1. In Definition XIII of chapter 12 a meaning was given to the statement that a1/h and a2/h are independent arguments. In Theorem (33) of chapter 14 it was shown that, if a1/h and a2/h are independent, a1a2/h = a1/h.a2/h. Thus where on given evidence there is independence between a1 and a2, the probability on this evidence of a1a2 jointly is the product of the probabilities of a1 and a2 separately. It is difficult to apply mathematical reasoning to the calculus of probabilities unless this condition is fulfilled; and the fulfilment of the condition has often been assumed too lightly. A good many of the most misleading fallacies in the theory of probability have been due to a use of the multiplication theorem in its simplified form in cases where this is illegitimate.

2. These fallacies have been partly due to the absence of a clear understanding as to what is meant by independence. Students of probability have thought of the independence of events, rather than of the independence of arguments or propositions. The one phraseology is, perhaps, as legitimate as the other; but when we speak of the dependence of events, we are led to believe that the question is one of direct causal dependence, two events being dependent if the occurrence of one is a part cause or a possible part cause of the occurrence of the other.