Borderlines and probabilities of borderlines: On the interconnection between vagueness and uncertainty

We describe an integrated approach to vagueness and uncertainty within a propositional logic setting and based on a combination of three valued logic and probability. Three valued valuations are employed in order to model explicitly borderline cases and in this context we give an axiomatic characterisation of two well known three valued models; supervaluations and Kleene valuations. We then demonstrate the close relationship between Kleene valuations and a sub-class of supervaluations. Belief pairs are lower and upper measures on the sentences of the language generated from a probability distribution defined over a finite set of three valued valuations. We describe links between these measures and other uncertainty theories and we show the close relationship between Kleene belief pairs and a sub-class of supervaluation belief pairs. Finally, a probabilistic approach to conditioning is explored within this framework.