From conditional events to conditional measures: A new axiomatic approach

Our starting point is a definition of conditional event E\H which differs from many seemingly "similar" ones adopted in the relevant literature since 1935, starting with de Finetti. In fact, if we do not assign the same "third" value it ("undetermined") to all conditional events, but make it depend on E\H, it turns out that this function t(E\H) can be taken as a general conditional uncertainty measure, and we get (through a suitable - in a sense, "compulsory" - choice of the relevant operations among, conditional events) the "natural" axioms for many different (besides probability) conditional measures.