Partial algebraic conditional spaces

Conditioning plays a central role, both from a theoretical and practical point of view, in domains such as logic and probability, or rule-based expert systems. In classical approaches to probability, there is the notion of "conditional probability" P(E\H), but usually there is no meaning given to E\H itself. In 1935 de Finetti (5) was the first to mention "conditional events" outside the function P. We shall refer to a concept of conditional event extensively discussed in (4), where the idea of de Finetti of looking at E\H, with H not equal theta (the impossible event), as a three-valued logical entity (true when both E and H are true, false when H is true and E is false, "undetermined" when H is false) is generalized (or better, in a sense, is given up) by letting the third "value" t(E, H) suitably depend on the given ordered pair (E, H) and not being just an undetermined common value for all pairs. Here an axiomatic definition is given of Partial Algebraic Conditional Spaces (PACS), that is a set of conditional events endowed with two partial operations (denoted by circle plus and circle dot): we then show that the structure discussed through a betting scheme in 4 (i.e., a class of particular random variables with suitable partial sum and product) is a "natural" model of a PCAS. Moreover, it turns out that the map t(E, H) can be looked on - with this choice of the two operations circle plus and circle dot - as a conditional probability (in its most general sense related to the concept of coherence) satisfying the classic de Finetti - Popper axioms.