The point that beliefs are held as probabilities requires amplification. In classical logic, if you have a set S={X,Y,Z,...} of n variables, your state of knowl

The point that beliefs are held as probabilities requires amplification. In classical logic, if you have a set S={X,Y,Z,...} of n variables, your state of knowledge is a map from S to {true, false}. Thus there are 2^n possibilities. The usual idea is to say that false=0 and true=1. Then one tries to model uncertainty with intermediate numerical values, replacing 2^n discrete possibilities with an n-dimensional space. This doesn't work and Bayesian reasoning does something else. With n variables there are 2^n basic conjunctions. This will be familiar from writing out truth tables, or perhaps from not writing out truth tables because they contain too many lines. The Bayesian approach assigns a probability to every basic conjunction and thus lives in a 2^n dimensional space. How can Bayes rule extend logic when it appears to be simpler? I think the answer is that it is working on a kind of "exploded" representation.