A second highly important kind of proposition is that which I propose to call *a partial identity*. When we say that “All mammalia are vertebrata,” we do not mean that mammalian animals are identical with vertebrate animals, but only that the mammalia form a *part of the class vertebrata*. Such a proposition was regarded in the old logic as asserting the inclusion of one class in another, or of an object in a class. It was called a universal affirmative proposition, because the attribute *vertebrate* was affirmed of the whole subject *mammalia*; but the attribute was said to be *undistributed*, because not all vertebrata were of necessity involved in the proposition. Aristotle, overlooking the importance of simple identities, and indeed almost denying their existence, unfortunately founded his system upon the notion of inclusion in a class, instead of adopting the basis of identity. He regarded inference as resting upon the rule that what is true of the containing class is true of the contained, in place of the vastly more general rule that what is true of a class or thing is true of the like. Thus he not only reduced logic to a fragment of its proper self, but destroyed the deep analogies which bind together logical and mathematical reasoning. Hence a crowd of defects, difficulties and errors which will long disfigure the first and simplest of the sciences. It is surely evident that the relation of inclusion rests upon the relation of identity. Mammalian animals cannot be included among vertebrates unless they be identical with part of the vertebrates. Cabinet Ministers are included almost always in the class Members of Parliament, because they are identical with some who sit in Parliament. We may indicate this identity with a part of the larger class in various ways; as for instance, Mammalia = part of the vertebrata. Diatomaceæ = a class of plants. Cabinet Ministers = some members of Parliament. Iron = a metal. In ordinary language the verbs *is* and *are* express mere inclusion more often than not. *Men are mortals*, means that *men* form a part of the class *mortal*; but great confusion exists between this sense of the verb and that in which it expresses identity, as in “The sun is the centre of the planetary system.” The introduction of the indefinite article *a* often expresses partiality; when we say “Iron is a metal” we clearly mean that iron is *one only* of several metals.