In this appendix, I will present some of the basic ideas of the mathematical theory of probability. As in the case of Appendix 1, this will not be a comprehensive or detailed survey; it is only intended to introduce the basic formal probability concepts and rules used in this book, and to clarify the terminology and notation used in this book. Here I will discuss only the abstract and formal calculus of probability; in Chapter 1, the question of interpretation is addressed.

A probability function, Pr, is any function (or rule of association) that assigns to (or associates with) each element X of some Boolean algebra B (see Appendix 1) a real number, Pr(X), in accordance with the following three conditions:

For all X and Y in B,

1. Pr(X) 0;

2. Pr(X) = 1, if X is a tautology (that is, if X is logically true, or X = 1 in B);

3. Pr(X∨Y) = Pr(X) + Pr(Y), if X&Y is a contradiction (that is, if X&Y is logically false, or X&Y = 0 in B).

These three conditions are the probability axioms, also called “the Kolmogorov axioms” (for Kolmogorov 1933). A function Pr that satisfies the axioms, relative to an algebra B, is said to be a probability function on B – that is, with “domain” B (that is, the set of propositions of B) and range the closed interval [0,1]. In what follows, reference to an assumed algebra B will be implicit.