A System of Logic, Ratiocinative and Inductive (Vol. 1 of 2)

by John Stuart Mill

To prove a negative, the argument must be capable of being expressed in this form:—

No one who is capable of self-control is necessarily vicious;
All negroes/Some negroes/Mr. A's negro are capable of self-control;
therefore
No negroes are/Some negroes are not/Mr. A's negro is not necessarily vicious.

Although all ratiocination admits of being thrown into one or the other of these forms, and sometimes gains considerably by the transformation, both in clearness and in the obviousness of its consequence; there are, no doubt, cases in which the argument falls more naturally into one of the other three figures, and in which its conclusiveness is more [pg 193] apparent at the first glance in those figures, than when reduced to the first. Thus, if the proposition were that pagans may be virtuous, and the evidence to prove it were the example of Aristides; a syllogism in the third figure,

Aristides was virtuous,
Aristides was a pagan,
therefore
Some pagan was virtuous,

would be a more natural mode of stating the argument, and would carry conviction more instantly home, than the same ratiocination strained into the first figure, thus—

Aristides was virtuous,
Some pagan was Aristides,
therefore
Some pagan was virtuous.

A German philosopher, Lambert, whose Neues Organon (published in the year 1764) contains among other things one of the most elaborate and complete expositions ever yet made of the syllogistic doctrine, has expressly examined what sorts of arguments fall most naturally and suitably into each of the four figures; and his solution is characterized by great ingenuity and clearness of thought.[29] The [pg 194] argument, however, is one and the same, in whichever figure it is expressed; since, as we have already seen, the premisses of a syllogism in the second, third, or fourth figure, and those of the syllogism in the first figure to which it may be reduced, are the same premisses in everything except language, or, at least, as much of them as contributes to the proof of the conclusion is the same. We are therefore at liberty, in conformity with the general opinion of logicians, to consider the two elementary forms of the first figure as the universal types of all correct ratiocination; the one, when the conclusion to be proved is affirmative, the other, when it is negative; even though certain arguments may have a tendency to clothe themselves in the forms of the second, third, and fourth figures; which, however, cannot possibly happen with the only class of arguments which are of first-rate [pg 195] scientific importance, those in which the conclusion is an universal affirmative, such conclusions being susceptible of proof in the first figure alone.

§ 2. On examining, then, these two general formulæ, we find that in both of them, one premiss, the major, is an universal proposition; and according as this is affirmative or negative, the conclusion is so too. All ratiocination, therefore, starts from a general proposition, principle, or assumption: a proposition in which a predicate is affirmed or denied of an entire class; that is, in which some attribute, or the negation of some attribute, is asserted of an indefinite number of objects distinguished by a common characteristic, and designated, in consequence, by a common name.

The other premiss is always affirmative, and asserts that something (which may be either an individual, a class, or [pg 196] part of a class) belongs to, or is included in, the class respecting which something was affirmed or denied in the major premiss. It follows that the attribute affirmed or denied of the entire class may (if there was truth in that affirmation or denial) be affirmed or denied of the object or objects alleged to be included in the class: and this is precisely the assertion made in the conclusion.

Whether or not the foregoing is an adequate account of the constituent parts of the syllogism, will be presently considered; but as far as it goes it is a true account. It has accordingly been generalized, and erected into a logical maxim, on which all ratiocination is said to be founded, insomuch that to reason, and to apply the maxim, are supposed to be one and the same thing. The maxim is, That whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo.