3.8 Conditioning Ranking Functions

Defining conditional ranking is straightforward, using an analogue of the properties CP1–3 that were used to characterize probabilistic conditioning. A conditional ranking function κ is a function mapping a Popper algebra 2^W ′ to ℕ* satisfying the following properties:

Note that + is the analogue for ranking functions to in probability (and min in possibility). I motivate this shortly.

Given an unconditional ranking function κ, the unique conditional ranking function with these properties with domain 2^W ′, where ′ = {U: κ (U) ≠ ∞}, is defined via

(Exercise 3.25). This definition of conditioning is consistent with the order-of-magnitude probabilistic interpretation of ranking functions. If μ(U ∩ V) is roughly ∊^k and μ(U) is roughly ∊^m, then μ(V | U) is roughly ∊^k−m. This, indeed, is the motivation for choosing + as the replacement for in CRk4.

Notice that there is an obvious analogue of Bayes' Rule for ranking functions: