Abstract
Lin and Kelly (J Philos Log 41(6):957–981, 2012) and Leitgeb (Ann Pure Appl Log 164(12):1338–1389, 2013, Philos Rev 123(2):131–171, 2014), offer similar solutions to the Lottery Paradox, defining acceptance rules which determine a rational agent’s beliefs in terms of broader features of her credal state than just her isolated credences in individual propositions. I express each proposal as a method for obtaining an ordering over a partition from a credence function, and then a belief set from the ordering. Although these proposals avoid the original Lottery Paradox, I raise a diachronic case which illustrates that neither satisfies both (i) Lin and Kelly’s constraint that the update on orderings track the update on credence functions, and (ii) the intuitive constraint that credence of at least 0.5 is necessary for rational belief. I conclude by suggesting that we reformulate these proposals in terms of orderings over entire algebras based on partitions rather than orderings just over the partitions themselves. Reformulating both rules in this way yields acceptance rules which avoid the Lottery Paradox while satisfying both the tracking and likeliness constraints.
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Notes
I take the ‘credal reductivist strategy’ label from Ross and Schroeder (2011).
When the agent is not important I omit the ‘a’. Eventually I will specify whose belief set is at issue by relativizing B instead to the credence function from which it is obtained.
Levi (1967) builds these norms into his fundamental principle of rational scientific inquiry, the “principle of deductive cogency”. Also these norms will hold in many approaches to rational belief which constrain belief sets using a normal epistemic logic, so that \(\phi \in B\) iff ‘\(\Box _{B}\ulcorner \phi \urcorner \)’ is true. If \(\Box _{B}\) is an operator abiding by axiom D then Consistency holds. If \(\Box _{B}\) abides by K and the rule of necessitation in a logic with modus ponens and all instantiations of propositional tautologies, then Closure holds. See chapter 2 of van Ditmarsch et al. (2008).
This axiom is commonly replaced with a principle of Countable Additivity, which identifies the value of countably infinite union of pairwise disjoint sets on P with the sum of the values of those sets on P. The problems I raise don’t turn on this difference.
Foley (1993) argues that having inconsistent beliefs need not lead to a plain violation of Consistency; one can rationally belief inconsistent propositions without thereby believing or being committed to believing everything whatsoever. He thus rejects closure, though not consistency as I’ve stated it. Kyburg (1961) also argues against Closure.
The received view concerning credences holds that, given some new information \(\psi \), a rational agent’s credence function P should “update by conditionalization” to another, \(P|\psi \), where \(P|\psi (-)\,\)=\(\,P(-|\psi )\), and for any \(\phi \) and \(\psi \), \(P(\phi |\psi )\), the probability of \(\phi \) conditional on \(\psi \) is defined as \(P(\phi \cap \psi )/P(\psi )\), if \(P(\psi )\ne 0\).
A potential objection to Tracking focuses on its requiring that one have maximally strong confidence in whatever one learns. Conditionalizing P on \(\phi \) models the strongest possible increase in the agent’s confidence in \(\phi \); but surely one can acquire a belief in \(\phi \) without such extreme confidence. So, the objection goes, there’s no reason to identify the result of acquiring new belief in \(\phi \) with the beliefs one has upon conditionalizing on \(\phi \). I think this is an important problem for any fully general theory of belief acquisition. However, if we restrict our attention on idealized cases in which one learns new information by becoming certain in it, then Tracking is quite plausible. And the examples I raise in Sects. 4 and 5 still make trouble for the belief revision methods I consider even on such an idealization. So I won’t strive to nuance Tracking here, though it is worth bearing in mind that in a final theory of belief acquisition some generalization of Tracking would be called for.
Thus \({\mathcal {Q}}\) is a countable set of mutually exclusive and exhaustive propositions \(\alpha _{1},\ldots , \alpha _{i}\) from \({{\mathcal {A}}}\). This follows Levi (1967), who defines his acceptance rule relative to a partition of answers, conceived of as a line of inquiry.
This closely relates to Contrastivist views in traditional epistemology. Schaffer (2005) defends contrastivism about knowledge, according to which whether or not an agent knows some proposition is relativized to a question which determines relevant alternatives against which the knowledge attributed is contrasted. Blaauw (2013) proposes such a contrastive view for outright belief itself.
Since answers to the relevant question represent the finest distinctions the agent is drawing in the context, there is an implicit restriction on what information can be learned by an agent for whom \({\mathcal {Q}}\) is at issue: the only learnable propositions are unions of answers in \({\mathcal {Q}}\). Thus for a learned \(\phi \), if \(\alpha \cap \phi \ne \emptyset \) then \(\alpha \subseteq \phi \). And if \(\alpha ,\beta \subseteq \phi \), \({P(\alpha )}\over {P(\beta )}\)=\({P(\alpha \cap \phi )}\over {P(\beta \cap \phi )}\)=\({{P(\alpha \cap \phi )}\over {P(\phi )}}\over {{P(\beta \cap \phi )}\over {P(\phi )}}\)=\({P(\alpha |\phi )}\over {P(\beta |\phi )}\).
It is worth mentioning that the Odds Rule’s belief revision method is not simply a function from belief set-proposition pairs (modeling prior beliefs and learned information) to belief sets. More structure is needed: the ordering of propositions given by \(\succ _{P}^{s}\) is a crucial component to the revision method and is never kicked away. Contrary to my simplified initial presentation, on this understanding of belief revision modeling an agent’s outright beliefs requires both a belief set and a plausibility ordering. This is not unique to L&K; even the classic belief revision operator of the AGM theory, due to Alchourron et al. (1985), requires a selection function defined in terms of a comparative relation over propositions.
Leitgeb does not bill his theory as a credal reductivist strategy, though he admits that the formal framework allows for such an interpretation. The problem cases I raise trouble either interpretation.
This follows from two claims proved in Leitgeb (2013) §2.5 and §2.6: (i) that for any two P-\(\hbox {stable}^{r}\) propositions \(\phi \) and \(\psi \), either \(\phi \subseteq \psi \) or \(\psi \subseteq \phi \), and (ii) there is no infinitely descending chain of P-\(\hbox {stable}^{r}\) propositions.
One might wonder why we should adopt a complicated rule like the Odds Rule or Stability Rule if we are allowed to invoke question-indexed thresholds in the first place. After all, the basic Lockean Thesis itself can avoid the original Lottery Puzzle if we set the question-indexed threshold to greater than \(1/(1-n)\). The problems I raise below for the Question-Indexed Odds Rule will be problems for any diachronic extension of such a Question-Indexed Lockean Thesis.
Given footnote 10, since \(\alpha \cap \phi , \beta \cap \phi \subseteq \phi \), \({P(\alpha \cap \phi )}\over {P(\beta \cap \phi )}\)=\({P(\alpha \cap \phi |\phi )}\over {P(\beta \cap \phi |\phi )}\). But \({P(\alpha \cap \phi |\phi )}\over {P(\beta \cap \phi |\phi )}\)=\({P(\alpha |\phi )}\over {P(\beta |\phi )}\).
So, if \(\alpha \) is in the nth most plausible group, it is in the \((n+1)\hbox {th}\) best group.
This kind of procedure is described by Leitgeb (2013).
Where there is more than one most plausible answer, the procedure works by imposing an arbitrary strict order over the most plausible.
References
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Acknowledgements
My thanks to Andrew Bacon, Kenny Easwaran, and two anonymous referees for helpful discussion.
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Wright, J. Rigged lotteries: a diachronic problem for reducing belief to credence. Synthese 195, 1355–1373 (2018). https://doi.org/10.1007/s11229-016-1275-x
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DOI: https://doi.org/10.1007/s11229-016-1275-x