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.

§ 5. 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. other premise, it must fulfil the function of a subject. general formula of all argument must be

Hence, in the
Hence the

M is P

S is M

... S is P;

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. Hence, corresponding to every syllogistic argument in the general form,

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

* This operation will be termed a contraposition of the premise and conclusion. VOL. VII.

34

meant

66

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 wellknown 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.

therefore, as well as representing propositions in general, particularly represents Universal Affirmative propositions; and thus the general form of syllogism

M is P; S is M;

S is P,

represents specially the syllogisms of the mood Barbara.

* What S is meant being generally undetermined.

§ 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:

and

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,

All not-M is P,

S is not M,

.. S is P;

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:

Then its two apagogical modifications are the second and third

figures.

It is customary to enumerate six moods of the third figure instead of four, and the moods Darapti and Felapton appear to be omitted. But a particular proposition is asserted (actually and not merely virtually) by the universal proposition which does not otherwise differ from it; and therefore Darapti is included both under Disamis and Datisi, and Felapton both under Bocardo and Ferison. (De Morgan.) The second figure, from the assertion of the rule and the denial of the result, infers the denial of the case; the third figure, from the denial of the result and assertion of the case, infers the denial of the rule. Hence we write the moods as follows, by allowing inferences only on the straight lines:

The symmetry of the system of moods of the three figures is also

exhibited in the following table.