There is a small number of useful elementary theorems about how con(–) works to which I have appealed, both in the body of the book and in appendix 2, without first proving that these principles follow from the supposition that con(–) satisfies the Kolmogorov axioms of probability. This appendix is designed to provide, for the sake of those readers who are somewhat unfamiliar with the workings of the probability calculus, the missing proofs.

Recall that con(–) satisfies the Kolmogorov axioms of probability if and only if

A. con(–) assigns a real number to every hypothesis P and hypothesis Q in such a way that

1. (i) con(P) ≥ 0;

2. (ii) if P is a tautology, then con(P) = 1; and

3. (iii) if P and Q are mutually exclusive, then con(P v Q) = con(P) + con(Q).

The theorems that will be proved are

1. T1. con(P) = 1 – con(~P);

2. T2. if P entails Q, then con(Q) ≥ con(P);

3. T3. if P and Q are logically equivalent, con(P) = con(Q);

4. T4. con(P & Q) = con(P) – con(P & ~Q); and

5. T5. con(P v Q) = con(P) + con(Q) – con(P & Q).