Zero-Probability and Coherent Betting: A Logical Point of View

The investigation reported in this paper aims at clarifying an important yet subtle distinction between (i) the logical objects on which measure theoretic probability can be defined, and (ii) the interpretation of the resulting values as rational degrees of belief. Our central result can be stated informally as follows. Whilst all subjective degrees of belief can be expressed in terms of a probability measure, the converse doesn’t hold: probability measures can be defined over linguistic objects which do not admit of a meaningful betting interpretation. The logical framework capable of expressing this will allow us to put forward a precise formalisation of de Finetti’s notion of event which lies at the heart of the Bayesian approach to uncertain reasoning.