Coherence and Fuzzy Reasoning

Upper and lower conditional previsions are defined by the Choquet integral with respect to the Hausdorff outer and inner measures when the conditioning events have positive and finite Hausdorff outer or inner measures in their dimension; otherwise, when conditioning events have infinite or zero Hausdorff outer or inner measures in their dimension, they are defined by a 0-1 valued finitely, but not countably additive probability. It is proven that, if we consider the restriction of the (outer) Haudorff measures to the Borel σ-field, these (upper) conditional and unconditional previsions satisfy the disintegration property in the sense of Dubins with respect to all countable partitions of Ω. This result is obtained as a consequence of the fact that non-disintegrability characterizes finitely as opposed to countably additive probability. Moreover upper and lower conditional previsions are proven to be coherent, in the sense of Walley, with the unconditional previsions. Properties related to the coherence of upper conditional probabilities are extended to the case where information is represented by fuzzy sets. In particular, given an infinite set Ω, a conditioning rule for possibility distribution is proposed so that it is coherent and it is coherent with the unconditional possibility distribution. Through this conditional possibility distribution, a conditional possibility measure with respect to the partition of all singletons of [0,1] is defined. It is proved it satisfies the conglomerative principle of de Finetti.