Probability theory developed as a means of analyzing the notion of chance and, as a mathematical discipline, it has developed in a rigorous manner based on a system of precise definitions and axioms. However, the syntax of probability theory exists independently of its use as a means of expressing and manipulating information and of quantifying the semantic notions of belief and likelihood regarding natural entities. In this book we employ the syntax of probability theory to quantify semantic notions that relate to the synthesis of artificial entities. In this appendix we present the basic notions of probability theory. However, since much of the terminology is motivated by the historical semantics, it will be necessary to supplement the standard treatment with terminology to render the definitions more relevant to our usage.

We begin by establishing the notation that is used in this book.

Definition C.1

A set is a collection of simple entities, called elements. If A is a set and the points ω1, ω2, … are its elements, we denote this relationship by the notation

A = {ω1, ω2, …}.

If ω is an element of the set A, we denote this relationship by the notation ω ∈ A, where ∈ is the element inclusion symbol.

Often, we will specify a set by the properties of the elements. Suppose A comprises the set of all points ω such that ω ∈ S possesses property P.