74 Ibid. I. xxxiii. p. 47, b. 16-40: αὕτη μὲν οὖν ἡ ἀπάτη γίνεται ἐν τῷ παρὰ μικρόν· ὠς γὰρ οὐδὲν διαφέρον εἰπεῖν τόδε τῷδε ὑπάρχειν, ἢ τόδε τῷδε παντὶ ὑπάρχειν, συγχωροῦμεν.
M. B. St. Hilaire observes in his note (p. 155): “L’erreur vient uniquement de ce qu’on confond l’universel et l’indeterminé séparés par une nuance très faible d’expression, qu’on ne doit pas cependant negliger.” Julius Pacius (p. 264) gives the same explanation at greater length; but the example chosen by Aristotle (ὁ Ἀριστομένης ἐστὶ διανοητὸς Ἀριστομένης) appears open to other objections besides.
75 Analyt. Prior. I. xxxiv. p. 48, a. 1-28.
76 Ibid. a. 2-23. See the Scholion of Alexander, p. 181, b. 16-27, Brandis.
Again, we must not suppose that we can always find one distinct and separate name belonging to each term. Sometimes one or all of the three terms can only be expressed by an entire phrase or proposition. In such cases it is very difficult to reduce the reasoning into regular syllogism. We may even be deceived into fancying that there are syllogisms without any middle term at all, because there is no single word to express it. For example, let A represent equal to two right angles; B, triangle; C, isosceles. Then we have a regular syllogism, with an explicit and single-worded middle term; A belongs first to B, and then to C through B as middle term (triangle). But how do we know that A belongs to B? We know it by demonstration; for it is a demonstrable truth that every triangle has its three angles equal to two right angles. Yet there is no other more general truth about triangles from which it is a deduction; it belongs to the triangle per se, and follows from the fundamental properties of the figure.77 There is, however, a middle term in the demonstration, though it is not single-worded and explicit; it is a declaratory proposition or a fact. We must not suppose that there can be any demonstration without a middle term, either single-worded or many-worded.
77 Ibid. I. xxxv. p. 48, a. 30-39: φανερὸν ὅτι τὸ μέσον οὐχ οὕτως ἀεὶ ληπτέον ὡς τόδε τι, ἀλλ’ ἐνίοτε λόγον, ὅπερ συμβαίνει κἀπὶ τοῦ λεχθέντος. A good Scholion of Philoponus is given, p. 181, b. 28-45, Brand.
When we are reducing any reasoning to a syllogistic form, and tracing out the three terms of which it is composed, we must expose or set out these terms in the nominative case; but when we actually construct the syllogism or put the terms into propositions, we shall find that one or other of the oblique cases, genitive, dative, &c., is required.78 Moreover, when we say, ‘this belongs to that,’ or ‘this may be truly predicated of that,’ we must recollect that there are many distinct varieties in the relation of predicate to subject. Each of the Categories has its own distinct relation to the subject; predication secundum quid is distinguished from predication simpliciter, simple from combined or compound, &c. This applies to negatives as well as affirmatives.79 There will be a material difference in setting out the terms of the syllogism, according as the predication is qualified (secundum quid) or absolute (simpliciter). If it be qualified, the qualification attaches to the predicate, not to the subject: when the major proposition is a qualified predication, we must consider the qualification as belonging, not to the middle term, but to the major term, and as destined to re-appear in the conclusion. If the qualification be attached to the middle term, it cannot appear in the conclusion, and any conclusion that embraces it will not be proved. Suppose the conclusion to be proved is. The wholesome is knowledge quatenus bonum or quod bonum est; the three terms of the syllogism must stand thus:—
Major — Bonum is knowable, quatenus bonum or quod bonum est.
Minor — The wholesome is bonum.
Ergo — The wholesome is knowable, quatenus bonum, &c.
For every syllogism in which the conclusion is qualified, the terms must be set out accordingly.80
78 Analyt. Prior. I. xxxvi. p. 48, a. 40-p. 49, a. 5. ἁπλῶς λέγομεν γὰρ τοῦτο κατὰ πάντων, ὅτι τοὺς μὲν ὅρους ἄει θετέον κατὰ τὰς κλήσεις τῶν ὀνομάτων — τὰς δὲ προτάσεις ληπτέον κατὰ τὰς ἑκάστου πτώσεις. Several examples are given of this precept.
79 Ibid. I. xxxvii. p. 49, a. 6-10. Alexander remarks in the Scholia (p. 183, a. 2) that the distinction between simple and compound predication has already been adverted to by Aristotle in De Interpretatione (see p. 20, b. 35); and that it was largely treated by Theophrastus in his work, Περὶ Καταφάσεως, not preserved.
80 Ibid. I. xxxviii. p. 49, a. 11-b. 2. φανερὸν οὖν ὅτι ἐν τοῖς ἐν μέρει συλλογισμοῖς οὕτω ληπτέον τοὺς ὅρους. Alexander explains οἱ ἐν μέρει συλλογισμοί (Schol. p. 183, b. 32, Br.) to be those in which the predicate has a qualifying adjunct tacked to it.
We are permitted, and it is often convenient, to exchange one phrase or term for another of equivalent signification, and also one word against any equivalent phrase. By doing this, we often facilitate the setting out of the terms. We must carefully note the different meanings of the same substantive noun, according as the definite article is or is not prefixed. We must not reckon it the same term, if it appears in one premiss with the definite article, and in the other without the definite article.81 Nor is it the same proposition to say B is predicable of C (indefinite), and B is predicable of all C (universal). In setting out the syllogism, it is not sufficient that the major premiss should be indefinite; the major premiss must be universal; and the minor premiss also, if the conclusion is to be universal. If the major premiss be universal, while the minor premiss is only affirmative indefinite, the conclusion cannot be universal, but will be no more than indefinite, that is, counting as particular.82
81 Analyt. Prior. I. xxxix.-xl. p. 49, b. 3-13. οὐ ταὐτὸν ἐστι τὸ εἶναι τὴν ἡδονὴν ἀγαθὸν καὶ τὸ εἶναι τὴν ἡδονὴν τὸ ἀγαθόν, &c.
82 Ibid. I. xli. p. 49, b. 14-32. The Scholion of Alexander (Schol. p. 184, a. 22-40) alludes to the peculiar mode, called by Theophrastus κατὰ πρόσληψιν, of stating the premisses of the syllogism: two terms only, the major and the middle, being enunciated, while the third or minor was included potentially, but not enunciated. Theophrastus, however, did not recognize the distinction of meaning to which Aristotle alludes in this chapter. He construed as an universal minor, what Aristotle treats as only an indefinite minor. The liability to mistake the Indefinite for an Universal is here again adverted to.
There is no fear of our being misled by setting out a particular case for the purpose of the general demonstration; for we never make reference to the specialties of the particular case, but deal with it as the geometer deals with the diagram that he draws. He calls the line A B, straight, a foot long, and without breadth, but he does not draw any conclusion from these assumptions. All that syllogistic demonstration either requires or employs, is, terms that are related to each other either as whole to part or as part to whole. Without this, no demonstration can be made: the exposition of the particular case is intended as an appeal to the senses, for facilitating the march of the student, but is not essential to demonstration.83
83 Ibid. I. xli. p. 50, a. 1: τῷ δ’ ἐκτίθεσθαι οὕτω χρώμεθα ὥσπερ καὶ τῷ αἰσθάνεσθαι τὸν μανθάνοντα λέγοντες· οὐ γὰρ οὕτως ὡς ἄνευ τούτων οὐχ οἷόν τ’ ἀποδειχθῆναι, ὥσπερ ἐξ ὧν ὁ συλλογισμός.
This chapter is a very remarkable statement of the Nominalistic doctrine; perceiving or conceiving all the real specialties of a particular case, but attending to, or reasoning upon, only a portion of them.
Plato treats it as a mark of the inferior scientific value of Geometry, as compared with true and pure Dialectic, that the geometer cannot demonstrate through Ideas and Universals alone, but is compelled to help himself by visible particular diagrams or illustrations. (Plato, Repub. vi. pp. 510-511, vii. p. 533, C.)
Aristotle reminds us once more of what he had before said, that in the Second and Third figures, not all varieties of conclusion are possible, but only some varieties; accordingly, when we are reducing a piece of reasoning to the syllogistic form, the nature of the conclusion will inform us which of the three figures we must look for. In the case where the question debated relates to a definition, and the reasoning which we are trying to reduce turns upon one part only of that definition, we must take care to look for our three terms only in regard to that particular part, and not in regard to the whole definition.84 All the modes of the Second and Third figures can be reduced to the First, by conversion of one or other of the premisses; except the fourth mode (Baroco) of the Second, and the fifth mode (Bocardo) of the Third, which can be proved only by Reductio ad Absurdum.85
84 Analyt. Prior. I. xlii., xliii. p. 50, a. 5-15. I follow here the explanation given by Philoponus and Julius Pacius, which M. Barthélemy St. Hilaire adopts. But the illustrative example given by Aristotle himself (the definition of water) does not convey much instruction.
85 Ibid. xlv. p. 50, b. 5-p. 51, b. 2.
No syllogisms from an Hypothesis, however, are reducible to any of the three figures; for they are not proved by syllogism alone: they require besides an extra-syllogistic assumption granted or understood between speaker and hearer. Suppose an hypothetical proposition given, with antecedent and consequent: you may perhaps prove or refute by syllogism either the antecedent separately, or the consequent separately, or both of them separately; but you cannot directly either prove or refute by syllogism the conjunction of the two asserted in the hypothetical. The speaker must ascertain beforehand that this will be granted to him; otherwise he cannot proceed.86 The same is true about the procedure by Reductio ad Absurdum, which involves an hypothesis over and above the syllogism. In employing such Reductio ad Absurdum, you prove syllogistically a certain conclusion from certain premisses; but the conclusion is manifestly false; therefore, one at least of the premisses from which it follows must be false also. But if this reasoning is to have force, the hearer must know aliunde that the conclusion is false; your syllogism has not shown it to be false, but has shown it to be hypothetically true; and unless the hearer is prepared to grant the conclusion to be false, your purpose is not attained. Sometimes he will grant it without being expressly asked, when the falsity is glaring: e.g. you prove that the diagonal of a square is incommensurable with the side, because if it were taken as commensurable, an odd number might be shown to be equal to an even number. Few disputants will hesitate to grant that this conclusion is false, and therefore that its contradictory is true; yet this last (viz. that the contradictory is true) has not been proved syllogistically; you must assume it by hypothesis, or depend upon the hearer to grant it.87
86 Ibid. xliv. p. 50, a. 16-28.
87 Analyt. Prior. I. xliv. p. 50, a. 29-38. See above, xxiii. p. 40, a. 25.
M. Barthélemy St. Hilaire remarks in the note to his translation of the Analytica Priora (p. 178): “Ce chapitre suffit à prouver qu’Aristote a distingué très-nettement les syllogismes par l’absurde, des syllogismes hypothétiques. Cette dernière dénomination est tout à fait pour lui ce qu’elle est pour nous.” Of these two statements, I think the latter is more than we can venture to affirm, considering that the general survey of hypothetical syllogisms, which Aristotle intended to draw up, either never was really completed, or at least has perished: the former appears to me incorrect. Aristotle decidedly reckons the Reductio ad Impossibile among hypothetical proofs. But he understands by Reductio ad Impossibile something rather wider than what the moderns understand by it. It now means only, that you take the contradictory of the conclusion together with one of the premisses, and by means of these two demonstrate a conclusion contradictory or contrary to the other premiss. But Aristotle understood by it this, and something more besides, namely, whenever, by taking the contradictory of the conclusion, together with some other incontestable premiss, you demonstrate, by means of the two, some new conclusion notoriously false. What I here say, is illustrated by the very example which he gives in this chapter. The incommensurability of the diagonal (with the side of the square) is demonstrated by Reductio ad Impossibile; because if it be supposed commensurable, you may demonstrate that an odd number is equal to an even number; a conclusion which every one will declare to be inadmissible, but which is not the contradictory of either of the premisses whereby the true proposition was demonstrated.
Here Aristotle expressly reserves for separate treatment the general subject of Syllogisms from Hypothesis.88
88 The expressions of Aristotle here are remarkable, Analyt. Prior. I. xliv. p. 50, a. 39-b. 3: πολλοὶ δὲ καὶ ἕτεροι περαίνονται ἐξ ὑποθέσεως, οὓς ἐπισκέψασθαι δεῖ καὶ διασημῆναι καθαρῶς. τίνες μὲν οὖν αἱ διαφοραὶ τούτων, καὶ ποσαχῶς γίνεται τὸ ἐξ ὑποθέσεως, ὕστερον ἐροῦμεν· νῦν δὲ τοσοῦντον ἡμῖν ἔστω φανερόν, ὅτι οὐκ ἔστιν ἀναλύειν εἰς τὰ σχήματα τοὺς τοιούτους συλλογισμούς. καὶ δι’ ἣν αἰτίαν, εἰρήκαμεν.
Syllogisms from Hypothesis were many and various, and Aristotle intended to treat them in a future treatise; but all that concerns the present treatise, in his opinion, is, to show that none of them can be reduced to the three Figures. Among the Syllogisms from Hypothesis, two varieties recognized by Aristotle (besides οἰ διὰ τοῦ ἀδυνάτου) were οἱ κατὰ μετάληψιν and οἱ κατὰ ποιότητα. The same proposition which Aristotle entitles κατὰ μετάληψιν, was afterwards designated by the Stoics κατὰ πρόσληψιν (Alexander ap. Schol. p. 178, b. 6-24).
It seems that Aristotle never realized this intended future treatise on Hypothetical Syllogisms; at least Alexander did not know it. The subject was handled more at large by Theophrastus and Eudêmus after Aristotle (Schol. p. 184, b. 45. Br.; Boethius, De Syllog. Hypothetico, pp. 606-607); and was still farther expanded by Chrysippus and the Stoics.
Compare Prantl, Geschichte der Logik, I. pp. 295, 377, seq. He treats the Hypothetical Syllogism as having no logical value, and commends Aristotle for declining to develop or formulate it; while Ritter (Gesch. Phil. iii. p. 93), and, to a certain extent, Ueberweg (System der Logik, sect. 121, p. 326), consider this to be a defect in Aristotle.
In the last chapter of the first book of the Analytica Priora, Aristotle returns to the point which we have already considered in the treatise De Interpretatione, viz. what is really a negative proposition; and how the adverb of negation must be placed in order to constitute one. We must place this adverb immediately before the copula and in conjunction with the copula: we must not place it after the copula and in conjunction with the predicate; for, if we do so, the proposition resulting will not be negative but affirmative (ἐκ μεταθέσεως, by transposition, according to the technical term introduced afterwards by Theophrastus). Thus of the four propositions:
| 1. Est bonum. | 2. Non est bonum. |
| 4. Non est non bonum. | 3. Est non bonum. |
No. 1 is affirmative; No. 3 is affirmative (ἐκ μεταθέσεως); Nos. 2 and 4 are negative. Wherever No. 1 is predicable, No. 4 will be predicable also; wherever No. 3 is predicable, No. 2 will be predicable also — but in neither case vice versâ.89 Mistakes often flow from incorrectly setting out the two contradictories.
89 Analyt. Prior. I. xlvi. p. 51, b. 5, ad finem. See above, Chap. IV. p. 118, seq.
CHAPTER VI.
ANALYTICA PRIORA II.
The Second Book of the Analytica Priora seems conceived with a view mainly to Dialectic and Sophistic, as the First Book bore more upon Demonstration.1 Aristotle begins the Second Book by shortly recapitulating what he had stated in the First; and then proceeds to touch upon some other properties of the Syllogism. Universal syllogisms (those in which the conclusion is universal) he says, have always more conclusions than one; particular syllogisms sometimes, but not always, have more conclusions than one. If the conclusion be universal, it may always be converted — simply, when it is negative, or per accidens, when it is affirmative; and its converse thus obtained will be proved by the same premisses. If the conclusion be particular, it will be convertible simply when affirmative, and its converse thus obtained will be proved by the same premisses; but it will not be convertible at all when negative, so that the conclusion proved will be only itself singly.2 Moreover, in the universal syllogisms of the First figure (Barbara, Celarent), any of the particulars comprehended under the minor term may be substituted in place of the minor term as subject of the conclusion, and the proof will hold good in regard to them. So, again, all or any of the particulars comprehended in the middle term may be introduced as subject of the conclusion in place of the minor term; and the conclusion will still remain true. In the Second figure, the change is admissible only in regard to those particulars comprehended under the subject of the conclusion or minor term, and not (at least upon the strength of the syllogism) in regard to those comprehended under the middle term. Finally, wherever the conclusion is particular, the change is admissible, though not by reason of the syllogism in regard to particulars comprehended under the middle term; it is not admissible as regards the minor term, which is itself particular.3
1 This is the remark of the ancient Scholiasts. See Schol. p. 188, a. 44, b. 11.
2 Analyt. Prior. II. i. p. 53, a. 3-14.
3 Analyt. Prior. II. i. p. 53, a. 14-35. M. Barthélemy St. Hilaire, following Pacius, justly remarks (note, p. 203 of his translation) that the rule as to particulars breaks down in the cases of Baroco, Disamis, and Bocardo.
On the chapter in general he remarks (note, p. 204):— “Cette théorie des conclusions diverses, soit patentes soit cachées, d’un même syllogisme, est surtout utile en dialectique, dans la discussion; où il faut faire la plus grande attention à ce qu’on accorde à l’adversaire, soit explicitement, soit implicitement.” This illustrates the observation cited in the preceding note from the Scholiasts.
Aristotle has hitherto regarded the Syllogism with a view to its formal characteristics: he now makes an important observation which bears upon its matter. Formally speaking, the two premisses are always assumed to be true; but in any real case of syllogism (form and matter combined) it is possible that either one or both may be false. Now, Aristotle remarks that if both the premisses are true (the syllogism being correct in form), the conclusion must of necessity be true; but that if either or both the premisses are false, the conclusion need not necessarily be false likewise. The premisses being false, the conclusion may nevertheless be true; but it will not be true because of or by reason of the premisses.4
4 Analyt. Prior. II. ii. p. 53, b. 5-10: ἐξ ἀληθῶν μὲν οὖν οὐκ ἔστι ψεῦδος συλλογίσασθαι, ἐκ ψευδῶν δ’ ἔστιν ἀληθές, πλὴν οὐ διότι ἀλλ’ ὅτι· τοῦ γὰρ διότι οὐκ ἔστιν ἐκ ψευδῶν συλλογισμός· δι’ ἣν δ’ αἰτίαν, ἐν τοῖς ἑπομένοις λεχθήσεται.
The true conclusion is not true by reason of these false premisses, but by reason of certain other premisses which are true, and which may be produced to demonstrate it. Compare Analyt. Poster. I. ii. p. 71, b. 19.
First, he would prove that if the premisses be true, the conclusion must be true also; but the proof that he gives does not seem more evident than the probandum itself. Assume that if A exists, B must exist also: it follows from hence (he argues) that if B does not exist, neither can A exist; which he announces as a reductio ad absurdum, seeing that it contradicts the fundamental supposition of the existence of A.5 Here the probans is indeed equally evident with the probandum, but not at all more evident; one who disputes the latter, will dispute the former also. Nothing is gained in the way of proof by making either of them dependent on the other. Both of them are alike self-evident; that is, if a man hesitates to admit either of them, you have no means of removing his scruples except by inviting him to try the general maxim upon as many particular cases as he chooses, and to see whether it does not hold good without a single exception.
5 Ibid. II. ii. p. 53, b. 11-16.
In regard to the case here put forward as illustration, Aristotle has an observation which shows his anxiety to maintain the characteristic principles of the Syllogism; one of which principles he had declared to be — That nothing less than three terms and two propositions, could warrant the inferential step from premisses to conclusion. In the present case he assumed, If A exists, then B must exist; giving only one premiss as ground for the inference. This (he adds) does not contravene what has been laid down before; for A in the case before us represents two propositions conceived in conjunction.6 Here he has given the type of hypothetical reasoning; not recognizing it as a variety per se, nor following it out into its different forms (as his successors did after him), but resolving it into the categorical syllogism.7 He however conveys very clearly the cardinal principle of all hypothetical inference — That if the antecedent be true, the consequent must be true also, but not vice versâ; if the consequent be false, the antecedent must be false also, but not vice versâ.
6 Analyt. Prior. II. ii. p. 53, b. 16-25. τὸ οὖν Ἀ ὥσπερ ἓν κεῖται, δύο προτάσεις συλληφθεῖσαι.
7 Aristotle, it should be remarked, uses the word κατηγορικός, not in the sense which it subsequently acquired, as the antithesis of ὑποθετικός in application to the proposition and syllogism, but in the sense of affirmative as opposed to στερητικός.
Having laid down the principle, that the conclusion may be true, though one or both the premisses are false, Aristotle proceeds, at great length, to illustrate it in its application to each of the three syllogistic figures.8 No portion of the Analytica is traced out more perspicuously than the exposition of this most important logical doctrine.
8 Analyt. Prior. II. ii.-iv. p. 53, b. 26-p. 57, b. 17. At the close (p. 57, a. 36-b. 17), the general doctrine is summed up.
It is possible (he then continues, again at considerable length) to invert the syllogism and to demonstrate in a circle. That is, you may take the conclusion as premiss for a new syllogism, together with one of the old premisses, transposing its terms; and thus you may demonstrate the other premiss. You may do this successively, first with the major, to demonstrate the minor; next, with the minor, to demonstrate the major. Each of the premisses will thus in turn be made a demonstrated conclusion; and the circle will be complete. But this can be done perfectly only in Barbara, and when, besides, all the three terms of the syllogism reciprocate with each other, or are co-extensive in import; so that each of the two premisses admits of being simply converted. In all other cases, the process of circular demonstration, where possible at all, is more or less imperfect.9
9 Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.
Having thus shown under what conditions the conclusion can be employed for the demonstration of the premisses, Aristotle proceeds to state by what transformation it can be employed for the refutation of them. This he calls converting the syllogism; a most inconvenient use of the term convert (ἀντιστρέφειν), since he had already assigned to that same term more than one other meaning, distinct and different, in logical procedure.10 What it here means is reversing the conclusion, so as to exchange it either for its contrary, or for its contradictory; then employing this reversed proposition as a new premiss, along with one of the previous premisses, so as to disprove the other of the previous premisses — i.e. to prove its contrary or contradictory. The result will here be different, according to the manner in which the conclusion is reversed; according as you exchange it for its contrary or its contradictory. Suppose that the syllogism demonstrated is: A belongs to all B, B belongs to all C; Ergo, A belongs to all C (Barbara). Now, if we reverse this conclusion by taking its contrary, A belongs to no C, and if we combine this as a new premiss with the major of the former syllogism, A belongs to all B, we shall obtain as a conclusion B belongs to no C; which is the contrary of the minor, in the form Camestres. If, on the other hand, we reverse the conclusion by taking its contradictory, A does not belong to all C, and combine this with the same major, we shall have as conclusion, B does not belong to all C; which is the contradictory of the minor, and in the form Baroco: though in the one case as in the other the minor is disproved. The major is contradictorily disproved, whether it be the contrary or the contradictory of the conclusion that is taken along with the minor to form the new syllogism; but still the form varies from Felapton to Bocardo. Aristotle shows farther how the same process applies to the other modes of the First, and to the modes of the Second and Third figures.11 The new syllogism, obtained by this process of reversal, is always in a different figure from the syllogism reversed. Thus syllogisms in the First figure are reversed by the Second and Third; those in the second, by the First and Third; those in the Third, by the First and Second.12
10 Schol. (ad Analyt. Prior. p. 59, b. 1), p. 190, b. 20, Brandis. Compare the notes of M. Barthélemy St. Hilaire, pp. 55, 242.
11 Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.
12 Ibid. x. p. 61, a. 7-15.
Of this reversing process, one variety is what is called the Reductio ad Absurdum; in which the conclusion is reversed by taking its contradictory (never its contrary), and then joining this last with one of the premisses, in order to prove the contradictory or contrary of the other premiss.13 The Reductio ad Absurdum is distinguished from the other modes of reversal by these characteristics: (1) That it takes the contradictory, and not the contrary, of the conclusion; (2) That it is destined to meet the case where an opponent declines to admit the conclusion; whereas the other cases of reversion are only intended as confirmatory evidence towards a person who already admits the conclusion; (3) That it does not appeal to or require any concession on the part of the opponent; for if he declines to admit the conclusion, you presume, as a matter of course, that he must adhere to the contradictory of the conclusion; and you therefore take this contradictory for granted (without asking his concurrence) as one of the bases of a new syllogism; (4) That it presumes as follows:— When, by the contradictory of the conclusion joined with one of the premisses, you have demonstrated the opposite of the other premiss, the original conclusion itself is shown to be beyond all impeachment on the score of form, i.e. beyond impeachment by any one who admits the premisses. You assume to be true, for the occasion, the very proposition which you mean finally to prove false; your purpose in the new syllogism is, not to demonstrate the original conclusion, but to prove it to be true by demonstrating its contradictory to be false.14
13 Analyt. Prior. II. xi. p. 61, a. 18, seq.
14 Ibid. p. 62, a. 11: φανερὸν οὖν ὅτι οὐ τὸ ἐναντίον, ἀλλὰ τὸ ἀντικείμενον, ὑποθετέον ἐν ἅπασι τοῖς συλλογισμοῖς. οὕτω γὰρ τὸ ἀναγκαῖον ἔσται καὶ τὸ ἀξίωμα ἔνδοξον. εἰ γὰρ κατὰ παντὸς ἢ κατάφασις ἢ ἀπόφασις, δειχθέντος ὅτι οὐχ ἡ ἀπόφασις, ἀνάγκη τὴν κατάφασιν ἀληθεύεσθαι. See Scholia, p. 190, b. 40, seq., Brand.
By the Reductio ad Absurdum you can in all the three figures demonstrate all the four varieties of conclusion, universal and particular, affirmative and negative; with the single exception, that you cannot by this method demonstrate in the First figure the Universal Affirmative.15 With this exception, every true conclusion admits of being demonstrated by either of the two ways, either directly and ostensively, or by reduction to the impossible.16