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connexion with the former one, seems to give much clearer information as to the conditions under which inference is possible than do the formulæ of inequalities, and I entertain no doubt at all that, even when an inference seems to be obtained without the use of an equation, a disguised substitution is really performed by the mind, exactly such as represented in the equations. But I can only assert my belief of this from the examination of the process in my own mind, and I must submit to the reader's judgment whether there are exceptions or not to the rule, that we always reason by means of identities or equalities.

19. Turning now to apply these considerations to the forms of logical inference, my proposed simplification of the rules of logic is founded upon an obvious extension of the one great process of substitution to all kinds of identity. The Latin word æqualis, which is the original of our equal, was not restricted in signification to similarity of quantities, but was often applied to anything which was unvaried or similar when compared with another. We have but to interpret the word equal in the older and wider sense of like or equivalent, in order to effect the long-desired union of logical and mathematical reasoning. For it For it is not difficult to show that all forms of reasoning consist in

repeated employment of the universal process of the substitution of equals, or, if the phrase be preferred, substitution of similars.

20. To prevent a confusion of mathematical and logical applications of the formula, it will be desirable to use large capital letters to denote the things compared in a logical sense, but the copula or sign of identity may remain as before. Thus the symbols

A = B

denote the identity of the things represented by the indefinite terms or names A and B. Thus A may be taken in one case to mean Iron, when B might mean the cheapest of the metals, or the most useful of the metals. In another example which we have used A would mean element, and B that which cannot be decomposed, and so on. The fundamental principle of reasoning authorizes us to substitute the term on one side of an identity for the other term, wherever this may be encountered, so that in whatever relation B stands to a third thing C, in the same relation A must stand to C. Or, using the sign to denote any possible or conceivable kind of relation, the formula

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represents a self-evident inference. Thus, If C be the father of B, C is father of A; If C be a compound of B, C is a compound of A ; If C be the absence of B, C is the absence of A; If C be identical with B, C is identical with A; and so on.

21. We may at once proceed to develop from this process of substitution all the forms of inference recognised by Aristotle, and many more. In the first place, there cannot be a simpler act of reasoning than the substitution of a definition for a term defined; and though this operation found no place in the old system of the syllogism, it ought to hold the first place in a true system. If we take the definition of element as

Element = undecomposable substance,

we are authorized to employ the terms element and undecomposable substance in lieu of each other in whatever relation either of them may be found. If we describe iron as a kind of element, it may also be described as a kind of undecomposable substance.

22. Sometimes we may have two definitions of the same term, and we may then equate these to each other. Thus, according to Mr. Senior,

(1) Wealth whatever has exchangeable value. (2) Wealth whatever is useful, transferable, and limited in supply.

We can employ either of these to make a substitution in the other, obtaining the equation,


Whatever has exchangeable value = whatever is useful, transferable, and limited in supply.

Where we have one affirmative proposition or equation, and one negative proposition, we still find the former sufficient for the process of inferThus


(1) Iron = the most useful metal.

(2) Iron


the metal most early used by primitive

By substituting in (2) by means of (1) we have the metal most early

The most useful metal

used by primitive nations.



23. But two negative propositions will of course give no result. Thus the two propositions,

the highest mountain in Great Britain, the highest mountain in the world,

do not allow of any substitution, and therefore do not give any means of inferring whether or not the highest mountain in Great Britain is the highest mountain in the world.

24. Postponing to a later part of this tract (§ 36) the consideration of negative forms of inference, I will now notice some inferences which involve combinations of terms. However many nouns, substantive or adjective, may be joined together, we may substitute for each its equivalent. Thus, if we have the propositions,

Square = equilateral rectangle,
Equilateral = equal-sided,

Rectangle = right-angled quadrilateral,
Quadrilateral four-sided figure,


we may by evident substitutions obtain

Square equal-sided, right-angled, four-sided



25. It is desirable at this point to draw attention to the fact that the order in which nouns adjective are stated is a matter of indifference. A foursided, equal-sided figure is identically the same as an equal-sided, four-sided figure; and even when it sometimes seems inelegant or difficult to alter the order of names describing a thing, it is grammatical usage, not logical necessity, which stands in the way. Hence, if A and B represent any two names or terms, their junction as in A B will be taken to

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