A number of extensions to basic probability theory have been discussed in the literature that address precisely this problem. We do not go into details here about these approaches (a nice summary and detailed comparison between different approaches can be found in Halpern (2010) and instead sketch the main ideas. The first approach is to use so-called Popper functions which take conditional probability measures as primitive. That is, for each non-empty event \(E\), there is a probability measure \(p_E(\cdot)\) satisfying the usual Kolmogrov axioms (relativized to \(E\), so for example \(p_E(E)=1\)). A second approach assigns to each agent a finite sequence of probability measures \((p_1,p_2,\ldots,p_n)\) called a lexicographic probability system. The idea is that to condition on \(F\), first find the first probability measure not assigning zero to \(F\) and use that measure to condition on \(F\). Roughly, one can see each of the probability measures in a lexicographic probability system as corresponding to a level of a plausibility ordering. We will return to these notions in Section 5.2.