It is popular to model one’s attitudes as probabilistic degrees of belief; and various arguments are given for this, such as the Dutch book argument or accuracy argument. However, this model and the corresponding justifications assume classical logic; e.g. someone should have degree of belief 1 that the wall is either red or not red – but perhaps the wall is vaguely red and some alternative attitude should be allowed.
We question what the attitudes of a rational believer should be like if the underlying logic is non-classical. And we argue that we need to more radically revise the picture of probabilistic belief, they should no longer simply assign numerical values to sentences.
We argue that for strong Kleene logics one’s beliefs are represented as a pair of numerical values, and for supervaluational logic, one should adopt sets of probabilities, i.e. so-called imprecise probabilities.
And we consider how the accuracy and Dutch book arguments would apply to these alternative models of belief.