Whatever is predicated of a term distributed whether affirmatively or negatively, may be predicated in like manner of everything contained under it.

Or more briefly :

What pertains to the higher class pertains also to the lower.

This merely means, in untechnical language, that what may be said of all the things of any sort or kind may be said of any one or any part of those things; and, secondly, what may be denied of all the things in a class may be denied of any one or any part of them. Whatever may be said of "All planets" may be said of Venus, the Earth, Jupiter, or any other planet; and, as they may all be said to revolve in elliptic orbits, it follows that this may be asserted of Venus, the Earth, Jupiter, or any other planet. Similarly, according to the negative part of the Dicta, we may deny that the planets are selfluminous, and knowing that Jupiter is a planet may deny that Jupiter is self-luminous. A little reflection would show that the affirmative Dictum is really the first of the Canons in a less complete and general form, and that the negative Dictum is similarly the second Canon. These Dicta in fact only apply to such cases of agreement bétween terms as consist in one being the name of a smaller class, and another of the larger class containing it. Logicians have for the most part strangely overlooked the important cases in which one term agrees with another to the extent of being identical with it; but this is a subject which we cannot fitly discuss here at any length. It is treated in my little work called The Substitution of Similars*.

Some logicians have held that in addition to the three laws which are called the Primary Laws of Thought,

* Macmillan and Co. 1869.

there is a fourth called "The Principle or Law of Sufficient Reason." It was stated by Leibnitz in the following words:

Nothing happens without a reason why it should be so rather than otherwise. For instance, if there be a pair of scales in every respect exactly alike on each side and with exactly equal weights in each scale, it must remain motionless and in equilibrium, because there is no reason why one side should go down more than the other. It is certainly a fundamental assumption in mechanical science that if a body is acted upon by two perfectly equal forces in different directions it will move equally between them, because there is no reason why it should move more to one side than the other. Mr Mansel, Sir W. Hamilton and others consider however that this law has no place in logic, even if it can be held self-evident at all; and the question which appears open to doubt need not be discussed here.

I have so freely used the word axiom in this lesson that it is desirable to clear up its meaning as far as possible. Philosophers do not perfectly agree about its derivation or exact meaning, but it certainly comes from the verb ağtów, which is rendered, to think worthy. It generally denotes a self-evident truth of so simple a character that it must be assumed to be true, and, as it cannot be proved by any simpler proposition, must itself be taken as the basis of reasoning. In mathematics it is clearly used in this sense.

See Hamilton's Lectures on Logic, Lectures 5 and 6.

LESSON XV.

THE RULES OF THE SYLLOGISM.

SYLLOGISM is the common name for Mediate Inference, or inference by a medium or middle term, and is to be distinguished from the process of Immediate Inference, or inference which is performed without the use of any third or middle term.

We are in the habit of employing a middle term or medium whenever we are prevented from comparing two things together directly, but can compare each of them with a certain third thing. We cannot compare the sizes of two halls by placing one in the other, but we can measure each by a foot rule or other suitable measure, which forms a common measure, and enables us to ascertain with any necessary degree of accuracy their relative dimensions. If we have two quantities of cotton goods and want to compare them, it is not necessary to bring the whole of one portion to the other, but a sample is cut off, which represents exactly the quality of one portion, and, according as this sample does or does not agree with the other portion, so must the two portions of goods agree or differ.

The use of a middle term in syllogism is closely parallel to what it is in the above instances, but not exactly the same. Suppose, as an example, that we wish to ascertain whether or not "Whales are viviparous," and that we had not an opportunity of observing the fact directly; we could yet show it to be so if we knew that "whales are mammalian animals," and that "all mam

malian animals are viviparous." It would follow that "whales are viviparous;" and so far as the inference is concerned it does not matter what is the meaning we attribute to the words viviparous and mammalian. In this case "mammalian animal" is the middle term.

The name Syllogism means the joining together in thought of two propositions, and is derived from the Greek words oúv, with, and λóyos, thought or reason. It is thus exactly the equivalent of the word Computation, which means thinking together (Latin con, together, puto, to think), or reckoning. In a syllogism we so unite in thought two premises, or propositions put forward, that we are enabled to draw from them or infer, by means of the middle term they contain, a third proposition called the conclusion. Syllogism may thus be defined as the act of thought by which from two given propositions we proceed to a third proposition, the truth of which necessarily follows from the truth of these given propositions. When the argument is fully expressed in language it is usual to call it concretely a syllogism.

The special rules of the syllogism are founded upon the Laws of Thought and the Canons considered in the previous Lesson. They serve to inform us exactly under what circumstances one proposition can be inferred from two other propositions, and are eight in number, as follows:

1. Every syllogism has three and only three terms. These terms are called the major term, the minor term, and the middle term.

2. Every syllogism contains three, and only three propositions.

These propositions are called the major premise, the minor premise, and the conclusion.

3. The middle term must be distributed once at least, and must not be ambiguous.

4. No term must be distributed in the conclusion which was not distributed in one of the premises.

5. From negative premises nothing can be inferred. 6. If one premise be negative, the conclusion must be negative; and vice versâ, to prove a negative conclusion one of the premises must be negative.

From the above rules may be deduced two subordinate rules, which it will nevertheless be convenient to state at once.

7. From two particular premises no conclusion can be drawn.

8. If one premise be particular, the conclusion must be particular.

All these rules are of such extreme importance that it will be desirable for the student not only to acquire a perfect comprehension of their meaning and truth, but to commit them to memory. During the remainder of this lesson we shall consider their meaning and force.

As the syllogism consists in comparing two terms by means of a middle term, there cannot of course be less than three terms, nor can there be more; for if there were four terms, say A, B, C, D, and we compared A with B and C with D, we should either have no common medium at all between A and D, or we should require a second syllogism, so as first to compare A and C with B, and then A and D with C.

The middle term may always be known by the fact that it does not occur in the conclusion. The major term is always the predicate of the conclusion, and the minor term the subject. These terms are thus called because in the universal affirmative proposition (A) the predicate is necessarily a wider or greater or major term than the subject; thus in "all men are mortals," the predicate includes all other animals as well as men, and is obviously a major term or wider term than men.