-zero risks of error? If we reacted to the Lottery Paradox by insisting that knowledge requires zero risk of error, that requirement seems to constrain us to denying that there is human knowledge by perception or memory, and more generally to force us into scepticism. 2 The example is of course from Dretske 1970; the other classic version of such a challenge to closure is Nozick 1981. For critical discussion of such objections to closure see Vogel 1990 and Hawthorne 2005. 3 See Hawthorne 2004 for discussion. P the amherst lecture in philosophy Lecture 4, 2009 Probability and Danger Timothy Williamson 4 Even beliefs about our own present mental states seem to carry some non-zero risk of error. But if knowledge of our contingent circumstances is unobtainable, the distinction between knowledge and ignorance loses most of its interest. A version of the Preface Paradox helps make the point vivid (Makinson 1965). Suppose that I compile a reference book containing large quantities of miscellaneous information. I take great care, and fortunately make not a single error. Indeed, by ordinary standards I know each individual item of information in the book. Still, I can reasonably acknowledge in the preface that since almost all such works contain errors, it is almost certain that mine does too. If I nevertheless believe the conjunction of all the individual items of information in the book (perhaps excluding the preface), the risk of error in that conjunctive belief seems so high that it is difficult to conceive it as knowledge. Thus MPC seems to fail, unless the standard for knowing is raised to sceptical heights. One advantage of the objection to MPC from the Preface Paradox over the generalization from the Lottery Paradox is that it avoids the unargued assumption that if a true belief that a given ticket will lose fails to constitute knowledge, the reason must be just that it has a non-zero risk of error. For whether a true belief constitutes knowledge might depend on all sorts of factors be