The need of some logical method more powerful and comprehensive than the old logic of Aristotle is strikingly illustrated by Mr. Venn in his most interesting and able article on Boole’s logic.[76] An easy example, originally got, as he says, by the aid of my method as simply described in the *Elementary Lessons in Logic*, was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic. It was answered by, at most, five or six of them. It was afterwards set, as an example on Boole’s method, to a small class who had attended a few lectures on the nature of these symbolic methods. It was readily answered by half or more of their number. [76] *Mind*; a Quarterly Review of Psychology and Philosophy; October, 1876, vol. i. p. 487. The problem was as follows:--“The members of a board were all of them either bondholders, or shareholders, but not both; and the bondholders as it happened, were all on the board. What conclusion can be drawn?” The conclusion wanted is, “No shareholders are bondholders.” Now, as Mr. Venn says, nothing can look simpler than the following reasoning, *when stated*:--“There can be no bondholders who are shareholders; for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements; nor off it, by the second.” Yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem. By symbolic statement the problem is instantly solved. Taking A = member of board B = bondholder C = shareholder the premises are evidently A = AB*c* ꖌ A*b*C B = AB. The class C or shareholders may in respect of A and B be developed into four alternatives, C = ABC ꖌ A*b*C ꖌ *a*BC ꖌ *ab*C. But substituting for A in the first and for B in the third alternative we get C = ABC*c* ꖌ AB*b*C ꖌ A*b*C ꖌ *a*ABC ꖌ *ab*C. The first, second, and fourth alternatives in the above are self-contradictory combinations, and only these; striking them out there remain C = A*b*C ꖌ *ab*C = *b*C, the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn’s reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method. *Abbreviation of the Process.* Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a thorough solution of a logical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are AB *a*B A*b* *ab*.