system of species between which there are unbridgeable chasms; evolution tends more and more to the final establishment of "real kinds," marked by the fact that there is no permanent possibility of cross-breeding between them. This makes it once more possible to distinguish between a "nominal" definition and a "real" definition. From an evolutionary point of view, a "real" definition would be one which specifies not merely enough characters to mark off the group defined from others, but selects also for the purpose those characters which indicate the line of historical development by which the group has successively separated itself from other groups descended from the same ancestors. We shall learn yet more of the significance of this conception of a "real kind" as we go on to make acquaintance with the outlines of First Philosophy. Over the rest of the formal logic of Aristotle we must be content to pass more rapidly. In connection with the doctrine of Propositions, Aristotle lays down the familiar distinction between the four types of proposition according to their quantity (as universal or particular) and quality (as affirmative or negative), and treats of their contrary and contradictory opposition in a way which still forms the basis of the handling of the subject in elementary works on formal logic. He also considers at great length a subject nowadays commonly excluded from the elementary books, the modal distinction between the Problematic proposition (x may be y), the Assertory (x is y), and the Necessary (x must be y), and the way in which all these forms may be contradicted. For him, modality is a formal distinction like quantity or quality, because he believes that contingency and necessity are not merely relative to the state of our knowledge, but represent real and objective features of the order of Nature.

In connection with the doctrine of Inference, it is worth while to give his definition of Syllogism or Inference (literally "computation") in his own words. "Syllogism is a discourse wherein certain things (viz. the premisses) being admitted, something else, different from what has been admitted, follows of necessity because the admissions are what they are." The last clause shows that Aristotle is aware that the all-important thing in an inference is not that the conclusion should be novel but that it should be proved. We may have known the conclusion as a fact before; what the inference does for us is to connect it with the rest of our knowledge, and thus to show why it is true. He also formulates the axiom upon which syllogistic inference rests, that "if A is predicated universally of B and B of C, A is necessarily predicated universally of C." Stated in the language of class-inclusion, and adapted to include the case where B is denied of C this becomes the formula, "whatever is asserted universally, whether positively or negatively, of a class B is asserted in like manner of any class C which is wholly contained in B," the axiom de omni et nullo of mediæval logic. The syllogism of the "first figure," to which this principle immediately applies, is accordingly regarded by Aristotle as the natural and perfect form of inference. Syllogisms of the second and third figures can only be shown to fall under the dictum by a process of "reduction" or transformation into corresponding arguments in the first "figure," and are therefore called "imperfect" or "incomplete," because they do not exhibit the conclusive force of the reasoning with equal clearness, and also because no universal affirmative conclusion can be proved in them, and the aim of science is always to establish such affirmatives. The list of "moods" of the three figures, and the doctrine of the methods by which each mood of the imperfect figures can be replaced by an equivalent mood of the first is worked out substantially as in our current text-books. The so-called "fourth" figure is not recognised, its moods being regarded merely as unnatural and distorted statements of those of the first figure.

Induction.--Of the use of "induction" in Aristotle's philosophy we shall speak under the head of "Theory of Knowledge." Formally it is called "the way of proceeding from particular facts to universals," and Aristotle insists that the conclusion is only proved if all the particulars have been examined. Thus he gives as an example the following argument, "x, y, z are long-lived species of animals; x, y, z are the only species which have no gall; ergo all animals which have no gall are long-lived." This is the "induction by simple enumeration" denounced by Francis Bacon on the ground that it may always be discredited by the production of a single "contrary instance," e.g. a single instance of an animal which has no gall and yet is not long-lived. Aristotle is quite aware that his "induction" does not establish its conclusion unless all the cases have been included in the examination. In fact, as his own example shows, an induction which gives certainty does not start with "particular facts" at all. It is a method of arguing that what has been proved true of each sub-class of a wider class will be true of the wider class as a whole. The premisses are strictly universal throughout. In general, Aristotle does not regard "induction" as proof at all. Historically "induction" is held by Aristotle to have been first made prominent in philosophy by Socrates, who constantly employed the method in his attempts to establish universal results in moral science. Thus he gives, as a characteristic argument for the famous Socratic doctrine that knowledge is the one thing needful, the "induction," "he who understands the theory of navigation is the best navigator, he who understands the theory of chariot-driving the best driver; from these examples we see that universally he who understands the theory of a thing is the best practitioner," where it is evident that all the relevant cases have not been examined, and consequently that the reasoning does not amount to proof. Mill's so-called reasoning from particulars to particulars finds a place in Aristotle's theory under the name of "arguing from an example." He gives as an illustration, "A war between Athens and Thebes will be a bad thing, for we see that the war between Thebes and Phocis was so." He is careful to point out that the whole force of the argument depends on the implied assumption of a universal proposition which covers both cases, such as "wars between neighbours are bad things." Hence he calls such appeals to example "rhetorical" reasoning, because the politician is accustomed to leave his hearers to supply the relevant universal consideration for themselves.

Theory of Knowledge.--Here, as everywhere in Aristotle's philosophy, we are confronted by an initial and insuperable difficulty. Aristotle is always anxious to insist on the difference between his own doctrines and those of Plato, and his bias in this direction regularly leads him to speak as though he held a thorough-going naturalistic and empirical theory with no "transcendental moonshine" about it. Yet his final conclusions on all points of importance are