Abstract
If we think, as Ramsey did, that a degree of belief that P is a stronger or weaker tendency to act as if P, then it is clear that not only uncertainty, but also vagueness, gives rise to degrees of belief. If I like hot coffee and do not know whether the coffee is hot or cold, I will have some tendency to reach for a cup; if I like hot coffee and know that the coffee is borderline hot, I will have some tendency to reach for a cup. Suppose that we take degrees of belief arising from uncertainty to obey the laws of probability and that we model vagueness using degrees of truth. We then encounter a problem: it does not look as though degrees of belief arising from vagueness should obey the laws of probability. One response would be to countenance two different sorts of degrees of belief: degrees of belief arising from uncertainty, which obey the laws of probability; and degrees of belief arising from vagueness, which obey a different set of laws. I argue, however, that if a degree of belief that P is a stronger or weaker tendency to act as if P, then this option is not open. Instead, I propose an account of the behaviour of degrees of belief that integrates subjective probabilities and degrees of truth. On this account, degrees of belief are expectations of degrees of truth. The account explains why degrees of belief behave in accordance with the laws of probability in cases involving only uncertainty, while also allowing degrees of belief to behave differently in cases involving only vagueness, and in mixed cases involving both uncertainty and vagueness. Justifications of the account are given both via Dutch books and in terms of epistemic accuracy.
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Notes
This brings us into the realm of fuzzy logics. At this stage, however, let us make no particular choice of fuzzy logic: no choice of particular operations on reals to associate with the connectives, no particular definition of logical consequence, and so on. So the discussion remains at a fairly abstract level, and goes through regardless of which particular choices we might make within the general realm of fuzzy logics.
For example, the degree of truth of A ∧ B is not, in general, the value that the probability calculus outputs for A ∧ B when given as inputs the degrees of truth of A and B.
Schiffer (2000) holds a view of this sort.
For details of these logics, see e.g. Smith (2012).
For proofs of the four propositions, see Smith (2010). The setting was less general in the earlier paper—but the proofs carry over, mutatis mutandis.
It need not hold if we define consequence in other ways.
It is the fact that s may be positive or negative that ensures that B(θ) is fair; cf. ‘You cut, I choose’ (de Finetti 1974, 86, n.\(\ddag\)). Note that if s is negative, then the agent ‘paying’ s × B(θ) to enter into the bet amounts to his being paid −s × B(θ) to enter into the bet, and the agent ‘being paid’ s amounts to his paying −s.
If θ turns out to be true—V(θ) = 1—the agent’s net return is the winnings—s—minus the price he paid to enter the bet—s × B(θ). If θ turns out to be false—V(θ) = 0—the agent’s net return is the ‘winnings’—0—minus the price he paid to enter the bet—s × B(θ).
It may be that two distinct worlds w and w′ determine the same point in \({\mathbb R^m: }\) this will happen if w and w′ assign the same degrees of truth to the propositions \(\theta_1\ldots\theta_m\).
f need not be one-one (cf. n.14).
Cf. de Finetti (1974, 58).
Of course, as far as this converse result is concerned, it need not be the probability measure that represents one’s own epistemic state.
This will not be defined if P(S n ) = 0. We must therefore assume that the agent does not start out with a probability measure that assigns measure zero to any set V of worlds such that she might subsequently get evidence that the actual world is in V.
See Dubois and Prade (1988).
See e.g. Smets (1981).
Versions of this paper were presented at a seminar at the Institute of Computer Science of the Academy of Sciences of the Czech Republic in Prague on 18 September 2009, at a Philosophy RSSS seminar at the Australian National University on 26 November 2009, at the Prague International Colloquium on Epistemic Aspects of Many-Valued Logic at the Institute of Philosophy of the Academy of Sciences of the Czech Republic on 15 September 2010, at the Metaphysical Indeterminacy Workshop II at the University of Leeds on 9 September 2011, at a joint session of the Australasian Association of Logic annual conference and the Twelfth Asian Logic Conference at Victoria University of Wellington on 15 December 2011, and at the Probability and Vagueness conference at the University of Tokyo on 21 March 2013; thanks to the audiences on those occasions for useful feedback. For helpful correspondence, I am grateful to Jeff Paris and Robbie Williams. Thanks to the two anonymous referees for their comments, and to the Australian Research Council for research support.
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Smith, N.J.J. Vagueness, Uncertainty and Degrees of Belief: Two Kinds of Indeterminacy—One Kind of Credence. Erkenn 79, 1027–1044 (2014). https://doi.org/10.1007/s10670-013-9588-3
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DOI: https://doi.org/10.1007/s10670-013-9588-3