Page:Mind (New Series) Volume 6.djvu/520

 504 HUGH MACCOLL: Many persons who are ready enough to admit the utility of mathematics profess scepticism as to the advantages of symbolic logic. To these I would remark, firstly, that the latter is the more general science and includes the former ; and, secondly, that, apart from its aid to accurate thinking in general, symbolic logic has already (as shown in my first paper in the Proceedings of the Mathematical Society) rendered important assistance in one of the most difficult and perplexing parts of mathematics. It was a true instinct that led Boole to attempt the construction of what he called a " General Method in Probabilities," founded on symbolic logic ; but unfortunately Boole allowed an essentially false principle to vitiate his whole reasoning, so that his elaborate " General Method in Probabilities," into whose service he presses the highest branches of mathematics, becomes, alas ! from this one flaw, a gigantic and imposing fallacy. His solution of the " challenge problem " which he proposed as a test of the power and efficacy of his method I have proved to be wrong in my fourth paper in the Proceedings of the Mathematical Society; and, in my controversy with Dr. MacAlister (see vol. xxxvii. of Mathe- matical Questions with their Solutions from the Educational Times), I think I have succeeded in laying bare the subjective fallacy over which Boole and many others after him have stumbled into error. " When the probabilities of events are given" says Boole, " but all information respecting their dependence withheld, the mind regards them as independent " (Laws of Thought, p. 256). And further on he says : " We must regard the events as inde- pendent of any connexion beside that of which we have information". In other words, when we have no informa- tion as to any connexion between A and B, but know (from observation or otherwise) that the chance of A happening is a and that of B happening is 6, we may infer that the chance of both happening is ab an utterly fallacious principle. Cases in which, from these data, the chance of the concurrence is ab, and cases in which it is not ab, may be exhibited to the eye by a simple geometrical con- struction, fixed and unvarying, as I have shown in my fourth paper in the Proceedings of the Mathematical Society, so that the chance remains always the same whether or not " the mind regards the events as independent ". No one can admire Boole's Laws of Thought more than I do. As a philosophical and speculative work it is brimful of profound thought and original suggestions, while its style is charmingly lucid and attractive ; but none the less must