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Uncertainty, indeterminacy and fuzziness: a probabilistic approach. (English) Zbl 1206.03023
Hosni, Hykel (ed.) et al., Probability, uncertainty and rationality. Pisa: Edizioni della Normale (ISBN 978-88-7642-347-5/hbk). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series (Nuova Serie) 10, 219-242 (2010).
From the introduction: To measure uncertainty we need probability while to measure indeterminacy we need imprecise probabilities. Following de Finetti, we adopt a behavioral interpretation both of probability and imprecise probability. However, under the specific conditions of ignorance that will be discussed in this paper, de Finetti’s key assumption ceases to be compelling. As we shall see, it is hardly possible to find, for any given event, a betting rate that we regard as fair, i.e. such that we are willing to accept a bet both on and against the event at that rate. This is bound to happen when ever we have little information on which to base our assessment so that our beliefs about the events of interest are indeterminate. In situations of this sort it seems more prudent and overall more ‘rational’ to give those events a probability interval. Whenever this happens, we speak about imprecise probabilities, that is an upper probability strictly greater than lower probability.
For the entire collection see [Zbl 1186.03005].

03B48 Probability and inductive logic
03B52 Fuzzy logic; logic of vagueness
68T37 Reasoning under uncertainty in the context of artificial intelligence