1 It might be more accurate to speak of ‘incompatible hypotheses with respect to any individual case’, or ‘mutually exclusive classes of events’.
2 The examples, of this kind, referring to human mortality are taken from the Carlisle tables. These differ considerably, as is well known, from other tables, but we have the high authority of De Morgan for regarding them as the best representative of the average mortality of the English middle classes at the present day.
3 I say, almost any proportion, because, as may easily be seen, arithmetic imposes certain restrictions upon the assumptions that can be made. We could not, for instance, suppose that all the black-haired men are short-sighted, for in any given batch of men the former are more numerous. But the range of these restrictions is limited, and their existence is not of importance in the above discussion.
4 Essay on Probabilities, p. 53. I have been reminded that in his article on Probability in the Encyclopædia Metropolitana he has stated that such rules involve no new principle.
5 This point will be fully discussed in a future chapter, after the general stand-point of an objective system of logic has been explained and illustrated.
6 Whitworth's Choice and Chance, Ed. II., p. 123. See also Boole's Laws of Thought, p. 370.
7 Opinions differ about the defence of such suppositions, as they do about the nature of them. Some writers, admitting the above assumption to be doubtful, call it the most impartial hypothesis. Others regard it as a sort of mean hypothesis.
8 Educational Times; Reprint, Vol.[ TN: space] xxxvii. p. 40. The question was proposed by Dr.[ TN: space] Macalister and gave rise to considerable controversy. As usual with problems of this inverse kind hardly any two of the writers were in agreement as to the assumptions to be made, or therefore as to the numerical estimate of the odds.
9 See Todhunter's History, pp. 333, 4.
There is an interesting discussion upon this question by the late J. D. Forbes in a paper in the Philosophical Magazine for Dec.[ TN: space] 1850. It was replied to in a subsequent number by Prof.[ TN: space] Donkin.