The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition of the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposition is true. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium chloride. If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, “I will give anybody their choice of these two truths, which amount to the same thing; He who loves himself best is the honestest man; or, The honestest man loves himself best.” Scotus Erigena exactly expresses this reciprocal character of identity in saying, “There are not two studies, one of philosophy and the other of religion; true philosophy is true religion, and true religion is true philosophy.” A mathematician would not think it worth while to mention that if *x* = *y* then also *y* = *x*. He would not consider these to be two equations at all, but one equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity there is no need for succession in order (see p. 33), each is simultaneously equal and identical to the other. These remarks will hold true both of logical and mathematical identity; so that I shall consider the two forms A = B and B = A to express exactly the same identity differently written. All need for rules of conversion disappears, and there will be no single proposition in the system which may not be written with either end foremost. Thus A = AB is the same as AB = A, *a*C = *b*C is the same as *b*C = *a*C, and so forth.