We argue that the model of probabilities needs revising when non-classical logics are considered. For strong-Kleene logic we suggest a belief-pair, and for supervaluational logic adopt imprecise probability.
We show that Moss's model of uncertainty is at least as expressively powerful as every other current imprecise probability framework. And we give a Dutch Book argument for certain failures of consistency.
Certain scenarios where one's beliefs undermine themselves can be paralleled with the liar paradox. An application of the supervaluational Kripke construction for the liar can be seen as leading to imprecise probabilities as the rational response.
We suggest that accuracy considerations should apply to the imprecise by using: what a set recommends is the set of what the individuals in it recommend. This results in a surprisingly nice picture of accuracy for the imprecise.