Conditional independence and natural conditional functions

The concept of conditional independence (CI) within the framework of natural conditional functions (NCFs) is studied. An NCF is a function ascribing natural numbers to possible states of the world; it is the central concept of Spohn's theory of deterministic epistemology. Basic properties of CI within this framework are recalled, and further results analogous to the results concerning probabilistic CI are proved. Firstly, the intersection of two CI-models is shown to be a CI-model. Using this, it is proved that Cl-models for NCFs have no finite complete axiomatic characterization (by means of a simple deductive system describing relationships among Cl-statements). The last part is devoted to the marginal problem for NCFs. It is shown that (pairwise) consonancy is equivalent to consistency iff the running intersection property holds. K E Y W O R D S : natural conditional function, conditional independence, axiomatic characterization, marginal problem, running intersection property