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]

[Footnote 1: This is the remark of the ancient Scholiasts. See Schol. p. 188, a. 44, b. 11.]

[Footnote 2: Analyt. Prior. II. i. p. 53, a. 3-14.]

[Footnote 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]

[Footnote 4: Analyt. Prior. II. ii. p. 53, b. 5-10: [Greek: e)x a)lêthô=n me\n ou)=n ou)k e)/sti pseu=dos sullogi/sasthai, e)k pseudô=n d' e)/stin a)lêthe/s, plê\n ou) dio/ti a)ll' o(/ti; tou= ga\r dio/ti ou)k e)/stin e)k pseudô=n sullogismo/s; di' ê(\n d' ai)ti/an, e)n toi=s e(pome/nois lechthê/setai.]

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.

[Footnote 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_.

[Footnote 6: Analyt. Prior. II. ii. p. 53, b. 16-25. [Greek: to\ ou)=n A ô(/sper e(\n kei=tai, du/o prota/seis sullêphthei=sai.]]

[Footnote 7: Aristotle, it should be remarked, uses the word [Greek: katêgoriko/s], not in the sense which it subsequently acquired, as the antithesis of [Greek: u(pothetiko/s] in application to the proposition and syllogism, but in the sense of affirmative as opposed to [Greek: sterêtiko/s].]

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.

[Footnote 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]

[Footnote 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_ ([Greek: a)ntistre/phein]), 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]

[Footnote 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.]

[Footnote 11: Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.]

[Footnote 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]**

[Footnote 13: Analyt. Prior. II. xi. p. 61, a. 18, seq.]

[Footnote 14: Ibid. p. 62, a. 11: [Greek: phanero\n ou)=n o(/ti ou) to\ e)nanti/on, a)lla\ to\ a)ntikei/menon, u(pothete/on e)n a(/pasi toi=s sullogismoi=s. ou(/tô ga\r to\ a)nagkai=on e)/stai kai\ to\ a)xi/ôma e)/ndoxon. ei) ga\r kata\ panto\s ê)\ kata/phasis ê)\ a)po/phasis, deichthe/ntos o(/ti ou)ch ê( a)po/phasis, a)na/gkê tê\n kata/phasin a)lêtheu/esthai.] 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]

[Footnote 15: Ibid. p. 61, a. 35-p. 62, b. 10; xii. p. 62, a. 21. Alexander, ap. Schol. p. 191, a. 17-36, Brand.]

[Footnote 16: Ibid. 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 (in _Camestres_), 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.[17] In 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]

[Footnote 17: Analyt. Prior. II. xv. p. 63, b. 22-p. 64, a. 32. Aristotle here declares _Subcontraries_ (as they were later called),--Some men are wise, Some men are not wise,--to be opposed only in expression or verbally ([Greek: kata\ tê\n le/xin mo/non]).]

[Footnote 18: Ibid. 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: [Greek: e)n de\ tô=s Topikoi=s paradi/dôsi metho/dous tô=n kru/pseôn di' a(\s tou=to dothê/setai] (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]

[Footnote 19: Ibid. II. xvi. p. 64, b. 30-35: [Greek: kai\ ga\r ei) o(/lôs mê\ sullogi/zetai, kai\ ei) di' a)gnôstote/rôn ê)\ o(moi/ôs a)gnô/stôn, kai\ ei) dia\ tô=n u(ste/rôn to\ pro/teron; ê( ga\r a)po/deixis e)k pistote/rôn te kai\ prote/rôn e)stin.... ta\ _me\n di' au(tô=n pe/phuke gnôri/zesthai, ta\ de\ di' a)/llôn_.]]

Distinct from all these three, however, Aristotle singles out and dwells upon another mode of error, which he calls _Petitio Principii_. Some truths, the _principia_, 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 a _Petitio Principii_. You may commit it either by assuming at once that which ought to be demonstrated, or by assuming, as if it were a _principium_, something else among those matters which in natural propriety would be demonstrated by means of a _principium_. 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]

[Footnote 20: Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9. _Petere principium_ is, in the phrase of Aristotle, not [Greek: tê\n a)rchê\n ai)tei=sthai], but [Greek: to\ e)n a)rchê=| ai)tei=sthai] or [Greek: to\ e)x a)rchê=s ai)tei=sthai] (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 commit _Petitio 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 commit _Petitio Principii_ by assuming that A may be predicated of B.[21] For 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 the _Petitio Principii_ is 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.[22] The essence of _Petitio Principii_ consists in this, that you exhibit as true _per se_ that which is not really true _per se_.[23] You 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]**

[Footnote 21: Ibid. p. 65, a. 1-10.]

[Footnote 22: Ibid. p. 65, a. 10: [Greek: ei) ou)=n tis, a)dê/lou o)/ntos o(/ti to\ A u(pa/rchei tô=| G, o(moi/ôs de\ kai\ o(/ti tô=| B, ai)toi=to tô=| B u(pa/rchein to\ A, ou(/pô dê=lon ei) to\ e)n a)rchê=| ai)tei=tai, a)ll' o(/ti ou)k a)podei/knusi, dê=lon; ou) ga\r a)rchê\ a)podei/xeôs to\ o(moi/ôs a)/dêlon. ei) me/ntoi to\ B pro\s to\ G ou(/tôs e)/chei ô(/ste tau)to\n ei)=nai, ê)\ dê=lon o(/ti a)ntistre/phousin, ê)\ u(pa/rchei tha/teron thate/rô|, to\ e)n a)rchê=| ai)tei=tai. kai\ ga\r a)/n, o(/ti tô=| B to\ A u(pa/rchei, di' e)kei/nôn deiknu/oi, ei) a)ntistre/phoi. nu=n de\ tou=to kôlu/ei, a)ll' ou)ch o( tro/pos. ei) de\ tou=to poioi=, to\ ei)rême/non a)\n poioi= kai\ a)ntistre/phoi ô(s dia\ triô=n.]

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 proposition _All B is A_ is identical with the proposition _All 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 proposition _All C is B_ is identical with the proposition _All C is A_. This can only be because B and A are identical. The identity of the terms is, their convertibility or their sequence ([Greek: u(pa/rchei, e(/petai]). This however requires some limitation; for as the major is always predicated ([Greek: u(pa/rchei, e(/petai]) 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." (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,