And now it is not difficult to guess why we employ such expressions as we do to indicate strict identity. If I habitually use the proposition "x is y" to indicate a relation between two things having similar elements and yet regarded as distinct, and look upon the proposition as justified by the similar elements, observing that, these remaining unchanged, the dissimilar elements may be very variable without affecting the truth of the proposition, what more natural than that I should go on using the propositional form when the dissimilar elements have diminished to zero--when the proposition has become "x is x"? To be sure, no one can take such a proposition literally, any more than one can soberly believe that one divided by zero results in infinity. Such expressions have their use and value, but they must be properly understood. If one uses the expression "x is x" to emphasize the fact that one is not to pass from x to any y or z--that one is to rule out all distinction or sense of difference, the use cannot be harmful. And the use of the propositional form has this great convenience: it puts a period, so to speak, to one's thinking, and prevents one from casting about for a completion of the thought. If one merely say to me "x," I shall probably take it as a subject and busy myself to find a predicate. If he say "x is x," he says really no more than x, but he makes me fix my thoughts upon x alone. SEC. 20. In the foregoing search for the element that the kinds of sameness have in common, I have had in mind chiefly the samenesses of things immediately known. It is not necessary to repeat the search in the field of the "external." We have but the seven kinds of sameness, and whatever may be the things that are the same in these several ways, the elements I have indicated must be present if our words are to be significant. But one thing remains for me to do in this part of my monograph, and that will not detain me long. I must distinguish between sameness and identity, or rather point out to what kinds of sameness this latter word is commonly applied.