conclusion may be regarded as a statement substituted for either of its premises, the substitution being justified by the other premises. Nothing is relevant to the other premises, except what is requisite to justify this substitution. Either, therefore, these other premises will by themselves yield a conclusion which, taken as a premise along with the first premise, justifies the final conclusion; or else some part of them, taken with the first premise, will yield a conclusion which, taken as a premise along with all the others, will again justify the final conclusion. In either case, it follows that every argument of more than two premises can be resolved into a series of arguments of two premises each. This justifies the distinction of simple and complex arguments.
§4. Of a General Type of Syllogistic Arguments
A valid, complete, simple argument will be designated as a syllogistic argument.
Every proposition may, in at least one way, be put into the form,
S is P;
the import of which is, that the objects to which S or the total subject applies have the characteristics attributed to every object to which P or the total predicate applies.
Every term has two powers or significations, according as it is subject or predicate. The former, which will here be termed its breadth, comprises the objects to which it is applied; while the latter, which will here be termed its depth, comprises the characters which are attributed to every one of the objects to which it can be applied. This breadth and depth must not be confounded with logical extension and comprehension, as these terms are usually taken.
Every substitution of one proposition for another must consist in the substitution of term for term. Such substitution can be justified only so far as the first term represents what is represented by the second. Hence the only possible substitutions are—
1st. The substitution for a term fulfilling the function of a subject of another whose breadth is included in that of the former; and
2d. The substitution for a term fulfilling the function of a predicate of another whose depth is included in that of the former.
If, therefore, in either premise a term appears as subject which does not appear in the conclusion as subject, then the other premise must declare that the breadth of that term includes the breadth of the term which replaces it in the conclusion. But this is to declare that every object of the latter term has every character of the former. The eliminated term, therefore, if it does not fulfil the function of predicate in one premise, does so in the other. But if the eliminated term fulfils the function of predicate in one premise, the other premise must declare that its depth includes that of the term which replaces it in the conclusion. Now, this is to declare that every character of the latter term belongs to every object of the former. Hence, in the other premise, it must fulfil the function of a subject. Hence the general formula of all argument must be
which is to be understood in this sense,—that the terms of every syllogistic argument fulfil functions of subject and predicate as here indicated, but not that the argument can be grammatically expressed in this way.
PART II. §1. Of Apagogical Forms
If C is true when P is, then P is false when C is. Hence it is always possible to substitute for any premise the denial of the conclusion, provided the denial of that premise be at the same time substituted for the conclusion.2 Hence, corresponding to every syllogistic argument in the general form,
There are two others:—
§2. Of Contradiction
The apagogical forms make it necessary to consider in what way propositions deny one another.
If a proposition be put into the general form,
S is P,
its contradictory has, 1st, as its subject, instead of S, “the S now meant”3 or “some S”; and has, 2d, as its predicate, instead of P, that which differs from P or “not P.“
From these relations of contradictories, from the necessities of the logic of apagogically related arguments, therefore, arises the need of the two divisions of propositions into affirmative and negative on the one hand, and into universal and particular on the other. The contradictory of a universal proposition is particular, and the contradictory of an affirmative proposition is negative. Contradiction is a reciprocal relation, and therefore the contradictory of a particular proposition is universal, and that of a negative proposition is affirmative. The contradiction of particular and negative propositions could not be brought under the general formula, were the distinctions of affirmative and negative absolute and not merely relative; but, in fact, not-not-P is the same as P. And, if it is said that “what is now meant of the part of S meant at another time, is P,” since the part of S meant at another time is left to be determined in whatever way the proposition made at another time may determine it, this can only be true if All S is P. Therefore, if one man says “some S is not P,” and another replies, “some of that same S is P,” this second person, since he allows the first man’s some S, which has not been defined, to remain undefined, in effect says that All S is P.
Whether contradictories differ in other respects than these well-known ones is an open question.
§3. Of Barbara
Since some S means “the part now meant of S,” a particular proposition is equivalent to a universal proposition with another subject; and in the same way a negative proposition is equivalent to an affirmative proposition with another predicate.
The form,
S is P,
therefore, as well as representing propositions in general, particularly represents Universal Affirmative propositions; and thus the general form of syllogism
represents specially the syllogisms of the mood Barbara.
§4. Of the First Figure
Since, in the general form, S may be any subject and P any predicate, it is possible to modify Barbara by making the major premise and conclusion negative, or by making the minor premise and conclusion particular, or in both these ways at once. Thus we obtain all the modes of the first figure.
It is also possible to have such arguments as these:—
Some M is P,
S has all the common characters of that part of M (whatever that part may be, and therefore of each and every M),
∴ S is P,
and
but as the theory of apagogical argument has not obliged us to take account of these peculiar modifications of subject and predicate, these arguments must be considered as belonging to Barbara. In this sense the major premise must always be universal, and the minor affirmative.
Three propositions which are related to one another as though major premise, minor premise, and conclusion of a syllogism of the first figure will be termed respectively Rule, Case, and Result.
§5. Second and Third Figures
Let the first figure be written thus:—
Fig. 1