have previously obtained, and an immense number of others or which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the class not-element or c. By the Law of Duality we can develop c into four alternatives, thus
If we substitute for A and B as before, we get
and, striking out the terms which break the Law of Contradiction, there remains
or what is not element is also not iron and not metal. This Indirect Method of Inference thus furnishes a complete solution of the following problem—Given any number of logical premises or conditions, required the description of any class of objects, or of any term, as governed by those conditions.
The steps of the process of inference may thus be concisely stated—
- By the Law of Duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises.
- For each term in these alternatives substitute its description as given in the premises.
- Strike out every alternative which is then found to break the Law of Contradiction.
- The remaining terms may be equated to the term in question as the desired description.
Mr. Venn’s Problem.
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.5 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.
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