Abstract
A common objection to theories of vagueness based on fuzzy logics centres on the idea that assigning a single numerical degree of truth—a real number between 0 and 1—to each vague statement is excessively precise. A common objection to Bayesian epistemology centres on the idea that assigning a single numerical degree of belief—a real number between 0 and 1—to each proposition is excessively precise. In this paper I explore possible parallels between these objections. In particular I argue that the only good objection along these lines to fuzzy theories of vagueness does not translate into a good objection to Bayesian epistemology. An important part of my argument consists in drawing a distinction between two different notions of degree of belief, which I call dispositional degree of belief and epistemic degree of belief.
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Notes
- 1.
Some authors would also like to include other kinds of facts, such as facts concerning the eligibility as referents of objects and sets, or facts concerning the simplicity/complexity of interpretations. Even on these views, however, usage plays a central role.
- 2.
Different Bayesians use different terms where I have used ‘belief state’—e.g. ‘credal state’, ‘credences’, ‘degrees of belief’, ‘degrees of confidence’, and ‘subjective probabilities’. At this point I use the term ‘belief state’ with scare quotes and without an explanation of what the term means, because a key issue below—beginning in Sect. 4—will be what, exactly, the agent’s ‘belief state’ is supposed to be.
- 3.
Taking propositions to be sets of possible worlds, the second tenet becomes the requirement that the assignment of real numbers to propositions constitutes a probability measure over the space of possible worlds.
- 4.
Given tenet 2, the agent’s assignment of real numbers to propositions can be referred to as a probability assignment. Tenet 3 then requires that the agent’s posterior probability assignment to a proposition P (after the evidence E comes in) is equal to her prior (before the evidence comes in) conditional probability of P given E.
- 5.
Some characterisations of Bayesianism (e.g. Joyce 2010, 282) add a fourth tenet: rational agents make decisions by maximising expected utility.
- 6.
Objections have also been made to tenets 2 and 3—but they are not my concern in this paper.
- 7.
The epistemic/dispositional distinction that I have just drawn is not the same as the traditional distinction between occurrent and dispositional belief: the latter is most naturally viewed as a distinction amongst epistemic beliefs.
- 8.
Note that this is a different kind of variation from the one we considered under (2b) above. There we were talking about a certain agent S in a certain context C, and imagining how S would set a betting ratio if asked to do so in C. Given that S is not actually asked in C, this means considering nearby worlds in which S is asked—and the thought was that S might not set the very same ratio in all of these worlds. Now the situation is different. We are imagining S first in actual context \(C_1\) and then in actual context \(C_2\), and we are imagining how S would set a betting ratio if asked to do so in each of these contexts. So now we are considering the ratio S sets in a counterfactual situation similar to \(C_1\) and the ratio S sets in a counterfactual situation similar to \(C_2\), where \(C_1\) and \(C_2\) are different contexts in the actual world (as opposed to the ratios S sets in two counterfactual situations, both of which are similar to the same actual context C).
- 9.
There is psychological evidence that agents’ explicit judgements or estimates of probability are variable and situation-dependent. This does not automatically mean, however, that agents’ behavioural dispositions are variable and context-dependent—that would follow only if degrees of belief (in the dispositional sense) are straightforwardly mirrored in estimates of probability. The possibility that I am considering now is that agents’ behavioural dispositions are variable and context-dependent.
- 10.
- 11.
Cf. also Kaplan (1996, 24.)
- 12.
Levi here refers to Peirce, Fisher, Neyman, Pearson and Kyburg as examples of earlier authors who held similar anti-Bayesian views. Cf. also Levi (1974, 394–5.)
- 13.
Thanks to Richard Dietz, Alan Hájek, Brian Hedden, Dan Lassiter, an anonymous referee, and audiences at a Current Projects seminar at the University of Sydney on 23 March 2017 and a Department of Philosophy seminar at the University of Adelaide on 7 September 2018 for helpful comments.
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Smith, N.J.J. (2019). Problems of Precision in Fuzzy Theories of Vagueness and Bayesian Epistemology. In: Dietz, R. (eds) Vagueness and Rationality in Language Use and Cognition. Language, Cognition, and Mind, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-15931-3_3
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