The axiom: ἐν ἅπαντι (συλλογισμῷ δεῖ κατηγορικόν τινα τῶν opov elval was enunciated by Aristotle.' Now there is of course a case in which a valid conclusion may be obtained from two negative premises. If the premises are given: What is not M is not P; and: S is not M-the inference follows: S is not P. But this inference does not fall under our above-given definition of the simple syllogism (§ 100), as a syllogism from three terms; for there are here four terms: S, P, M, and not-M (that, which is not M). If this is reduced to a simple syllogism, the minor premise must (by means of an immediate inference per aequipollentiam, cf. § 96) take the form: S is a not-M. But then it is according to quality an affirmative judgment (§ 69), and the rule, that from merely negative premises nothing can follow in a simple syllogism, may remain unchanged. This reduction is not an artificial mean, contrived in order to violently reconcile an actual exception to a rule falsely considered to be universally valid. We only arrive naturally at the conclusion, when we think the minor premise in the form: S falls under the notion of those beings which are not M.

6

The old logicians have already noticed this case, and have sought to solve the difficulty by this very reduction. Boëthius says: Sed fuerunt, qui hoc quum ex multis aliis, tum ex aliquo Platonis syllogismo colligerent ;-in quodam enim dialogo Plato huiusmodi interrogat syllogismum: sensus, inquit, non contingit rationem substantiae; quod non contingit rationem substantiae, ipsius veritatis notionem non contingit; sensus igitur veritatis notionem non contingit. Videtur enim ex omnibus negativis fecisse syllogismum, quod fieri non potest, atque ideo aiunt, infinitum verbum, quod est: non-contingit, pro participio infinito posuisse, id est: non-contingens est;—et id quidem Alexander Aphrodisieus arbitratur ceterique complures.' It is not improbable that the doctrine of qualitative Aequipollence between two judgments owes its origin to the explanation of the syllogistic case.

1 Anal. Pri. i. 24.

2 Ad Arist. de Interpr. p. 403; Prantl, Gesch. der Log. i. § 555.

In the Middle Ages Duns Scotus combated the universal validity of the rule: Ex mere negativis nihil sequitur, on the ground of that case.

2

Wolff enunciates the axiom:' si terminus medius fuerit negativus, propositio minor infinita est (negandi particula non refertur ad copulam, sed ad praedicatum), and remarks:3 equidem non ignoro, esse qui sibi persuadeant, steriles esse nugas, quae de propositionibus infinitis aliisque aequipollentibus in doctrina syllogistica dicuntur, eum in finem excogitatas, ut per praecipitantiam statutae regulae salventur; but justly repels this view because his doctrine necessarily follows from the notions of the terms. The later logicians have superficially passed over this question.

According to the rule established in the foregoing paragraphs, the premises in the following forms of combination cannot lead to valid inferences:

The sixteen possible forms of combination are therefore already reduced (§ 104), in so far as the combination consists of those only from which an inference can be obtained, to the following twelve:

According to other criteria certain other forms must be eliminated.

§ 107. In all figures of the simple categorical syllogism, no valid inference results if both premises are particular. Ex mere particularibus nihil sequitur.' For

Log. § 377.

2 Ibid. § 208.

3 Ibid. § 377.

с с

(a) If both are particularly affirmative, then only an indefinite part of the sphere of the middle notion is united with an indefinite part of the spheres of either of the two remaining terms. If the middle term is the subject in any one of the premises, or in both, the assertion holds good, according to the particular form of the judgment for an indefinite part only of the sphere of the middle notion. If it is predicate, the same indefiniteness arises from a more universal reason, because in every affirmative judgment it remains unexpressed whether the sphere of the predicate wholly or only partially coincides with the sphere of the subject (cf. § 71). Hence it remains indefinite whether the same part of the middle notion or a different part is united with the two other terms in the two premises, and it is also uncertain in what relation they stand to each other. Hence no conclusion is obtained.

(b) If the one premise is particularly affirmative and the other particularly negative, it is also indefinite with what part of the sphere of the middle notion the one extreme is particularly connected, and from what part of this sphere (if the middle notion is the subject in the other premise) the other extreme is separated, or whether the middle notion (if it makes the predicate in the other premise), while it is quite separated from a part of the sphere of the other extreme, is also, wholly, in part, or not at all, separated from the other part of this sphere. If it is also uncertain, whether the two extremes have any definite relation to one and the same part of the middle notion or not, the relation in which they stand to each other is the more uncertain. Hence, again, no definite conclusion can be reached.

(c) If both premises are particularly negative, then, partly because of the indefiniteness which lies in the particularity of both premises, and partly because of the negative nature of both premises (§ 106), no valid inference results.

Since the ground of the proof of the invalidity lies in the indefiniteness of the parts of the spheres, it follows that one can apply to it the axiom of the paragraph on those singular judgments whose subject is something denoted by its universal notion, but is an individual left indeterminate, i.e. those singular judgments which (§ 70) fall under the wider notion of the particular. This indefiniteness, however, has nothing to do with those judgments whose subject is an individual designated individually (e.g. by a proper name), i.e. with those which are not to be reckoned individuals but generals.

Aristotle expressed the axiom that no syllogism can be without a universal premise in these words: ἐν ἅπαντι συλλογισμῷ δεῖ τὸ καθόλου ὑπάρχειν. Later logicians have more universally based the proof which Aristotle adduced in examples of individuals only, on the relations of the spheres.

The forms of combination which must be rejected according to this rule, besides o o, which has been already eliminated by the rule of the preceding paragraph, are the three following :

So that according to this the following nine forms remain :

But all of them do not lead to valid conclusions.

1 Anal. Pr. i. 24.

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§ 108. Lastly, in all figures the combination of a particular major premise with a negative minor premise does not lead to a valid inference. For—

(a) If the major premise is particularly affirmative, and the minor premise universally negative, the middle notion M, according to the major premise, whether it form its subject or predicate, is connected with an indefinite part of the sphere of one extreme a (cf. § 71; cf. 107), but, according to the minor premise, is completely separated from the other extreme B, according to the following Schema:

Here there is a conclusion whose subject is a and whose predicate is B: (At least) some A, viz. those which coincide with M, are not в, because it is quite separated from all M, and must, therefore, also be separated from those A which coincide with M. There is no conclusion, however, whose subject is B and whose predicate is a, because it remains undetermined, according to the premises, whether B is also quite separated from the remaining A, and therefore from the whole sphere of the notion A, or partly coincides with it, or finally falls