Reasoning about Probability via Continuous Functions.

For a functional representation of a logic we mean a representation of its formulas by means of (possibly real-valued) functions. Functional representations have shown to be a key tool for the study of (algebraizable) non-classical logics, since they allow one to approach the study of typical proof theoretical properties via the functional semantics. Interestingly, the functional representation for Łukasiewicz logic has been very recently shown to have an impact outside the purely logical realm; indeed, it can be applied to study properties of artificial neural networks. In this contribution we will provide a functional representation for a probability modal logic, FP(Ł), that builds on Łukasiewicz calculus by adding to it a unary operator P reading "it is probable that". While this logic is not algebraizable in the usual sense, we still can provide a functional representation for its probability formulas. Our contribution will present two ways of providing a functional representation of the formulas of the modal logic FP(Ł): a local one, that relies on de Finetti's coherence argument; and a global one that, instead, relies on probability distributions on a finite domain.