Abstract
The idea that beliefs may be stake-sensitive is explored. This is the idea that the strength with which a single, persistent belief is held may vary and depend upon what the believer takes to be at stake. The stakes in question are tied to the truth of the belief—not, as in Pascal’s wager and other cases, to the belief’s presence. Categorical beliefs and degrees of belief are considered; both kinds of account typically exclude the idea and treat belief as stake-invariant, though an exception is briefly described. The role of the assumption of stake-invariance in familiar accounts of degrees of belief is also discussed, and morals are drawn concerning finite and countable Dutch book arguments.
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Notes
I do not offer a taxonomy of accounts of degrees of belief here. We can distinguish multiple accounts descending from Ramsey (1926) and de Finetti (1937); some of the criteria that Good (1971) employs are relevant, as are others. But I will lapse into using singular phrases like ‘the Ramsey-de Finetti model of action-guiding belief’ without attempting to explicitly specify which variants count.
And no doubt also on his background and other beliefs. Beliefs are influenced by other beliefs, and their historical backgrounds influence our doxastic states. On a given occasion of believing that p, what the believer takes to be at stake on the truth of p is a fact about the believer. But it is not necessarily a fact about what the believer takes to be evidence for p, and typically it is a fact that is connected with features of the circumstances in which the believer finds himself.
For examples: Stalnaker (1984, Chapter 5) regards belief as one sort of acceptance concept; other sorts of acceptance involve acting as if one believes, in a variety of ways that may be for a purpose, compartmentalized, and direct products of decision. Similarly Bratman (1992). Note that Stalnaker distinguishes a belief from a belief state; the latter is a state of having a total set of beliefs. Van Fraassen (1980) takes acceptance of p (a theory T) to involve belief in a related p* (that T is empirically adequate), together with pragmatic commitments. Kaplan (1996) ties categorical belief that p to preference for asserting p in a particular sort of circumstance, a context of inquiry; this account is a descendent of one in which accepting that p was treated in a similar way. In both accounts Kaplan contrasts acceptance/belief with degrees of confidence. Many other accounts make use of an idea of acceptance.
For examples, Lewis (1996), Cohen (1999), and Fantl and McGrath (2002). There is an extensive literature on the ways context influences our standards and judgments about justification and knowledge. The stakes can matter—directly, or by influencing the salience of possible error, or perhaps in other ways.
As might happen when the stakes are raised by John’s grocer Daniel, who offers a large wager that there are peanut products in the dinner.
I take responsiveness-to-altered-stakes to be the believer’s responsiveness to his revised assessment of stakes (not to their mere presence, of which he might be unaware); this includes such cases as when one gives little or no occurrent thought to the stakes and then suddenly understands them to be high. The sudden understanding might be taken to be a revision of a dispositional doxastic state that differed from the new belief that the stakes are high, provoking a change of mind.
Ramsey (1926), p. 67; from here on when I speak of choice-guiding degrees of belief, it is the Ramsey-de Finetti model that I have in mind.
Here I focus on models of precise degrees of belief. About vague degrees of belief we can also raise questions of stake-sensitivity. Does what is at stake shift the range that characterizes a vague degree of belief? Does it make vague degrees of belief sharper or vaguer? These are models and questions that I set aside in this paper, except to suggest that the first sort of influence is at odds with usual ways of thinking of categorical belief in epistemology.
In Armendt (2008) several examples of relevant views expressed in recent epistemological work are briefly raised and discussed. Though interesting, they do not amount to endorsements of epistemic accounts involving stake-sensitive belief.
And it applies equally to Ramsey’s degrees of belief.
When p, q are incompatible, dob(p) + dob(q) = dob(p v q). The dobs represent what the believer would pay for a bet that pays $1. If the rule is violated, the cunning bettor makes money from the believer by selling the high side to him and buying back the low side.
“We find, therefore, that a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency. They do not depend for their meaning on any degree of belief in a proposition being uniquely determined as the rational one; they merely distinguish those sets of beliefs which obey them as consistent ones.
Having any definite degree of belief implies a certain measure of consistency, namely willingness to bet on a given proposition at the same odds for any stake, the stakes being measured in terms of ultimate values. Having degrees of belief obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you.” (Ramsey 1926, pp. 78–79).
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Acknowledgements
It is a great pleasure to contribute to this occasion honoring my teacher and friend Brian Skyrms. Thanks to SkyrmsFest participants Persi Diaconis, Patrick Suppes, Bill Harms, and particularly Kevin Zollman for helpful comments and discussion. I have also benefited from conversations about these ideas with Alan Hajek, Jim Joyce, Bernie Kobes, Stewart Cohen, Brian Skyrms, Brad Monton, Michael Tooley, Bas van Fraassen, Paul Weirich, Joel Press, and Melissa Baker.
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Armendt, B. Stakes and beliefs. Philos Stud 147, 71–87 (2010). https://doi.org/10.1007/s11098-009-9451-1
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DOI: https://doi.org/10.1007/s11098-009-9451-1