Reasoning about Probability via Continuous Functions

Abstract

For functional representation in an algebraizable logic we mean a representation of the algebras of formulas of the logic by means of (possibly real-valued) functions. Functional representations have shown to be a key tool for the study of non-classical logics, since they allow to regard formulas as functions and, by means of them, to approach the study of typical proof theoretical properties of the logics by means of their functional semantics. In the realm of (algebraizable) fuzzy logics, possibly the most well-known result in this respect is McNaughton theorem that shows formulas of the infinite-valued Lukasiewicz calculus to correspond, up to logical equivalence, to real valued continuous and piecewise linear functions. The functional representation for many-valued logics has been very recently shown in a paper by the second author to have an impact outside the purely logical realm and, in particular, they can be applied to study properties of artificial neural networks. In this contribution, we will provide a functional representation for the probability modal logic FP(L) that builds on Lukasiewicz calculus by adding to it a unary operator P that reads “it is probable that”. While the logic FP(L) is not algebraizable, at least not in the usual sense due to Blok and Pigozzi, we still can provide a functional representation result for its modal formulas. In order to do so, we adapt the usual universal algebraic methods to this peculiar setting, and moreover we make use of some techniques developed in a recent paper by two of the authors where a class of purely algebraic models for FP(L) based on de Finetti's coherence criterion have been introduced and studied. Our contribution will present two ways of providing a functional representation of the algebras of formulas of the modal logic FP(L): 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.