*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.