Towards an algebraic framework for many-valued conditional probability

We investigate probability functions defined over many valued conditional events. During the last decade or so considerable research effort has been directed towards the understanding of what (subjective) conditional probability might look like in the context of many-valued logics. Given its centrality in the area, a particularly well-studied case is that of Lukasiewicz’s infinite-valued logic L∞ (cf. [1, 5]). Among the initial attempts to define conditionals on such logics let us recall Gerla’s [4] and Kroupa’s [6, 7], whilst more recent characterizations are due to Mundici [12, 13] and Montagna [10]. A particularly hard problem in defining MV-conditional probability originates from the fact that MV-logics have no unique notion of conjunction. So, if we are to define conditional probability via the so-called rule of compound probabilities P (θ | φ) = P (θ∧φ) P (φ) we face the problem of justifying one particular choice of conjunction. This is by no means trivial unless we are happy to settle with a purely conventional (i.e. arbitrary) choice. Yet conditionals do have an intuitive semantics and our work aims at providing a formalization of probabilistic conditionals which goes some way towards capturing those intuitions. The main idea of our work consists in treating conditional probability as a simple (i.e. unconditional) probability on a conditional algebra. In particular we are interested in defining a notion of probability satisfying two sorts of intuitive constraints. The former concerns the semantics of the underlying logic which we assume is many-valued. This allows us to model the concept of conditional many-valued event. The latter sort of constraints we shall be looking at, concerns the properties of the probability function, which we take to represent an agent’s subjective degree of belief in the spirit of de Finetti’s work [2, 3]. This opens the problem of defining a suitable notion conditional bet. We propose the rejection constraint according to