Accept & reject statement-based uncertainty models

•We develop a framework for modelling and reasoning based on a pair of gamble sets.•We translate the framework to one in terms of a pair of partial preference orders.•Our framework generalises most frameworks based on a linear utility assumption.•Particularly, it generalises the theories of coherent linear and lower previsions.•We show the framework's usefulness with a symmetry judgements modelling example.

We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss—as a bridge to existing frameworks—a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.