Abstract
This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method \({\sf B}_{p}\), which specifies an initial belief state \({\sf B}_{p}(\top)\) that is revised to the new propositional belief state \({\sf B}(E)\) upon receipt of information E. An acceptance rule tracks Bayesian conditioning when \({\sf B}_{p}(E) = {\sf B}_{p|_{E}}(\top)\), for every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The “Lockean” rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1–2):221–262, 1975; Alchourron et al., J Symb Log 50:510–530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987).
Highlights
There are two widespread practices for modeling the doxastic state of a subject— as a probability measure over propositions or as a single proposition corresponding to the conjunction of all propositions the subject believes
One natural idea, called the Lockean rule in honor of John Locke, who proposed something like it, is to accept all and only the logical consequences of the set of all sufficiently probable propositions, whose probabilities are no less than some fixed threshold t strictly less than one
The threshold for acceptance is set, the Lockean rule leads to acceptance of inconsistency, a difficulty known as the lottery paradox (Kyburg 1961)
Summary
There are two widespread practices for modeling the doxastic state of a subject— as a probability measure over propositions or as a single proposition corresponding to the conjunction of all propositions the subject believes. That skeptical approach severely restricts the scope and practical relevance of propositional reasoning, so it is natural to seek a more liberal standard for acceptance. One natural idea, called the Lockean rule in honor of John Locke, who proposed something like it, is to accept all and only the logical consequences of the set of all sufficiently probable propositions, whose probabilities are no less than some fixed threshold t strictly less than one. The threshold for acceptance is set, the Lockean rule leads to acceptance of inconsistency, a difficulty known as the lottery paradox (Kyburg 1961). With probability one some ticket wins, so that proposition is accepted. If t is the threshold, a lottery with more than 1/(1 − t) tickets suffices for acceptance of inconsistency
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