between the conditioning and the conditioned. And this is not to be received as an arbitrary hypothesis, but as a scientific truth. It is an unquestionable fact, in spite of the contradiction of Waitz' and of Prantl,2 that Aristotle did not formally comprehend under his notion of inferences ¿§ úπoléσɛws, hypothetical inferences in the later sense, and that his syllogistic therefore required this enlargement. He reckoned indirect proof among the syllogisms hypothetical in his sense-Toû δ ̓ ἐξ ὑποθέσεως μέρος τὸ διὰ τοῦ ἀδυνάτου —because in it a false proposition, viz. the contradictory opposite of the proposition to be proved, is hypothetically taken as true, in the meaning of the (real or feigned) opponent who might assert it, and so serves as an úπóbeσis, and forms the basis of a syllogism by means of which something evidently untrue is inferred, because its contradictory is already recognised to be true, with the aim and in order to destroy the false hypothesis itself by showing the falsehood of its consequence.

The remark of Aristotle :4 πολλοὶ δὲ καὶ ἕτεροι περαίνονται ἐξ ὑποθέσεως, οὓς ἐπισκέψασθαι δεῖ καὶ διασημῆναι καθαρῶς, appears to have induced Theophrastus and Eudemus to reconstruct more accurately the theory of hypothetical inference. Boëthius says that, in the doctrine of the hypothetical syllogisms, Theophrastus rerum tantum summas exsequitur, Eudemus latiorem docendi graditur viam.' Theophrastus treats in particular of the thoroughly hypothetical syllogisms, in which the premises are of the same form with each other and with the conclusion (οἱ δι' ὅλου οι δι' ὅλων ὑποθετικοί, διὰ τριῶν ὑποθετικοί, called also by Theophrastus συλλογισμοί κατ ̓ ἀναXoyíav), and have three syllogistic figures like categorical syllogisms. He appears to have made the hypothetical proposition (εἰ τὸ Α, τὸ Β) parallel with the categorical (τὸ Α κατὰ TOû B), the condition (ɛi Tò A) with the predicate (τò A), and what is conditioned (Tò B) with the subject (Kaтà TOû B). This at least is the only explanation of the fact that he considered

1 Ad Arist. Org. i. 433. 2 Gesch. der Log. i. 272, 295.

3 Anal. Pri. i. 23.

4 Ibid. i. 44.

6 According to the accounts of Alex. Ad Prantl, Gesch. der Log. i. 381.

5 De Syl. Hyp. p. 606. Anal. Pri. fol. 134; cf.

G G

that form of inference to be the Second Figure of the hypothetical syllogism, in which the premises beginning with the same condition end with a different conditioned: ei Tò A, Tò B. εἰ μὴ τὸ Α, τὸ Γ· εἰ ἄρα μὴ τὸ Β, τὸ Γ; and that to be the Third Figure in which the premises beginning with a different condition and ending with the same conditioned: ɛi tò A, tò Γ· εἰ τὸ Β, οὐ τὸ Γ· εἰ ἄρα τὸ Α, οὐ τὸ B. Theophrastus conΒ. trives to find, by this way of paralleling, the most complete analogy between the First Figure of the Hypothetical inference and the First Figure of the Categorical, according to the following position of the premises: εἰ τὸ Α, τὸ Β· εἰ τὸ Β, τὸ Γ· εi apa тò A, Tò г. This opinion of Theophrastus may have determined his choice of the letters, since the early logicians, and Aristotle himself, made the letter which stood first in the alphabet denote the most general term, or what stands in the same relation as the general term.

But this way of paralleling the two kinds of inference is false. The condition is rather to be considered as analogous to the subject of the categorical proposition, and what is conditioned to the predicate. For the sphere of the cases in which the condition exists is not equal to the sphere of the predicate, which is the wider, but is equal to the sphere of the subject, which is either narrower or equal to the sphere of the conditioned.

Alexander of Aphrodisias showed the true relation.' He properly recognised the Third Figure in that hypothetical inference which Theophrastus made the Second, and the Second in that which he made the Third.

The Stoics have paid special attention to hypothetical syllogism.

Boëthius (in his writing De Syllogismo hypothetico) represents the possible forms of conditional inferences with superabundant detail.

Kant refers hypothetical inference, as well as the hypothetical judgment, to the category of Dependence.

[Hamilton, in his earlier writings, followed Kant's opinion;

1 Ad Arist. Anal. Pri. fol. 134.

latterly he believed that all hypothetical inference could be classed under immediate inference.']

We agree with the opinion that the logical distinction between the categorical and hypothetical mode of inference rests on the metaphysical distinction between the categories of inherence and dependence. It is not to be considered, as some later logicians have done, only or almost only a difference in the verbal expression. Cf. §§ 68, 85, and 94.

§ 122. MIXED INFERENCES are those whose premises are judgments which have different relations. HYPOTHETICOCATEGORICAL inferences belong to this class. From the combination of an hypothetical premise with a categorical, which either asserts the fact of the condition or denies the fact of the conditioned, there follows in the first case the categorical affirmation of what is conditioned (modus ponens), in the other case the categorical negation of the condition (modus tollens). The modus. ponens corresponds to the First Figure of categorical inferences, the modus tollens to the Second. Different modifications, which correspond to the moods in the first two figures, result by the admission of negation into the second member of the hypothetical premise, as well as by that of the distinction of quantity (in some cases-in all cases). If the negation occurs in the first member of the hypothetical premise, the case corresponds to categorical inferences of the same figures which have a negative subject-notion in the major premise. No form of these inferences can agree with the Third and Fourth Figures of the categorical syllogism (in whose minor premise the middle

[1 Cf. Lectures on Logic, ii. 376 ff.]

notion is subject). For the condition in the hypothetical corresponds to the subject of the categorical judgments, and the subject does not occur in the minor premise where a categorical takes the place of a conditioned assertion. Hence in such an assertion the part mediating the inference would be wanting.

The Scheme of the modus ponens, in the fundamental form which corresponds to the mood Barbara, and is more accurately called the modus ponendo ponens,' is: If A is, в is; A is: Therefore B is. Its formula was, as given by the older Logicians: posita conditione ponatur conditionatum. The modus ponendo tollens corresponds to the mood Celarent; If A is, B is not; A is: ..B is not. These moods pass over into Darii and Ferio, if the minor premise is: Sometimes or in some cases A is, and accordingly the conclusion is: .. B is in certain cases, and B is not in certain cases. If the major premise runs: If A is not, B is, or B is not; and the minor premise: A is not, the existence or non-existence of в follows by a modus tollendo ponens or tollendo tollens. The scheme of the modus tollens in its fundamental form, which corresponds to the mood Camestres, and may more strictly be called modus tollendo tollens, is: If A is, B is; now B is not: .. A is not. Its formula is as follows: sublato conditionato, tollatur conditio. The modus ponendo tollens corresponds to the mood Cesare: If A is, в is not; B is.. A is not. The moods Baroco and Festino can in this case be formed in a way quite analogous to the construction of Darii and Ferio. When the negation occurs in the first member of the hypothetical major premise, a modus tollendo ponens: If A is not, B is; now B is not: .. A is; and a modus ponendo ponens: If A is not, B is not; now B is: .. A is-may be formed. A conclusion from the conditioned to the condition is unjustifiable: If A is, B is; now B is: .'. A is (just as a categorical inference in the Second Figure from two affirmative premises is false); for the sphere of the cases where B is may be more extensive than the sphere of the case where a is,

1 Drobisch, 3rd ed. § 98.

so that B can exist where A is not. For the same reason the inference: If A is, B is; now A is not: .. B is not, is false (just as a categorical inference in the First Figure with a negative minor premise is not valid).

In this case also, because of the thorough-going analogy, a few examples will suffice. Böckh' concludes, in opposition to Gruppe, in the modus ponendo ponens (and in the modus ponendo tollens) after the manner of the First Figure: If Plato in the Timaeus teaches the daily motion of the heavens from the east to the west, he must deny the daily rotation of the earth on its axis from west to east (and so cannot teach the rotation of the earth on its axis); but he teaches the former: Therefore he must deny the latter (and cannot teach it). With equal correctness, Böckh, in the same book, in opposition to Stallbaum, argues in the modus tollendo tollens after the manner of the Second Figure: If Plato teaches the rotation of the earth about the axis of the universe, he must also accept the rotation of the earth round its own axis (for the one axis is only the lengthening of the other); but he denies the latter rotation: Therefore he denies the former also.

Inferences of this kind, though only one of the premises is hypothetical and the other categorical, are commonly called hypothetical inferences, and so explained. The older Peripatetics (more particularly Theophrastus and Eudemus) have already made use of this opinion. They call the hypothetical major premise τὸ συνημμένον, its conditioning member τὸ ἡγούμενον, the conditioned Tò Tóμevov, the categorical minor premise μɛTáλnis, because it repeats categorically, or changes to this form, what was already asserted in the hypothetical major premise as a member, and, lastly, the conclusion σvμnépаoμa.

The Stoics change the terminology without, as it appears, essentially advancing the doctrine. They call the hypothetical major premise Tò троTIKÓν, or the major premise generally Tò λῆμμα, its members τὸ ἡγούμενον and τὸ λήγον, the categorical minor premise πρóïìηis, and, lastly, the conclusion, as in general, ἐπιφορά.

In his Untersuchungen über das Kosmische Systems des Plato, 1852. 2 Cf. Philop. Ad Anal. Pr. fol. lx. a.