Middle Term unequivocal.—The middle term must not be equivocal.—This rule is violated in the following syllogism. Nothing is heavier than lead; feathers are heavier than nothing; therefore, feathers are heavier than lead. The middle term, nothing, is here used in different senses in the two premises.
Middle Term to be distributed.—Essentially the same thing occurs when the middle term is not, at least once, in the premises, used in its most complete and comprehensive sense, or, as the logicians express it, distributed. As, for example, when I say, White is a color, the term color is not here distributed, for it properly includes many things besides white. If now I introduce into another proposition the same term in a similar manner, as Black is a color, I evidently include under the term, as now used, some part of the class of things denoted by the general word color, which was not included under the same term as first used. The color which is affirmed to agree with black, is not the same color which is affirmed to agree with white. The term, in fact, denotes one thing in the one proposition, and another in the other. A syllogism thus constructed, is invalid. Hence the rule, that the middle term must be distributed, or taken in its completeness, to include the whole class which it properly denotes, at least once in the premises. This is done either by making it the subject of an affirmative, or the predicate of a negative proposition; as, All men are mortal, or, No vice is useful. Here the term man in the one case, and the term useful in the other, are each distributed or taken in their completeness. There is no individual to whom the term man can properly be applied, who is not included in the expression, all men, nor is there any useful thing which is not here denied of vice.
What distributed in the Conclusion.—On the same principle, no term must be distributed in the conclusion which was not distributed in one of the premises. This rule is violated in the following syllogism, All birds are bipeds; no man is a bird; therefore, no man is a biped. Here the term biped, in the major premiss, is not taken in its completeness, since many creatures besides birds are bipeds. Birds are only one sort of bipeds. In the conclusion, however, the term biped, being the predicate of a negative proposition, is distributed, the whole class of bipeds is spoken of, and man is excluded from the whole class. The syllogism is, of course, invalid.
Law of negative Premiss.—It is further a law of the syllogism, that from negative premises nothing can be inferred. Also, that if one premiss is negative, the conclusion will be negative.
Law of particular Premiss.—From two particular premises nothing follows, but if one premiss is particular, the conclusion will be so.
These rules are too obvious, and too easily verified, to require illustration.
IV. Different Kinds of Syllogism.
Syllogisms differ.—We have mentioned as yet only those properties of the syllogism which universally belong to it. There are differences, however, which require to be noticed, and which constitute a distinction of some importance, presenting, in fact, two distinct kinds of syllogism.
Two Modes of procedure.—There are manifestly two entirely distinct modes of procedure in reasoning. We may infer from the whole to the parts, or from the parts to the whole. The former is called deductive, the latter inductive reasoning. The one is precisely the reverse of the other in method of procedure. Each is a perfectly valid method of reasoning, and each is, in itself, a distinct and valid kind of syllogism. Each requires the other. The deductive is wholly dependent on the inductive for its major premiss, which is only the conclusion of a previous induction, while, on the other hand, the induction is valuable chiefly as preparing the way for subsequent deduction. Each has equal claims with the other to be regarded as a distinct and independent form of syllogism. They have not, however, been so treated by logicians, but, on the contrary, the inductive method has been regarded, almost universally, as a mere appendage of the deductive, an imperfect form of one or another of the several figures of the syllogism deductive. Of this we shall have occasion to speak more fully in the historical sketch.