Abstract
Until recently, it seemed like no theory about the relationship between rational credence and rational outright belief could reconcile three independently plausible assumptions: that our beliefs should be logically consistent, that our degrees of belief should be probabilistic, and that a rational agent believes something just in case she is sufficiently confident in it. Recently a new formal framework has been proposed that can accommodate these three assumptions, which is known as “the stability theory of belief” or “high probability cores.” In this paper, I examine whether the stability theory of belief can meet two further constraints that have been proposed in the literature: that it is irrational to outright believe lottery propositions, and that it is irrational to hold outright beliefs based on purely statistical evidence. I argue that these two further constraints create a dilemma for a proponent of the stability theory: she must either deny that her theory is meant to give an account of the common epistemic notion of outright belief, or supplement the theory with further constraints on rational belief that render the stability theory explanatorily idle. This result sheds light on the general prospects for a purely formal theory of the relationship between rational credence and belief, i.e. a theory that does not take into account belief content. I argue that it is doubtful that any such theory could properly account for these two constraints, and hence play an important role in characterizing our common epistemic notion of outright belief.
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Notes
Leitgeb’s presentation of the view differs from Arló-Costa and Pedersen’s version in a number of ways, some of which will become important later in the paper. Leitgeb starts from the standard Kolmogorov axiomatization of probability, whereas Arló-Costa and Pedersen prefer a version of the view that is based on primitive conditional probabilities. Also, Leitgeb (2014) highlights the importance of the fineness with which the space of possibilities is partitioned, which isn’t emphasized in his earlier paper, or by Arló-Costa and Pedersen. The authors also differ on the way the threshold for rational belief is set in a context. Arló-Costa and Pedersen, following Leitgeb (2013), claim that the threshold is set by the strongest available proposition (2012, Definition 3.11), whereas Leitgeb (2014, 2015) leaves this open.
I am omitting some of the precise formal details of Leitgeb’s theory in order to convey the essence of the view to the reader. For a fully precise statement of the view, readers are invited to consult Leitgeb’s work directly. The same holds for Arló-Costa and Pedersen’s view.
Footnote 5 in Leitgeb (2014) gives a sketch of the algorithm by which one can determine which propositions have stably high credence in a given context: “Assume that W = {w1,…,wn}, and P({w1}) ≥ P({w2}) ≥ ··· ≥ P({wn}). If P({w1}) > P({w2}) + ··· + P({wn}), then {w1} is the first, and least, non-empty P-stable set, and one moves on to the list P({w2}),…, P({wn}); e.g. if P({w2}) > P({w3}) + ··· + P({wn}), then {w1, w2} would be the next P-stable set. On the other hand, if P({w1}) ≤ P({w2}) + ··· + P({wn}) then consider P({w2}): if it is greater than P({w3}) + ··· + P ({wn}) then {w1, w2} is the first P-stable set, and one moves on to the list P({w3}),…, P({wn}); but if P({w2}) is less than or equal to P({w3}) + ··· + P({wn}) then consider P({w1}), P({w2}), P({w3}): and so forth. The procedure is terminated when the least subset of W of probability 1 is reached.”
For example, consider a fair lottery with ten tickets, in which there is only a 10 % chance that the lottery will have any winner. No inconsistency ensues from believing that every ticket (each of which has a 1 % chance of winning) is a loser, because the agent might simply believe that there will be no winner (as long as a credence of 0.9 is sufficiently high to rationalize belief. But we can of course change the numbers to accommodate other thresholds). It seems intuitively plausible to say that even in this case, belief in the lottery propositions is irrational, even though the agent’s beliefs aren’t inconsistent.
For the claim that it shouldn’t matter whether a lottery has an even probability distribution or not, see for example: McKinnon (2013), Christensen (2004, p. 58), Hawthorne (2004, pp. 8–9), Nelkin (2000), Williamson (2000, p. 248), DeRose (1996) and Vogel (1990).
Some authors discuss lottery propositions in the context of whether they can be known, rather than believed. Yet, some supporters of knowledge norms, such as Williamson, endorse the further claim that one ought not to believe something unless one knows it. It follows that one ought not to believe lottery propositions. (Williamson 2000, p. 255)
Some of the propositions that Hawthorne classifies as lottery propositions could also be subsumed under (5), because they are propositions for which we have purely statistical evidence, for example “I will not die in an accident in the next year.” Or “Not all of these golfers will get a perfect score.”
Buchak’s paper is not the first to make the distinction between statistical and nonstatistical evidence, and to discuss its relation to rational belief. For further references, I invite the reader to consult the extensive bibliography in her paper.
This problem with the partition-sensitivity of Leitgeb’s view is pointed out independently by Fitelson (2015) and Schurz (2015). Fitelson shows how coin-toss expansions of an agent’s doxastic space render initially permissible beliefs impermissible, and he uses these results to demonstrate that the stability theory’s verdicts look bad in light of confirmation theory.
Schurz proves a general theorem about the kinds of epistemic expansions that always render initially permissible beliefs impermissible: take any analytically independent expansion of an agent’s doxastic space, where each possibility of the old partition is split up into the same number of new possibilities. Let n be the number of doxastic possibilities in the set that is used to expand the agent’s original doxastic space, and let t be the threshold for rational belief before expanding the agent’s doxastic space. According to Schurz’ Theorem 3, the set of initially permissible beliefs will not remain permissible after an analytically independent expansion where n ≥ t/(1 − t).
Of course, we could have a theory that only allows for rational belief if the agent has a rational credence of 1. But this view is not a serious contender in the first place.
References
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Acknowledgments
I would like to thank Fay Edwards, Branden Fitelson, Alan Hájek, Hannes Leitgeb, Hanti Lin, Richard Pettigrew, Ryan Platte, Brian Talbot, and an anonymous referee for helpful comments and discussion.
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Staffel, J. Beliefs, buses and lotteries: Why rational belief can’t be stably high credence. Philos Stud 173, 1721–1734 (2016). https://doi.org/10.1007/s11098-015-0574-2
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DOI: https://doi.org/10.1007/s11098-015-0574-2