86Ibid. xliv. p. 50, a. 16-28.
86Ibid. xliv. p. 50, a. 16-28.
87Analyt. 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 thelatteris 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: theformerappears to me incorrect. Aristotle decidedly reckons theReductio ad Impossibileamong hypothetical proofs. But he understands byReductio ad Impossibilesomething 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 byReductio 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.
87Analyt. 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 thelatteris 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: theformerappears to me incorrect. Aristotle decidedly reckons theReductio ad Impossibileamong hypothetical proofs. But he understands byReductio ad Impossibilesomething 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 byReductio 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
88The 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.
88The 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 anegativeproposition; 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, accordingto 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.
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 casevice versâ.89Mistakes often flow from incorrectly setting out the two contradictories.
89Analyt. Prior. I. xlvi. p. 51, b. 5, ad finem. See above, Chap. IV.p. 118, seq.
89Analyt. Prior. I. xlvi. p. 51, b. 5, ad finem. See above, Chap. IV.p. 118, seq.
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.1Aristotle 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, orper 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.2Moreover, 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
1This is the remark of the ancient Scholiasts. See Schol. p. 188, a. 44, b. 11.
1This is the remark of the ancient Scholiasts. See Schol. p. 188, a. 44, b. 11.
2Analyt. Prior. II. i. p. 53, a. 3-14.
2Analyt. Prior. II. i. p. 53, a. 3-14.
3Analyt. 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 ofBaroco,Disamis, andBocardo.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.
3Analyt. 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 ofBaroco,Disamis, andBocardo.
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 itsformalcharacteristics: he now makes an important observation which bears upon itsmatter. Formally speaking,thetwo 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
4Analyt. 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.
4Analyt. 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 theprobandumitself. 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 areductio ad absurdum, seeing that it contradicts the fundamental supposition of the existence of A.5Here theprobansis indeed equally evident with theprobandum, 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.
5Ibid. II. ii. p. 53, b. 11-16.
5Ibid. 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 maintainthe 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.6Here he has given the type of hypothetical reasoning; not recognizing it as a varietyper se, nor following it out into its different forms (as his successors did after him), but resolving it into the categorical syllogism.7He however conveys very clearly the cardinal principle of all hypothetical inference — That if the antecedent be true, the consequent must be true also, but notvice versâ; if the consequent be false, the antecedent must be false also, but notvice versâ.
6Analyt. Prior. II. ii. p. 53, b. 16-25. τὸ οὖν Ἀ ὥσπερ ἓν κεῖται, δύο προτάσεις συλληφθεῖσαι.
6Analyt. Prior. II. ii. p. 53, b. 16-25. τὸ οὖν Ἀ ὥσπερ ἓν κεῖται, δύο προτάσεις συλληφθεῖσαι.
7Aristotle, 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.
7Aristotle, 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.8No portion of the Analytica is traced out more perspicuously than the exposition of this most important logical doctrine.
8Analyt. 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.
8Analyt. 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 demonstratein 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 inBarbara, 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
9Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.
9Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.
Having thus shown under what conditions the conclusioncan 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 callsconvertingthe syllogism; a most inconvenient use of the termconvert(ἀντιστρέφειν), since he had already assigned to that same term more than one other meaning, distinct and different, in logical procedure.10What it here means isreversingthe 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 itscontrary, 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 thecontraryof the minor, in the formCamestres. If, on the other hand, we reverse the conclusion by taking itscontradictory, 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 thecontradictoryof the minor, and in the formBaroco: though in the one case as in the other the minor is disproved. The major iscontradictorilydisproved, 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 fromFelaptontoBocardo. 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.11The 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
10Schol. (ad Analyt. Prior. p. 59, b. 1), p. 190, b. 20, Brandis. Compare the notes of M. Barthélemy St. Hilaire, pp. 55, 242.
10Schol. (ad Analyt. Prior. p. 59, b. 1), p. 190, b. 20, Brandis. Compare the notes of M. Barthélemy St. Hilaire, pp. 55, 242.
11Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.
11Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.
12Ibid. x. p. 61, a. 7-15.
12Ibid. x. p. 61, a. 7-15.
Of this reversing process, one variety is what is called theReductio 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 contradictoryor contrary of the other premiss.13TheReductio ad Absurdumis 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
13Analyt. Prior. II. xi. p. 61, a. 18, seq.
13Analyt. Prior. II. xi. p. 61, a. 18, seq.
14Ibid. 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.
14Ibid. 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 theReductio ad Absurdumyou 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.15With 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
15Ibid. p. 61, a. 35-p. 62, b. 10; xii. p. 62, a. 21. Alexander, ap. Schol. p. 191, a. 17-36, Brand.
15Ibid. p. 61, a. 35-p. 62, b. 10; xii. p. 62, a. 21. Alexander, ap. Schol. p. 191, a. 17-36, Brand.
16Ibid. xiv. p. 63, b. 12-21.
16Ibid. xiv. p. 63, b. 12-21.
In the Second and Third figures, though not in the First, it is possible to obtain conclusions even from two premisses which are contradictory or contrary to each other; but the conclusion will, as a matter of course, be a self-contradictory one. Thus if in the Second figure you have the two premisses — All Science is good; No Science is good — you get the conclusion (inCamestres), No Science is Science. In opposed propositions,the same predicate must be affirmed and denied of the same subject in one of the three different forms — All and None, All and Not All, Some and None. This shows why such conclusions cannot be obtained in the First figure; for it is the characteristic of that figure that the middle term must be predicate in one premiss, and subject in the other.17In dialectic discussion it will hardly be possible to get contrary or contradictory premisses conceded by the adversary immediately after each other, because he will be sure to perceive the contradiction: you must mask your purpose by asking the two questions not in immediate succession, but by introducing other questions between the two, or by other indirect means as suggested in the Topica.18
17Analyt. Prior. II. xv. p. 63, b. 22-p. 64, a. 32. Aristotle here declaresSubcontraries(as they were later called), — Some men are wise, Some men are not wise, — to be opposed only in expression or verbally (κατὰ τὴν λέξιν μόνον).
17Analyt. Prior. II. xv. p. 63, b. 22-p. 64, a. 32. Aristotle here declaresSubcontraries(as they were later called), — Some men are wise, Some men are not wise, — to be opposed only in expression or verbally (κατὰ τὴν λέξιν μόνον).
18Ibid. II. xv. p. 64, a. 33-37. See Topica, VIII. i. p. 155, a. 26; Julius Pacius, p. 372, note. In the Topica, Aristotle suggests modes of concealing the purpose of the questioner and driving the adversary to contradict himself: ἐν δὲ τῶς Τοπικοῖς παραδίδωσι μεθόδους τῶν κρύψεων δι’ ἃς τοῦτο δοθήσεται (Schol. p. 192, a. 18, Br.). Compare also Analyt. Prior. II. xix. p. 66, a. 33.
18Ibid. II. xv. p. 64, a. 33-37. See Topica, VIII. i. p. 155, a. 26; Julius Pacius, p. 372, note. In the Topica, Aristotle suggests modes of concealing the purpose of the questioner and driving the adversary to contradict himself: ἐν δὲ τῶς Τοπικοῖς παραδίδωσι μεθόδους τῶν κρύψεων δι’ ἃς τοῦτο δοθήσεται (Schol. p. 192, a. 18, Br.). Compare also Analyt. Prior. II. xix. p. 66, a. 33.
Aristotle now passes to certain general heads of Fallacy, or general liabilities to Error, with which the syllogizing process is beset. What the reasoner undertakes is, to demonstrate the conclusion before him, and to demonstrate it in the natural and appropriate way; that is, from premisses both more evident in themselves and logically prior to the conclusion. Whenever he fails thus to demonstrate, there is error of some kind; but he may err in several ways: (1) He may produce a defective or informal syllogism; (2) His premisses may be more unknowable than his conclusion, or equally unknowable; (3) His premisses, instead of being logically prior to the conclusion, may be logically posterior to it.19
19Ibid. II. xvi. p. 64, b. 30-35: καὶ γὰρ εἰ ὅλως μὴ συλλογίζεται, καὶ εἰ δι’ ἀγνωστοτέρων ἢ ὁμοίως ἀγνώστων, καὶ εἰ διὰ τῶν ὑστέρων τὸ πρότερον· ἡ γὰρ ἀπόδειξις ἐκ πιστοτέρων τε καὶ προτέρων ἐστιν.… τὰμὲν δι’ αὑτῶν πέφυκε γνωρίζεσθαι,τὰ δὲ δι’ ἄλλων.
19Ibid. II. xvi. p. 64, b. 30-35: καὶ γὰρ εἰ ὅλως μὴ συλλογίζεται, καὶ εἰ δι’ ἀγνωστοτέρων ἢ ὁμοίως ἀγνώστων, καὶ εἰ διὰ τῶν ὑστέρων τὸ πρότερον· ἡ γὰρ ἀπόδειξις ἐκ πιστοτέρων τε καὶ προτέρων ἐστιν.… τὰμὲν δι’ αὑτῶν πέφυκε γνωρίζεσθαι,τὰ δὲ δι’ ἄλλων.
Distinct from all these three, however, Aristotle singles out and dwells upon another mode of error, which he callsPetitio Principii. Some truths, theprincipia, are by nature knowable through or in themselves, others are knowable only through other things. If you confound this distinction, and ask or assume something of the latter class as if it belonged to the former, you commit aPetitio Principii. You may commit it either by assuming at once that which ought to be demonstrated, or by assuming, as if it were aprincipium, something else among those matters which in natural propriety would be demonstratedby means of aprincipium. Thus, there is (let us suppose) a natural propriety that C shall be demonstrated through A; but you, overlooking this, demonstrate B through C, and A through B. By thus inverting the legitimate order, you do what is tantamount to demonstrating A through itself; for your demonstration will not hold unless you assume A at the beginning, in order to arrive at C. This is a mistake made not unfrequently, and especially by some who define parallel lines; for they give a definition which cannot be understood unless parallel lines be presupposed.20
20Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9.Petere principiumis, in the phrase of Aristotle, not τὴν ἀρχὴν αἰτεῖσθαι, but τὸ ἐν ἀρχῇ αἰτεῖσθαι or τὸ ἐξ ἀρχῆς αἰτεῖσθαι (xvi. p. 64, b. 28, 34).
20Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9.Petere principiumis, in the phrase of Aristotle, not τὴν ἀρχὴν αἰτεῖσθαι, but τὸ ἐν ἀρχῇ αἰτεῖσθαι or τὸ ἐξ ἀρχῆς αἰτεῖσθαι (xvi. p. 64, b. 28, 34).
When the problem is such, that it is uncertain whether A can be predicated either of C or of B, if you then assume that A is predicable of B, you may perhaps not commitPetitio Principii, but you certainly fail in demonstrating the problem; for no demonstration will hold where the premiss is equally uncertain with the conclusion. But if, besides, the case be such, that B is identical with C, that is, either co-extensive and reciprocally convertible with C, or related to C as genus or species, — in either of these cases you commitPetitio Principiiby assuming that A may be predicated of B.21For seeing that B reciprocates with C, you might just as well demonstrate that A is predicable of B, because it is predicable of C; that is, you might demonstrate the major premiss by means of the minor and the conclusion, as well as you can demonstrate the conclusion by means of the major and the minor premiss. If you cannot so demonstrate the major premiss, this is not because the structure of the syllogism forbids it, but because the predicate of the major premiss is more extensive than the subject thereof. If it be co-extensive and convertible with the subject, we shall have a circular proof of three propositions in which each may be alternately premiss and conclusion. The like will be the case, if thePetitio Principiiis in the minor premiss and not in the major. In the First syllogistic figure it may be in either of the premisses; in the Second figure it can only be in the minor premiss, and that only in one mode (Camestres) of the figure.22The essence ofPetitio Principiiconsists in this, that you exhibit as trueper sethat which is not really trueper se.23You may commit this fault either in Demonstration, when you assume for true what is not really true, or in Dialectic, when you assume as probable and conformable to authoritative opinion what is not really so.24
21Ibid. p. 65, a. 1-10.
21Ibid. p. 65, a. 1-10.
22Ibid. p. 65, a. 10: εἰ οὖν τις, ἀδήλου ὄντος ὅτι τὸ Ἀ ὑπάρχει τῷ Γ, ὁμοίως δὲ καὶ ὅτι τῷ Β, αἰτοῖτο τῷ Β ὑπάρχειν τὸ Ἀ, οὕπω δῆλον εἰ τὸ ἐν ἀρχῇ αἰτεῖται, ἀλλ’ ὅτι οὐκ ἀποδείκνυσι, δῆλον· οὐ γὰρ ἀρχὴ ἀποδείξεως τὸ ὁμοίως ἄδηλον. εἰ μέντοι τὸ Β πρὸς τὸ Γ οὕτως ἔχει ὥστε ταὐτὸν εἶναι, ἢ δῆλον ὅτι ἀντιστρέφουσιν, ἢ ὑπάρχει θάτερον θατέρῳ, τὸ ἐν ἀρχῇ αἰτεῖται. καὶ γὰρ ἄν, ὅτι τῷ Β τὸ Ἀ ὑπάρχει, δι’ ἐκείνων δεικνύοι, εἰ ἀντιστρέφοι. νῦν δὲ τοῦτο κωλύει, ἀλλ’ οὐχ ὁ τρόπος. εἰ δὲ τοῦτο ποιοῖ, τὸ εἰρημένον ἂν ποιοῖ καὶ ἀντιστρέφοι ὡς διὰ τριῶν.This chapter, in which Aristotle declares the nature of Petitio Principii, is obscure and difficult to follow. It has been explained at some length, first by Philoponus in the Scholia (p. 192, a. 35, b. 24), afterwards by Julius Pacius (p. 376, whose explanation is followed by M. B. St. Hilaire, p. 288), and by Waitz, (I. p. 514). But the translation and comment given by Mr. Poste appear to me the best: “Assuming the conclusion to be affirmative, let us examine a syllogism in Barbara:—All B is A.All C is B.∴ All C is A.And let us first suppose that the major premiss is a Petitio Principii;i.e.that the propositionAll B is Ais identical with the propositionAll C is A. This can only be because the terms B and C are identical. Next, let us suppose that the minor premiss is a Petitio Principii:i.e.that the propositionAll C is Bis identical with the propositionAll C is A. This can only be because B and A are identical. The identity of the terms is, their convertibility or their sequence (ὑπάρχει, ἕπεται). This however requires some limitation; for as the major is always predicated (ὑπάρχει, ἕπεται) of the middle, and the middle of the minor, if this were enough to constitute Petitio Principii, every syllogism with a problematical premiss would be a Petitio Principii.â€� (See the Appendix A, pp. 178-183, attached to Mr. Poste’s edition of Aristotle’s Sophistici Elenchi.)Compare, about Petitio Principii, Aristot. Topic. VIII. xiii. p. 162, b. 34, in which passage Aristotle gives to the fallacy called Petitio Principii a still larger sweep than what he assigns to it in the Analytica Priora. Mr. Poste’s remark is perfectly just, that according to the above passage in the Analytica, every syllogism with a problematical (i.e.real as opposed to verbal) premiss would be a Petitio Principii; that is, all real deductive reasoning, in the syllogistic form, would be a Petitio Principii. To this we may add, that, from the passage above referred to in the Topica, all inductive reasoning also (reasoning from parts to whole) would involve Petitio Principii.Mr. Poste’s explanation of this difficult passage brings into view the original and valuable exposition made by Mr. John Stuart Mill of the Functions and Logical Value of the Syllogism. — System of Logic, Book II. ch. iii. sect 2:— â€�It must be granted, that in every syllogism, considered as an argument to prove the conclusion, there is a Petitio Principii,â€� &c.Petitio Principii, if ranked among the Fallacies, can hardly be extended beyond the first of the five distinct varieties enumerated in the Topica, VIII. xiii.
22Ibid. p. 65, a. 10: εἰ οὖν τις, ἀδήλου ὄντος ὅτι τὸ Ἀ ὑπάρχει τῷ Γ, ὁμοίως δὲ καὶ ὅτι τῷ Β, αἰτοῖτο τῷ Β ὑπάρχειν τὸ Ἀ, οὕπω δῆλον εἰ τὸ ἐν ἀρχῇ αἰτεῖται, ἀλλ’ ὅτι οὐκ ἀποδείκνυσι, δῆλον· οὐ γὰρ ἀρχὴ ἀποδείξεως τὸ ὁμοίως ἄδηλον. εἰ μέντοι τὸ Β πρὸς τὸ Γ οὕτως ἔχει ὥστε ταὐτὸν εἶναι, ἢ δῆλον ὅτι ἀντιστρέφουσιν, ἢ ὑπάρχει θάτερον θατέρῳ, τὸ ἐν ἀρχῇ αἰτεῖται. καὶ γὰρ ἄν, ὅτι τῷ Β τὸ Ἀ ὑπάρχει, δι’ ἐκείνων δεικνύοι, εἰ ἀντιστρέφοι. νῦν δὲ τοῦτο κωλύει, ἀλλ’ οὐχ ὁ τρόπος. εἰ δὲ τοῦτο ποιοῖ, τὸ εἰρημένον ἂν ποιοῖ καὶ ἀντιστρέφοι ὡς διὰ τριῶν.
This chapter, in which Aristotle declares the nature of Petitio Principii, is obscure and difficult to follow. It has been explained at some length, first by Philoponus in the Scholia (p. 192, a. 35, b. 24), afterwards by Julius Pacius (p. 376, whose explanation is followed by M. B. St. Hilaire, p. 288), and by Waitz, (I. p. 514). But the translation and comment given by Mr. Poste appear to me the best: “Assuming the conclusion to be affirmative, let us examine a syllogism in Barbara:—
All B is A.All C is B.∴ All C is A.
And let us first suppose that the major premiss is a Petitio Principii;i.e.that the propositionAll B is Ais identical with the propositionAll C is A. This can only be because the terms B and C are identical. Next, let us suppose that the minor premiss is a Petitio Principii:i.e.that the propositionAll C is Bis identical with the propositionAll C is A. This can only be because B and A are identical. The identity of the terms is, their convertibility or their sequence (ὑπάρχει, ἕπεται). This however requires some limitation; for as the major is always predicated (ὑπάρχει, ἕπεται) of the middle, and the middle of the minor, if this were enough to constitute Petitio Principii, every syllogism with a problematical premiss would be a Petitio Principii.â€� (See the Appendix A, pp. 178-183, attached to Mr. Poste’s edition of Aristotle’s Sophistici Elenchi.)
Compare, about Petitio Principii, Aristot. Topic. VIII. xiii. p. 162, b. 34, in which passage Aristotle gives to the fallacy called Petitio Principii a still larger sweep than what he assigns to it in the Analytica Priora. Mr. Poste’s remark is perfectly just, that according to the above passage in the Analytica, every syllogism with a problematical (i.e.real as opposed to verbal) premiss would be a Petitio Principii; that is, all real deductive reasoning, in the syllogistic form, would be a Petitio Principii. To this we may add, that, from the passage above referred to in the Topica, all inductive reasoning also (reasoning from parts to whole) would involve Petitio Principii.
Mr. Poste’s explanation of this difficult passage brings into view the original and valuable exposition made by Mr. John Stuart Mill of the Functions and Logical Value of the Syllogism. — System of Logic, Book II. ch. iii. sect 2:— �It must be granted, that in every syllogism, considered as an argument to prove the conclusion, there is a Petitio Principii,� &c.
Petitio Principii, if ranked among the Fallacies, can hardly be extended beyond the first of the five distinct varieties enumerated in the Topica, VIII. xiii.
23Analyt. Prior. II. xvi. p. 65, a. 23-27: τὸ γὰρ ἐξ ἀρχῆς τί δύναται, εἴρηται ἡμῖν, ὅτι τὸ δι’ αὑτοῦ δεικνύναι τὸ μὴ δι’ αὑτοῦ δῆλον. — τοῦτο δ’ ἔστι, τὸ μὴ δεικνύναι.The meaning of some lines in this chapter (p. 65, a. 17-18) is to me very obscure, after all the explanations of commentators.
23Analyt. Prior. II. xvi. p. 65, a. 23-27: τὸ γὰρ ἐξ ἀρχῆς τί δύναται, εἴρηται ἡμῖν, ὅτι τὸ δι’ αὑτοῦ δεικνύναι τὸ μὴ δι’ αὑτοῦ δῆλον. — τοῦτο δ’ ἔστι, τὸ μὴ δεικνύναι.
The meaning of some lines in this chapter (p. 65, a. 17-18) is to me very obscure, after all the explanations of commentators.
24Ibid. p. 65, a. 35; Topic. VIII. xiii. p. 162, b. 31.
24Ibid. p. 65, a. 35; Topic. VIII. xiii. p. 162, b. 31.
We must be careful to note, that when Aristotle speaks of aprincipiumas knowable in itself, or true in itself, he does not mean that it is innate, or that it starts up in the mind ready made without any gradual building up or preparation. What he means is, that it is not demonstrable deductively from anything else prior or more knowable by nature than itself. He declares (as we shall see) thatprincipiaare acquired, and mainly by Induction.
Next toPetitio Principii, Aristotle indicates another fallacious or erroneous procedure in dialectic debate; misconception ormisstatement of the real grounds on which a conclusion rests —Non per Hoc. You may impugn the thesis (set up by the respondent) directly, by proving syllogistically its contrary or contradictory; or you may also impugn it indirectly byReductio ad Absurdum;i.e.you prove by syllogism some absurd conclusion, which you contend to be necessarily true, if the thesis is admitted. Suppose you impugn it in the first method, or directly, by a syllogism containing only two premisses and a conclusion:Non per Hocis inapplicable here, for if either premiss is disallowed, the conclusion is unproved; the respondent cannot meet you except by questioning one or both of the premisses of your impugning syllogism.25But if you proceed by the second method or indirectly,Non per Hocmay become applicable; for there may then be more than two premisses, and he may, while granting that the absurd conclusion is correctly made out, contend that the truth or falsehood of his thesis is noway implicated in it. He declares (in Aristotle’s phrase) that the absurdity or falsehood just made out does not follow as a consequence from his thesis, but from other premisses independent thereof; that it would stand equally proved, even though his thesis were withdrawn.26In establishing the falsehood or absurdity you must take care that it shall be one implicated with or dependent upon his thesis. It is this last condition that he (the respondent) affirms to be wanting.27
25Analyt. Prior. II. xvii. p. 65, b. 4: ὅταν ἀναιρέθῃ τι δεικτικως διὰ τῶν Α, Β, Γ, &c.; xviii. 66, a. 17: ἢ γὰρ ἐκ τῶν δύο προτάσεων ἢ ἐκ πλειόνων πᾶς ἐστὶ συλλογισμός· εἰ μὲν οὖν ἐκ τῶν δύο, τούτων ἀνάγκη τὴν μὲν ἑτέραν ἢ καὶ ἀμφοτέρας εἶναι ψευδεῖς· &c. Whoever would understand this difficult chapter xvii., will do well to study it with the notes of Julius Pacius (p. 360), and also the valuable exposition of Mr. Poste, who has extracted and illustrated it in Appendix B. (p. 190) of the notes to his edition of the Sophistici Elenchi. The six illustrative diagrams given by Julius Pacius afford great help, though the two first of them appear to me incorrectly printed, as to the brackets connecting the different propositions.
25Analyt. Prior. II. xvii. p. 65, b. 4: ὅταν ἀναιρέθῃ τι δεικτικως διὰ τῶν Α, Β, Γ, &c.; xviii. 66, a. 17: ἢ γὰρ ἐκ τῶν δύο προτάσεων ἢ ἐκ πλειόνων πᾶς ἐστὶ συλλογισμός· εἰ μὲν οὖν ἐκ τῶν δύο, τούτων ἀνάγκη τὴν μὲν ἑτέραν ἢ καὶ ἀμφοτέρας εἶναι ψευδεῖς· &c. Whoever would understand this difficult chapter xvii., will do well to study it with the notes of Julius Pacius (p. 360), and also the valuable exposition of Mr. Poste, who has extracted and illustrated it in Appendix B. (p. 190) of the notes to his edition of the Sophistici Elenchi. The six illustrative diagrams given by Julius Pacius afford great help, though the two first of them appear to me incorrectly printed, as to the brackets connecting the different propositions.
26Ibid. II. xvii. p. 65, b. 38, b. 14, p. 66, a. 2, 7: τὸ μὴπαρὰ τοῦτοσυμβαίνειν τὸ ψεῦδος — τοῦ μὴπαρὰ τὴν θέσινεἶναι τὸ ψεῦδος — οὐπαρὰ τὴν θέσινσυμβαίνει τὸ ψεῦδος — οὐκ ἂν εἴηπαρὰ τὴν θέσιν.Instead of the preposition παρά, Aristotle on two occasions employs διά â€” οὕτω γὰρ ἔσταιδιὰ τὴν ὑπόθεσιν— p. 65, b. 33, p. 66, a. 3.The preposition παρά, with acc. case, meanson account of,owing to, &c. See Matthiæ and Kühner’s Grammars, and the passage of Thucydides i. 141; καὶ ἕκαστοςοὐ παρὰ τὴν ἑαυτοῦ ἀμέλειανοἰεται βλάψειν, μέλειν δέ τινι καὶ ἄλλῳ ὑπὲρ ἑαυτοῦ τι προϊδεῖν, &c., which I transcribe partly on account of Dr. Arnold’s note, who says about παρὰ here:— “This is exactly expressed in vulgar English,all along ofhis own neglect,i. e.owing to his own neglect.â€�
26Ibid. II. xvii. p. 65, b. 38, b. 14, p. 66, a. 2, 7: τὸ μὴπαρὰ τοῦτοσυμβαίνειν τὸ ψεῦδος — τοῦ μὴπαρὰ τὴν θέσινεἶναι τὸ ψεῦδος — οὐπαρὰ τὴν θέσινσυμβαίνει τὸ ψεῦδος — οὐκ ἂν εἴηπαρὰ τὴν θέσιν.
Instead of the preposition παρά, Aristotle on two occasions employs διά â€” οὕτω γὰρ ἔσταιδιὰ τὴν ὑπόθεσιν— p. 65, b. 33, p. 66, a. 3.
The preposition παρά, with acc. case, meanson account of,owing to, &c. See Matthiæ and Kühner’s Grammars, and the passage of Thucydides i. 141; καὶ ἕκαστοςοὐ παρὰ τὴν ἑαυτοῦ ἀμέλειανοἰεται βλάψειν, μέλειν δέ τινι καὶ ἄλλῳ ὑπὲρ ἑαυτοῦ τι προϊδεῖν, &c., which I transcribe partly on account of Dr. Arnold’s note, who says about παρὰ here:— “This is exactly expressed in vulgar English,all along ofhis own neglect,i. e.owing to his own neglect.â€�
27Ibid. II. xvii. p. 65, b. 33: δεῖ πρὸς τοὺς ἐξ ἀρχῆς ὅρους συνάπτειν τὸ ἀδύνατον· οὕτω γὰρ ἔσται διὰ τὴν ὑπόθεσιν.
27Ibid. II. xvii. p. 65, b. 33: δεῖ πρὸς τοὺς ἐξ ἀρχῆς ὅρους συνάπτειν τὸ ἀδύνατον· οὕτω γὰρ ἔσται διὰ τὴν ὑπόθεσιν.
Aristotle tells us that this was a precaution which the defender of a thesis was obliged often to employ in dialectic debate, in order to guard against abuse or misapplication ofReductio ad Absurdumon the part of opponents, who (it appears) sometimestook credit for success, when they had introduced and demonstrated some absurd conclusion that had little or no connection with the thesis.28But even when the absurd conclusion is connected with the thesis continuously, by a series of propositions each having a common term with the preceding, in either the ascending or the descending scale, we have here more than three propositions, and the absurd conclusion may perhaps be proved by the other premisses, without involving the thesis. In this case the respondent will meet you withNon per Hoc:29he will point out that his thesis is not one of the premisses requisite for demonstrating your conclusion, and is therefore not overthrown by the absurdity thereof. Perhaps the thesis may be false, but you have not shown it to be so, since it is not among the premisses necessary for proving yourabsurdum. Anabsurdummay sometimes admit of being demonstrated by several lines of premisses,30each involving distinct falsehood. Every false conclusion implies falsity in one or more syllogistic or prosyllogistic premisses that have preceded it, and isowing toor occasioned by this first falsehood.31