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14 Waitz, in his note (p. 374), endeavours, but I think without success, to show that Aristotles proof is not open to the criticism here advanced. He admits that it is obscurely indicated, but the amplification of it given by himself still remains exposed to the same objection.

Even the friends and companions of Aristotle were not satisfied with his manner of establishing this fundamental rule as to the conversion of propositions. Eudęmus is said to have given a different proof; and Theophrastus assumed as self-evident, without any proof, that the Universal Negative might always be converted simply.15 It appears to me that no other or better evidence of it can be offered, than the trial upon particular cases, that is to say, Induction.16 Nothing is gained by dividing (as Aristotle does) the whole A into parts, one of which is C; nor can I agree with Theophrastus in thinking that every learner would assent to it at first hearing, especially at a time when no universal maxims respecting the logical value of propositions had ever been proclaimed. Still less would a Megaric dialectician, if he had never heard the maxim before, be satisfied to stand upon an alleged ŕ priori necessity without asking for evidence. Now there is no other evidence except by exemplifying the formula, No A is B, in separate propositions already known to the learner as true or false, and by challenging him to produce any one case, in which, when it is true to say No A is B, it is not equally true to say, No B is A; the universality of the maxim being liable to be overthrown by any one contradictory instance.17 If this proof does not convince him, no better can be produced. In a short time, doubtless, he will acquiesce in the general formula at first hearing, and he may even come to regard it as self-evident. It will recall to his memory an aggregate of separate cases each individually forgotten, summing up their united effect under the same aspect, and thus impressing upon him the general truth as if it were not only authoritative but self-authorized.

15 See the Scholia of Alexander on this passage, p. 148, a. 30–45, Brandis; Eudemi Fragm. ci.-cv. pp. 145–149, ed. Spengel.

16 We find Aristotle declaring in Topica, II. viii. p. 113, b. 15, that in converting a true Universal Affirmative proposition, the negative of the Subject of the convertend is always true of the negative of the Predicate of the convertend; e.g. If every man is an animal, every thing which is not an animal is not a man. This is to be assumed (he says) upon the evidence of Induction uncontradicted iteration of particular cases, extended to all cases universally λαμβάνειν δ ἐξ ἐπαγωγῆς, οἷον εἰ ὁ ἄνθρωπος ζῷον, τὸ μὴ ζῷον οὐκ ἄνθρωποςˇ ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων. ἐπὶ πάντων οὖν τὸ τοιοῦτον ἀξιωτέον.

The rule for the simple conversion of the Universal Negative rests upon the same evidence of Induction, never contradicted.

17 Dr. Wallis, in one of his acute controversial treatises against Hobbes, remarks upon this as the process pursued by Euclid in his demonstrations: You tell us next that an Induction, without enumeration of all the particulars, is not sufficient to infer a conclusion. Yes, Sir, if after the enumeration of some particulars, there comes a general clause, and the like in other cases (as here it doth), this may pass for a proofe till there be a possibility of giving some instance to the contrary, which here you will never be able to doe. And if such an Induction may not pass for proofe, there is never a proposition in Euclid demonstrated. For all along he takes no other course, or at least grounds his Demonstrations on Propositions no otherwise demonstrated. As, for instance, he proposeth it in general (i. c. 1.) To make an equilateral triangle on a line given. And then he shows you how to do it upon the line A B, which he there shows you, and leaves you to supply: And the same, by the like means, may be done upon any other strait line; and then infers his general conclusion. Yet I have not heard any man object that the Induction was not sufficient, because he did not actually performe it in all lines possible. (Wallis, Due Correction to Mr. Hobbes, Oxon. 1656, sect. v. p. 42.) This is induction by parity of reasoning.

So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: τὸ καθόλου δὲ ὑπάρχει τότε, ὅταν ἐπὶ τοῦ τυχόντος καὶ πρώτου δεικνύηται.

Aristotle passes next to Affirmatives, both Universal and Particular. First, if A can be predicated of all B, then B can be predicated of some A; for if B cannot be predicated of any A, then (by the rule for the Universal Negative) neither can A be predicated of any B. Again, if A can be predicated of some B, in this case also, and for the same reason, B can be predicated of some A.18 Here the rule for the Universal Negative, supposed already established, is applied legitimately to prove the rules for Affirmatives. But in the first case, that of the Universal, it fails to prove some in the sense of not-all or some-at-most, which is required; whereas, the rules for both cases can be proved by Induction, like the formula about the Universal Negative. When we come to the Particular Negative, Aristotle lays down the position, that it does not admit of being necessarily converted in any way. He gives no proof of this, beyond one single exemplification: If some animal is not a man, you are not thereby warranted in asserting the converse, that some man is not an animal.19 It is plain that such an exemplification is only an appeal to Induction: you produce one particular example, which is entering on the track of Induction; and one example alone is sufficient to establish the negative of an universal proposition.20 The converse of a Particular Negative is not in all cases true, though it may be true in many cases.

18 Aristot. Analyt. Prior. I. ii. p. 25, a. 17–22.

19 Ibid. p. 25, a. 22–26.

20 Though some may fancy that the rule for converting the Universal Negative is intuitively known, yet every one must see that the rule for converting the Universal Affirmative is not thus self-evident, or derived from natural intuition. In fact, I believe that every learner at first hears it with great surprise. Some are apt to fancy that the Universal Affirmative (like the Particular Affirmative) may be converted simply. Indeed this error is not unfrequently committed in actual reasoning; all the more easily, because there is a class of cases (with subject and predicate co-extensive) where the converse of the Universal Affirmative is really true. Also, in the case of the Particular Negative, there are many true propositions in which the simple converse is true. A novice might incautiously generalize upon those instances, and conclude that both were convertible simply. Nor could you convince him of his error except by producing examples in which, when a true proposition of this kind is converted simply, the resulting converse is notoriously false. The appeal to various separate cases is the only basis on which we can rest for testing the correctness or incorrectness of all these maxims proclaimed as universal.

From one proposition taken singly, no new proposition can be inferred; for purposes of inference, two propositions at least are required.21 This brings us to the rules of the Syllogism, where two propositions as premisses conduct us to a third which necessarily follows from them; and we are introduced to the well-known three Figures with their various Modes.22 To form a valid Syllogism, there must be three terms and no more; the two, which appear as Subject and Predicate of the conclusion, are called the minor term (or minor extreme) and the major term (or major extreme) respectively; while the third or middle term must appear in each of the premisses, but not in the conclusion. These terms are called extremes and middle, from the position which they occupy in every perfect Syllogism that is in what Aristotle ranks as the First among the three figures. In his way of enunciating the Syllogism, this middle position formed a conspicuous feature; whereas the modern arrangement disguises it, though the denomination middle term is still retained. Aristotle usually employs letters of the alphabet, which he was the first to select as abbreviations for exposition;23 and he has two ways (conforming to what he had said in the first chapter of the present treatise) of enunciating the modes of the First figure. In one way, he begins with the major extreme (Predicate of the conclusion): A may be predicated of all B, B may be predicated of all C; therefore, A may be predicated of all C (Universal Affirmative). Again, A cannot be predicated of any B, B can be predicated of all C; therefore, A cannot be predicated of any C (Universal Negative). In the other way, he begins with the minor term (Subject of the conclusion): C is in the whole B, B is in the whole A; therefore, C is in the whole A (Universal Affirmative). And, C is in the whole B, B is not in the whole A; therefore, C is not in the whole A (Universal Negative). We see thus that in Aristotles way of enunciating the First figure, the middle term is really placed between the two extremes,24 though this is not so in the Second and Third figures. In the modern way of enunciating these figures, the middle term is never placed between the two extremes; yet the denomination middle still remains.

21 Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b. 35; Analyt. Poster. I. iii. p. 73, a. 7.

22 Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.

23 M. Barthélemy St. Hilaire (Logique dAristote, vol. ii. p. 7, n.), referring to the examples of Conversion in chap. ii., observes: Voici le prémier usage des lettres représentant des idées; cest un procédé tout ŕ fait algébrique, cest ŕ dire, de généralisation. Déjŕ, dans lHerméneia, ch. 13, § 1 et suiv., Aristote a fait usage de tableaux pour représenter sa pensée relativement ŕ la consécution des modales. Il parle encore spécialement de figures explicatives, liv. 2. des Derniers Analytiques, ch. 17, § 7. Vingt passages de lHistoire des Animaux attestent quil joignait des dessins ŕ ses observations et ŕ ses théories zoologiques. Les illustrations pittoresques datent donc de fort loin. Lemploi symbolique des lettres a été appliqué aussi par Aristote ŕ la Physique. Il lavait emprunté, sans doute, aux procédés des mathématiciens.

We may remark, however, that when Aristotle proceeds to specify those combinations of propositions which do not give a valid conclusion, he is not satisfied with giving letters of the alphabet; he superadds special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12, 34, 38).

24 Aristot. Analyt. Prior. I. iv. p. 25, b. 35: καλῶ δὲ μέσον, ὃ καὶ αὐτὸ ἐν ἄλλῳ καὶ ἄλλο ἐν τούτῳ ἐστίν, ὃ καὶ τῇ θέσει γίνεται μέσον.

The Modes of each figure are distinguished by the different character and relation of the two premisses, according as these are either affirmative or negative, either universal or particular. Accordingly, there are four possible varieties of each, and sixteen possible modes or varieties of combinations between the two. Aristotle goes through most of the sixteen modes, and shows that in the first Figure there are only four among them that are legitimate, carrying with them a necessary conclusion. He shows, farther, that in all the four there are two conditions observed, and that both these conditions are indispensable in the First figure: (1) The major proposition must be universal, either affirmative or negative; (2) The minor proposition must be affirmative, either universal or particular or indefinite. Such must be the character of the premisses, in the first Figure, wherever the conclusion is valid and necessary; and vice versâ, the conclusion will be valid and necessary, when such is the character of the premisses.25

25 Aristot. Analyt. Prior. I. iv. p. 26, b. 26, et sup.

In regard to the four valid modes (Barbara, Celarent, Darii, Ferio, as we read in the scholastic Logic) Aristotle declares at once in general language that the conclusion follows necessarily; which he illustrates by setting down in alphabetical letters the skeleton of a syllogism in Barbara. If A is predicated of all B, and B of all C, A must necessarily be predicated of all C. But he does not justify it by any real example; he produces no special syllogism with real terms, and with a conclusion known beforehand to be true. He seems to think that the general doctrine will be accepted as evident without any such corroboration. He counts upon the learners memory and phantasy for supplying, out of the past discourse of common life, propositions conforming to the conditions in which the symbolical letters have been placed, and for not supplying any contradictory examples. This might suffice for a treatise; but we may reasonably believe that Aristotle, when teaching in his school, would superadd illustrative examples; for the doctrine was then novel, and he is not unmindful of the errors into which learners often fall spontaneously.26

26 Analyt. Poster. I. xxiv. p. 85, b. 21.

When he deals with the remaining or invalid modes of the First figure, his manner of showing their invalidity is different, and in itself somewhat curious. If (he says) the major term is affirmed of all the middle, while the middle is denied of all the minor, no necessary consequence follows from such being the fact, nor will there be any syllogism of the two extremes; for it is equally possible, either that the major term may be affirmed of all the minor, or that it may be denied of all the minor; so that no conclusion, either universal or particular, is necessary in all cases.27 Examples of such double possibility are then exhibited: first, of three terms arranged in two propositions (A and E), in which, from the terms specially chosen, the major happens to be truly affirmable of all the minor; so that the third proposition is an universal Affirmative:

Major and Middle.	}	Animal is predicable of every Man;
Middle and Minor	}	Man is not predicable of any Horse;
Major and Minor	}	Animal is predicable of every Horse.

Next, a second example is set out with new terms, in which the major happens not to be truly predicable of any of the minor; thus exhibiting as third proposition an universal Negative:

Major and Middle.	}	Animal is predicable of every Man;
Middle and Minor	}	Man is not predicable of any Stone;
Major and Minor	}	Animal is not predicable of any Stone.

Here we see that the full exposition of a syllogism is indicated with real terms common and familiar to every one; alphabetical symbols would not have sufficed, for the learner must himself recognize the one conclusion as true, the other as false. Hence we are taught that, after two premisses thus conditioned, if we venture to join together the major and minor so as to form a pretended conclusion, we may in some cases obtain a true proposition universally Affirmative, in other cases a true proposition universally Negative. Therefore (Aristotle argues) there is no one necessary conclusion, the same in all cases, derivable from such premisses; in other words, this mode of syllogism is invalid and proves nothing. He applies the like reasoning to all the other invalid modes of the first Figure; setting them aside in the same way, and producing examples wherein double and opposite conclusions (improperly so called), both true, are obtained in different cases from the like arrangement of premisses.

27 Analyt. Prior. I. iv. p. 26, a. 2, seq.

This mode of reasoning plainly depends upon an appeal to prior experience. The validity or invalidity of each mode of the First figure is tested by applying it to different particular cases, each of which is familiar and known to the learner aliunde; in one case, the conjunction of the major and minor terms in the third proposition makes an universal Affirmative which he knows to be true; in another case, the like conjunction makes an universal Negative, which he also knows to be true; so that there is no one necessary (i.e. no one uniform and trustworthy) conclusion derivable from such premisses.28 In other words, these modes of the First figure are not valid or available in form; the negation being sufficiently proved by one single undisputed example.

28 Though M. Barthélemy St. Hilaire (note, p. 19) declares Aristotles exposition to be a model of analysis, it appears to me that the grounds for disallowing this invalid mode of the First figure (A E A, or A E E) are not clearly set forth by Aristotle himself, while they are rendered still darker by some of his best commentators. Thus Waitz says (p. 381): Per exempla allata probat (Aristoteles) quod demonstrare debebat ex ipsâ ratione quam singuli termini inter se habeant: est enim proprium artis logicć, ut terminorum rationem cognoscat, dum res ignoret. Num de Caio prćdicetur animal nescit, scit de Caio prćdicari animal, si animal de homine et homo de Caio prćdicetur.

This comment of Waitz appears to me founded in error. Aristotle had no means of shewing the invalidity of the mode A E in the First figure, except by an appeal to particular examples. The invalidity of the invalid modes, and the validity of the valid modes, rest alike upon this ultimate reference to examples of propositions known to be true or false, by prior experience of the learner. The valid modes are those which will stand this trial and verification; the invalid modes are those which will not stand it. Not till such verification has been made, is one warranted in generalizing the result, and enunciating a formula applicable to unknown particulars (rationem terminorum cognoscere, dum res ignoret). It was impossible for Aristotle to do what Waitz requires of him. I take the opposite ground, and regret that he did not set forth the fundamental test of appeal to example and experience, in a more emphatic and unmistakeable manner.

M. Barthélemy St. Hilaire (in the note to his translation, p. 14) does not lend any additional clearness, when he talks of the conclusion from the propositions A and E in the First figure. Julius Pacius says (p. 134): Si tamen conclusio dici debet, quć non colligitur ex propositionibus, &c. Moreover, M. St. Hilaire (p. 19) slurs over the legitimate foundation, the appeal to experience, much as Aristotle himself does: Puis prenant des exemples oů la conclusion est de toute évidence, Aristote les applique successivement ŕ chacune de ces combinaisons; celles qui donnent la conclusion fournie dailleurs par le bon sens, sont concluantes ou syllogistiques, les autres sont asyllogistiques.

We are now introduced to the Second figure, in which each of the two premisses has the middle term as Predicate.29 To give a legitimate conclusion in this figure, one or other of the premisses must be negative, and the major premiss must be universal; moreover no affirmative conclusions can ever be obtained in it none but negative conclusions, universal or particular. In this Second figure too, Aristotle recognizes four valid modes; setting aside the other possible modes as invalid30 (in the same way as he had done in the First figure), because the third proposition or conjunction of the major term with the minor, might in some cases be a true universal affirmative, in other cases a true universal negative. As to the third and fourth of the valid modes, he demonstrates them by assuming the contradictory of the conclusion, together with the major premiss, and then showing that these two premisses form a new syllogism, which leads to a conclusion contradicting the minor premiss. This method, called Reductio ad Impossibile, is here employed for the first time; and employed without being ushered in or defined, as if it were familiarly known.31

29 Analyt. Prior. I. v. p. 26, b. 34. As Aristotle enunciates a proposition by putting the predicate before the subject, he says that in this Second figure the middle term comes πρῶτον τῇ θέσει. In the Third figure, for the same reason, he calls it ἔσχατον τῇ θέσει, vi. p. 28, a. 15.

30 Analyt. Prior. I. v. p. 27, a. 18. In these invalid modes, Aristotle says there is no syllogism; therefore we cannot properly speak of a conclusion, but only of a third proposition, conjoining the major with the minor.

31 Ibid. p. 27, a. 15, 26, seq. It is said to involve ὑπόθεσις, p. 28, a. 7; to be ἐξ ὑποθέσεως xxiii. p. 41, a. 25; to be τοῦ ἐξ ὑποθέσεως, as opposed to δεικτικός, xxiii. p. 40, b. 25.

M. B. St. Hilaire remarks justly, that Aristotle might be expected to define or explain what it is, on first mentioning it (note, p. 22).

Lastly, we have the Third figure, wherein the middle term is the Subject in both premisses. Here one at least of the premisses must be universal, either affirmative or negative. But no universal conclusions can be obtained in this figure; all the conclusions are particular. Aristotle recognizes six legitimate modes; in all of which the conclusions are particular, four of them being affirmative, two negative. The other possible modes he sets aside as in the two preceding figures.32

32 Ibid. I. vi. p. 28, a. 10-p. 29, a. 18.

But Aristotle assigns to the First figure a marked superiority as compared with the Second and Third. It is the only one that yields perfect syllogisms; those furnished by the other two are all imperfect. The cardinal principle of syllogistic proof, as he conceives it, is That whatever can be affirmed or denied of a whole, can be affirmed or denied of any part thereof.33 The major proposition affirms or denies something universally respecting a certain whole; the minor proposition declares a certain part to be included in that whole. To this principle the four modes of the First figure manifestly and unmistakably conform, without any transformation of their premisses. But in the other figures such conformity does not obviously appear, and must be demonstrated by reducing their syllogisms to the First figure; either ostensively by exposition of a particular case, and conversion of the premisses, or by Reductio ad Impossibile. Aristotle, accordingly, claims authority for the Second and Third figures only so far as they can be reduced to the First.34 We must, however, observe that in this process of reduction no new evidence is taken in; the matter of evidence remains unchanged, and the form alone is altered, according to laws of logical conversion which Aristotle has already laid down and justified. Another ground of the superiority and perfection which he claims for the First figure, is, that it is the only one in which every variety of conclusion can be proved; and especially the only one in which the Universal Affirmative can be proved the great aim of scientific research. Whereas, in the Second figure we can prove only negative conclusions, universal or particular; and in the Third figure only particular conclusions, affirmative or negative.35

33 Ibid. I. xli. p. 49, b. 37: ὅλως γὰρ ὃ μή ἐστιν ὡς ὅλον πρὸς μέρος καὶ ἄλλο πρὸς τοῦτο ὡς μέρος πρὸς ὅλον, ἐξ οὐδενὸς τῶν τοιούτων δείκνυσιν ὁ δεικνύων, ὥστε οὐδὲ γίνεται συλλογισμός.

He had before said this about the relation of the three terms in the Syllogism, I. iv. p. 25, b. 32: ὅταν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι τῷ μέσῳ καὶ τὸν μέσον ἐν ὅλῳ τῷ πρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι συλλογισμὸν τέλειον (Dictum de Omni et Nullo).

34 Analyt. Prior. I. vii. p. 29, a. 30-b. 25.

35 Ibid. I. iv. p. 26, b. 30, p. 27, a. 1, p. 28, a. 9, p. 29, a. 15. An admissible syllogism in the Second or Third figure is sometimes called δυνατὸς as opposed to τέλειος, p. 41, b. 33. Compare Kampe, Die Erkenntniss-Theorie des Aristoteles, p. 245, Leipzig, 1870.

Such are the main principles of syllogistic inference and rules for syllogistic reasoning, as laid down by Aristotle. During the medićval period, they were allowed to ramify into endless subtle technicalities, and to absorb the attention of teachers and studious men, long after the time when other useful branches of science and literature were pressing for attention. Through such prolonged monopoly which Aristotle, among the most encyclopedical of all writers, never thought of claiming for them they have become so discredited, that it is difficult to call back attention to them as they stood in the Aristotelian age. We have to remind the reader, again, that though language was then used with great ability for rhetorical and dialectical purposes, there existed as yet hardly any systematic or scientific study of it in either of these branches. The scheme and the terminology of any such science were alike unknown, and Aristotle was obliged to construct it himself from the foundation. The rhetorical and dialectical teaching as then given (he tells us) was mere unscientific routine, prescribing specimens of art to be committed to memory: respecting syllogism (or the conditions of legitimate deductive inference) absolutely nothing had been said.36 Under these circumstances, his theory of names, notions, and propositions as employed for purposes of exposition and ratiocination, is a remarkable example of original inventive power. He had to work it out by patient and laborious research. No way was open to him except the diligent comparison and analysis of propositions. And though all students have now become familiar with the various classes of terms and propositions, together with their principal characteristics and relations, yet to frame and designate such classes for the first time without any precedent to follow, to determine for each the rules and conditions of logical convertibility, to put together the constituents of the Syllogism, with its graduation of Figures and difference of Modes, and with a selection, justified by reasons given, between the valid and the invalid modes all this implies a high order of original systematizing genius, and must have required the most laborious and multiplied comparisons between propositions in detail.

36 Aristot. Sophist. Elench. p. 184, a. 1, b. 2: διόπερ ταχεῖα μὲν ἄτεχνος δ ἦν ἡ διδασκαλία τοῖς μανθάνουσι παρ αὐτῶνˇ οὐ γὰρ τέχνην ἀλλὰ τὰ ἀπὸ τῆς τέχνης διδόντες παιδεύειν ὑπελάμβανον περὶ δὲ τοῦ συλλογίζεσθαι παντελῶς οὐδὲν εἴχομεν πρότερον ἄλλο λέγειν, ἀλλ ἢ τριβῇ ζητοῦντες πολὺν χρόνον ἐπονοῦμεν.

The preceding abridgment of Aristotles exposition of the Syllogism applies only to propositions simply affirmative or simply negative. But Aristotle himself, as already remarked, complicates the exposition by putting the Modal propositions (Possible, Necessary) upon the same line as the above-mentioned Simple propositions. I have noticed, in dealing with the treatise De Interpretatione, the confusion that has arisen from thus elevating the Modals into a line of classification co-ordinate with propositions simply Assertory. In the Analytica, this confusion is still more sensibly felt, from the introduction of syllogisms in which one of the premisses is necessary, while the other is only possible. We may remark, however, that, in the Analytica, Aristotle is stricter in defining the Possible than he has been in the De Interpretatione; for he now disjoins the Possible altogether from the Necessary, making it equivalent to the Problematical (not merely may be, but may be or may not be).37 In the middle, too, of his diffuse exposition of the Modals, he inserts one important remark, respecting universal propositions generally, which belongs quite as much to the preceding exposition about propositions simply assertory. He observes that universal propositions have nothing to do with time, present, past, or future; but are to be understood in a sense absolute and unqualified.38

37 Analyt. Prior. I. viii. p. 29, a. 32; xiii. p. 32, a. 20–36: τὸ γὰρ ἀναγκαῖον ὁμωνύμως ἐνδέχεσθαι λέγομεν. In xiv. p. 33, b. 22, he excludes this equivocal meaning of τὸ ἐνδεχόμενον δεῖ δὲ τὸ ἐνδέχεσθα λαμβάνειν μὴ ἐν τοῖς ἀναγκαίοις, ἀλλὰ κατὰ τὸν εἰρημένον διορισμόν. See xiii. p. 32, a. 33, where τὸ ἐνδέχεσθαι ὑπάρχειν is asserted to be equivalent to or convertible with τὸ ἐνδέχεσθαι μὴ ὑπάρχειν; and xix. p. 38, a. 35: τὸ ἐξ ἀνάγκης οὐκ ἦν ἐνδεχόμενον. Theophrastus and Eudemus differed from Aristotle about his theory of the Modals in several points (Scholia ad Analyt. Priora, pp. 161, b. 30; 162, b. 23; 166, a. 12, b. 15, Brand.). Respecting the want of clearness in Aristotle about τὸ ἐνδεχόμενον, see Waitzs note ad p. 32, b. 16. Moreover, he sometimes uses ὑπάρχον in the widest sense, including ἐνδεχόμενον and ἀναγκαῖον, xxiii. p. 40, b. 24.

38 Analyt. Prior. I. xv. p. 34, b. 7.

Having finished with the Modals, Aristotle proceeds to lay it down, that all demonstration must fall under one or other of the three figures just described; and therefore that all may be reduced ultimately to the two first modes of the First figure. You cannot proceed a step with two terms only and one proposition only. You must have two propositions including three terms; the middle term occupying the place assigned to it in one or other of the three figures.39 This is obviously true when you demonstrate by direct or ostensive syllogism; and it is no less true when you proceed by Reductio ad Impossibile. This last is one mode of syllogizing from an hypothesis or assumption:40 your conclusion being disputed, you prove it indirectly, by assuming its contradictory to be true, and constructing a new syllogism by means of that contradictory together with a second premiss admitted to be true; the conclusion of this new syllogism being a proposition obviously false or known beforehand to be false. Your demonstration must be conducted by a regular syllogism, as it is when you proceed directly and ostensively. The difference is, that the conclusion which you obtain is not that which you wish ultimately to arrive at, but something notoriously false. But as this false conclusion arises from your assumption or hypothesis that the contradictory of the conclusion originally disputed was true, you have indirectly made out your case that this contradictory must have been false, and therefore that the conclusion originally disputed was true. All this, however, has been demonstration by regular syllogism, but starting from an hypothesis assumed and admitted as one of the premisses.41