from which, according to Celarent in the First Figure follows:

PeS

from which, lastly, by conversio simplex, comes:

S e P

Instead of this Reduction, modern logicians (as for example Wolf') used another, by the Contraposition of the major premise. It is to be preferred because it does away with the Conversion of the minor premise, which is unnatural in many cases. It is, however, inferior in value to the direct comparison of spheres.

In Festino, as in Cesare, the major premise only is converted, and the conclusion is drawn in Ferio.

In Baroco the ductio per impossibile or the apagogical demonstration is applied. In order to prove that from the premises:

Pa M

the conclusion:

SOM

S o P

necessarily follows, it is shown that the contradictory opposite of the conclusion, viz. S a P, could not co-exist with the premises. For if S a P is thought along with the major premise P a M, S a M follows in the First Figure according to Barbara, which is the contradictory opposite of the given minor premise, and therefore must be as false as So M is true. Hence the admission which led to this false result must also be

false, i.e. S a P must be false. Hence the contradictory Log. § 384; cf. § 399.

1

opposite So P must be true; which was to be proved. This Reduction is not so unnatural as it at first may appear to be. If (according to Trendelenburg) thought at first sight deduces from the given judgments: all squares are parallelograms: some regular rectilineal figures are not parallelograms, the inference: some regular rectilineal figures are not squaresanalysis might discover in this process of thought the implied hidden flective consciousness, which, only slightly modified, is brought to the light of consciousness by the AristotelianScholastic Reduction:-for if they were squares they would be parallelograms, which they are not. This Reduction is at least as natural as the Wolffian, by the Contraposition of the major premise-e.g. in that example: What is not a parallelogram is not a square.

Baroco may also be referred to Camestres and Festino to Cesare, when those (some) S of which the minor premise is true be placed under a special notion and denoted by S'. Then the conclusion must hold good universally of S', and consequently particularly of S. Aristotle calls such a procedure ExOσis. Cf. § 115. Demonstration of every kind of reduction by the immediate comparison of spheres is to be preferred. [For Hamilton's views upon Reduction, cf. App. A.]

114. In the Third Figure, whose general scheme is the following (§ 103)

the minor premise must be affirmative. For if it is nega tive (Me S or Mo S) where, according to the general rules (§§ 106-108), the major premise must be universally affirmative (M a P), then it remains uncertain whether the S, which (in Me S) is thought to be separated from

1 Arist. Anal. Pri. i. 6.

the whole sphere of M, or at least (in M o S) from a part of that sphere, may perhaps fall within another part of the sphere of P (perhaps as a species-notion coordinate with M under the genus P), whether it crosses the sphere of P, or whether it lies wholly without that sphere. (If S and P were understood to be only the indifferent signs of the two Extremes both in Me S and in M o S, an inference of the form P S, viz. P o S, would result. But then, in reference to this conclusion, the negative premise is no longer the minor but the major premise. For the S has become the major notion, the praedicatum conclusionis, and the universal affirmative premise is the minor premise. It occurs in the moods Felapton and Bocardo.)

The forms of combination which are rejected by this are:

a e and а о

Hence of the eight combinations whose validity stands the test of the general rules (§§ 106-108), the following six remain

over:

a a

e a

i a a i

оа e i

It must now be shown that these lead to truly valid inferences.

§ 115. The valid moods of the Third Figure have the forms a a i, e a o, i a i, a i i, o a o, e io. They take the names Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison. The vowels in them denote in succession the form of the major and minor premises and the conclusion. The consonants refer to the Aristotelian-Scholastic Reduction. Here also the proof of the validity results on immediate comparison of spheres.

takes in the mood Darapti the more definite form:

According to the premises the sphere of M is a common part of the spheres of P and S. Hence these latter must also coincide with each other in this part; while the relation of their other parts, if there be any, remains indefinite. Hence the conclusion holds good: At least one part of the sphere of S belongs to the predicate P.

In every example where both extremes may be made. substantive, a double inference may be deduced from the same premises; e.g. when these terms are A and B, A i B as well as в i A. But since in both cases the major premise in each is of a universal affirmative form, and the same thing is done with the minor premise in each, there result, as has been remarked above (§ 113), two different examples of the same syllogistic mood, not, as in Cesare and Camestres, two different moods. The mood Felapton has the form

M e Р

M a S

So P

The proof of its validity lies in this, that those S with which M coincides must together with M itself be separated from P. Hence (at least) some S are

not P.

The form of the mood Disamis is the following:-

If the spheres of M and P are partially united, and if M falls wholly within S, then S must also be partially united with P, viz. in that part at least with which the portion of M falling within P coincides. (If only some M are P and others not, all S cannot be P, but in this case also only some S are P; cf. under Bocardo.)

The mood Datisi is of a wholly similar kind. The conclusion is made from the same premises as in Disamis. The proposition which is the converse of the conclusion of Disamis is taken as the conclusion. The particular proposition which there was major premise is here the minor, and the universal proposition is the major premise. The form of this mood is

Those S with which a part of M coincides, because this part along with the whole sphere of M falls within the sphere of P, must be included with it in this same sphere. Hence at least some S must have the predicate P. (If only some M are S, all S may yet be P.)