Reasoning with uncertainty and context-dependent languages

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where imprecise probabilities and imprecise probabilistic reasoning can be represented. In this way we obtain a mathematical de¯nition of imprecise-probabilistic sets. A valid and complete proof system, w.r.t. the intended semantics of imprecise probabilities, is described using the internal language of the topos.