Every argument by inequalities may then be thrown into the form of an equality; but the converse is not true. We cannot possibly prove that two quantities are equal by merely asserting that they are both greater or both less than another quantity. From *e* > *f* and *g* > *f*, or *e* < *f* and *g* < *f*, we can infer no relation between *e* and *g*. And if the reader take the equations *x* = *y* = 3 and attempt to prove that therefore *x* = 3, by throwing them into inequalities, he will find it impossible to do so. From these considerations I gather that reasoning in arithmetic or algebra by so-called inequalities, is only an imperfectly expressed reasoning by equalities, and when we want to exhibit exactly and clearly the conditions of reasoning, we are obliged to use equalities explicitly. Just as in pure logic a negative proposition, as expressing mere difference, cannot be the means of inference, so inequality can never really be the true ground of inference. I do not deny that affirmation and negation, agreement and difference, equality and inequality, are pairs of equally fundamental relations, but I assert that inference is possible only where affirmation, agreement, or equality, some species of identity in fact, is present, explicitly or implicitly. *Arithmetical Reasoning.* It may seem somewhat inconsistent that I assert number to arise out of difference or discrimination, and yet hold that no reasoning can be grounded on difference. Number, of course, opens a most wide sphere for inference, and a little consideration shows that this is due to the unlimited series of identities which spring up out of numerical abstraction. If six people are sitting on six chairs, there is no resemblance between the chairs and the people in logical character. But if we overlook all the qualities both of a chair and a person and merely remember that there are marks by which each of six chairs may be discriminated from the others, and similarly with the people, then there arises a resemblance between the chairs and the people, and this resemblance in number may be the ground of inference. If on another occasion the chairs are filled by people again, we may infer that these people resemble the others in number though they need not resemble them in any other points.