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of metals belong to elements. But it is hardly too much to say that Aristotle committed the greatest and most lamentable of all mistakes in the history of science when he took this kind of proposition as the true type of all propositions, and founded thereon his system. It was by a mere fallacy of accident that he was misled; but the fallacy once committed by a master-mind became so rooted in the minds of all succeeding logicians, by the influence of authority, that twenty centuries have thereby been rendered a blank in the history of logic.
13. Aristotle ignored the existence of an infinite number of definitions and other propositions which do not share the peculiarity of the example we have taken. If we define elements as substances which cannot be decomposed, this definition is of the form
Elements undecomposable substances;
and since the term element does not occur in the second member, we may apply the dictum usefully in both directions. Whatever we know of the term element we may assert of the distinct term undecomposable substance; and, vice versa, whatever we know of the term undecomposable substance we may assert of element.
1 In strictness we should add, by our present means.
Iron is the most useful of the metals; hardly needs quantification of the predicate, for it is evidently of the form
Iron = the most useful of the metals,
the terms being both singular terms, and convertible with each other. We may evidently infer of both terms what we know of either. If we join to the above the similar proposition,
we are easily enabled to infer that the cheapest of the metals = the most useful of the metals, since by the dictum we know of iron that it is the cheapest of the metals; and this we are enabled to assert of the most useful, and vice versa. These are almost self-evident forms of reasoning, and yet they were neither the foundation of Aristotle's system, nor were they included in the superstructure of that system. His syllogism was therefore an edifice in which the corner-stone itself was omitted, and the true system is to be created by supplying this omission, and re-erecting the edifice from the very foundation.
14. I am thus led to take the equation as the fundamental form of reasoning, and to modify
Aristotle's dictum in accordance therewith. It may then be formulated somewhat as follows:
Whatever is known of a term may be stated of its equal or equivalent.
Or in other words,
Whatever is true of a thing is true of its like.
I must beg of the reader not to prejudge the value of this very evident axiom. It is derived from Aristotle's dictum by omitting the distinction of the subject and predicate; and it may seem to have become thereby even a more transparent truism than the original, which has been condemned as such by Mr. J. S. Mill and some others. But the value of the formula must be judged by its results; and I do not hesitate to assert that it not only brings into harmony all the branches of logical doctrine, but that it unites them in close analogy to the corresponding parts of mathematical method. All acts of mathematical reasoning may, I believe, be considered but as applications of a corresponding axiom of quantity; and the force of the axiom may best be illustrated in the first place by looking at it in its mathematical aspect.
15. The axiom indeed with which Euclid begins to build presents at first sight little or no resem
blance to the modified dictum. The axiom asserts
Things equal to the same thing are equal to each
a = b = c
gives a = c.
Here two equations are apparently necessary in order that an inference may be evolved; and there is something peculiar about the threefold symmetrical character of the formula which attracts the attention, and prevents the true nature of the process of mind from being discovered. We get hold of the true secret by considering that an inference is equally possible by the use of a single equation, but that when there is no equation no inference at all can be drawn. Thus if we use the sign to denote the existence of an inequality or difference, then one equality and one inequality, as in
a = b c,
enable us to infer an inequality
Two inequalities, on the other hand, as in
a b c
do not enable us to make any inference concerning the relation of a and c; for if these quantities are
equal, they may both differ from b, and so they may if they are unequal. The axiom of Euclid thus requires to be supplemented by two other axioms, which can only be expressed in somewhat awkward language, as follows:—
If the first of three things be equal to the second, but the second be unequal to the third, the first is unequal to the third.
And again :
If two things be both unequal to a third common thing, they may or may not be equal to each other.
16. Reflection upon the force of these axioms and their relations to each other will show, I think, that the deductive power always resides in an equality, and that difference as such is incapable of affording any inference. My meaning will be more plainly exhibited by placing the symbols in the following form:
Here the inference is seen to be obtained by substituting a for 6 by virtue of their equality as expressed in the first equation a = b, the second equation b c being that in which substitution is