the remarkable process known as Eratosthenes’ Sieve is the only mode by which we can select the prime numbers.1 It bears a strong analogy to the indirect method here to be described. We can prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing that the contrary supposition inevitably leads to contradiction.2 Many other demonstrations in various branches of the mathematical sciences proceed upon a like method. Now, if there is only one important truth which must be, and can only be, proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its comparative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment.
Simple Illustrations.
In tracing out the powers and results of this method, we will begin with the simplest possible instance. Let us take a proposition of the common form, A = AB, say,
and let us investigate its full meaning. Any person who has had the least logical training, is aware that we can draw from the above proposition an apparently different one, namely,
While some logicians, as for instance De Morgan,3 have considered the relation of these two propositions to be purely self-evident, and neither needing nor allowing analysis, a great many more persons, as I have observed while teaching logic, are at first unable to perceive the close connection between them. I believe that a true and complete system of logic will furnish a clear analysis of this process, which has been called Contrapositive Conversion; the full process is as follows:—
Firstly, by the Law of Duality we know that
If it be metal, we know that it is by the premise an element; we should thus be supposing that the same thing is an element and a not-element, which is in opposition to the Law of Contradiction. According to the only other alternative, then, the not-element must be a not-metal.
To represent this process of inference symbolically we take the premise in the form
A = AB. (1)
We observe that by the Law of Duality the term not-B is thus described
b = A b ꖌ ab . (2)
For A in this proposition we substitute its description as given in (1), obtaining
But according to the Law of Contradiction the term ABb must be excluded from thought, or
Hence it results that b is either nothing at all, or it is ab; and the