 When one wants to compare the probability of different events, say of selecting a black ball and selecting a white ball, it may be more convenient to consider probabilities to be terms in their own right. That is, an expression \(Px(\phi)\) is interpreted as referring to some rational number. Then one can extend the language with arithmetical operations such as addition and multiplication, and with operators such as equality and inequalities to compare probability terms. One can then say that one is twice as likely to select a black ball compared to a white ball as \(Px(B(x))=2 \times Px (W(x))\). Such an extension requires that the language contains two separate classes of terms: one for probabilities, numbers and the results of arithmetical operations on such terms, and one for the domain of discourse which the probabilistic operators quantify over. We will not present such a language and semantics in detail here. One can find such a system in Bacchus (1990).