20Though some may fancy that the rule for converting the Universal Negative is intuitively known, yet every one must see that the rule for converting the Universal Affirmative is not thus self-evident, or derived from natural intuition. In fact, I believe that every learner at first hears it with great surprise. Some are apt to fancy that the Universal Affirmative (like the Particular Affirmative) may be convertedsimply. Indeed this error is not unfrequently committed in actual reasoning; all the more easily, because there is a class of cases (with subject and predicate co-extensive) where the converse of the Universal Affirmativeisreally true. Also, in the case of the Particular Negative, there are many true propositions in which the simple converse is true. A novice might incautiously generalize upon those instances, and conclude that both were convertible simply. Nor could you convince him of his error except by producing examples in which, when a true proposition of this kind is converted simply, the resulting converse is notoriously false. The appeal to various separate cases is the only basis on which we can rest for testing the correctness or incorrectness of all these maxims proclaimed as universal.
20Though some may fancy that the rule for converting the Universal Negative is intuitively known, yet every one must see that the rule for converting the Universal Affirmative is not thus self-evident, or derived from natural intuition. In fact, I believe that every learner at first hears it with great surprise. Some are apt to fancy that the Universal Affirmative (like the Particular Affirmative) may be convertedsimply. Indeed this error is not unfrequently committed in actual reasoning; all the more easily, because there is a class of cases (with subject and predicate co-extensive) where the converse of the Universal Affirmativeisreally true. Also, in the case of the Particular Negative, there are many true propositions in which the simple converse is true. A novice might incautiously generalize upon those instances, and conclude that both were convertible simply. Nor could you convince him of his error except by producing examples in which, when a true proposition of this kind is converted simply, the resulting converse is notoriously false. The appeal to various separate cases is the only basis on which we can rest for testing the correctness or incorrectness of all these maxims proclaimed as universal.
From one proposition taken singly, no new proposition can be inferred; for purposes of inference, two propositions at least are required.21This brings us to the rules of the Syllogism, where two propositions as premisses conduct us to a third which necessarily follows from them; and we are introduced to the well-known three Figures with their various Modes.22To form a valid Syllogism, there must be three terms and no more; the two, which appear as Subject and Predicate of the conclusion, are called theminorterm (or minor extreme) and themajorterm (or major extreme) respectively; while the third ormiddleterm must appear in each of the premisses, but not in the conclusion. These terms are calledextremesandmiddle, from the position which they occupy in every perfect Syllogism — that is in what Aristotle ranks as the First among the three figures. Inhisway of enunciating the Syllogism, this middle position formed a conspicuous feature; whereas the modern arrangement disguises it, though the denominationmiddleterm is still retained. Aristotle usually employs letters of the alphabet, which he was the first to select as abbreviations for exposition;23and he has two ways (conforming to what he had said in the first chapter of the present treatise) of enunciating the modes of the First figure. In one way, he begins with the major extreme (Predicate of the conclusion): A may be predicated of all B, B may be predicated of all C; therefore, A may be predicated of all C (Universal Affirmative). Again, A cannot be predicated of any B, B can be predicated of all C; therefore, A cannot be predicated of any C (Universal Negative). In the other way, he begins with the minor term (Subject of the conclusion): C is in the whole B, B is in the whole A; therefore, C is in the whole A (Universal Affirmative). And, C is in the whole B, B is not in the whole A; therefore, C is not in the whole A (Universal Negative). We see thus that in Aristotle’s way of enunciating the First figure, the middleterm is really placed between the two extremes,24though this is not so in the Second and Third figures. In the modern way of enunciating these figures, the middle term is never placed between the two extremes; yet the denominationmiddlestill remains.
21Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b. 35; Analyt. Poster. I. iii. p. 73, a. 7.
21Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b. 35; Analyt. Poster. I. iii. p. 73, a. 7.
22Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.
22Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.
23M. Barthélemy St. Hilaire (Logique d’Aristote, vol. ii. p. 7, n.), referring to the examples of Conversion in chap. ii., observes:— “Voici le prémier usage des lettres représentant des idées; c’est un procédé tout à fait algébrique, c’est à dire, de généralisation. Déjà , dans l’Herméneia, ch. 13, § 1 et suiv., Aristote a fait usage de tableaux pour représenter sa pensée relativement à la consécution des modales. Il parle encore spécialement de figures explicatives, liv. 2. des Derniers Analytiques, ch. 17, § 7. Vingt passages de l’Histoire des Animaux attestent qu’il joignait des dessins à ses observations et à ses théories zoologiques. Les illustrations pittoresques datent donc de fort loin. L’emploi symbolique des lettres a été appliqué aussi par Aristote à la Physique. Il l’avait emprunté, sans doute, aux procédés des mathématiciens.�We may remark, however, that when Aristotle proceeds to specify those combinations of propositions whichdo notgive a valid conclusion, he is not satisfied with giving letters of the alphabet; he superadds special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12, 34, 38).
23M. Barthélemy St. Hilaire (Logique d’Aristote, vol. ii. p. 7, n.), referring to the examples of Conversion in chap. ii., observes:— “Voici le prémier usage des lettres représentant des idées; c’est un procédé tout à fait algébrique, c’est à dire, de généralisation. Déjà , dans l’Herméneia, ch. 13, § 1 et suiv., Aristote a fait usage de tableaux pour représenter sa pensée relativement à la consécution des modales. Il parle encore spécialement de figures explicatives, liv. 2. des Derniers Analytiques, ch. 17, § 7. Vingt passages de l’Histoire des Animaux attestent qu’il joignait des dessins à ses observations et à ses théories zoologiques. Les illustrations pittoresques datent donc de fort loin. L’emploi symbolique des lettres a été appliqué aussi par Aristote à la Physique. Il l’avait emprunté, sans doute, aux procédés des mathématiciens.�
We may remark, however, that when Aristotle proceeds to specify those combinations of propositions whichdo notgive a valid conclusion, he is not satisfied with giving letters of the alphabet; he superadds special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12, 34, 38).
24Aristot. Analyt. Prior. I. iv. p. 25, b. 35: καλῶ δὲμέσον, ὃ καὶ αὐτὸ ἐν ἄλλῳ καὶ ἄλλο ἐν τούτῳ ἐστίν, ὃ καὶ τῇ θέσει γίνεται μέσον.
24Aristot. Analyt. Prior. I. iv. p. 25, b. 35: καλῶ δὲμέσον, ὃ καὶ αὐτὸ ἐν ἄλλῳ καὶ ἄλλο ἐν τούτῳ ἐστίν, ὃ καὶ τῇ θέσει γίνεται μέσον.
The Modes of each figure are distinguished by the different character and relation of the two premisses, according as these are either affirmative or negative, either universal or particular. Accordingly, there are four possible varieties of each, and sixteen possible modes or varieties of combinations between the two. Aristotle goes through most of the sixteen modes, and shows that in the first Figure there are only four among them that are legitimate, carrying with them a necessary conclusion. He shows, farther, that in all the four there are two conditions observed, and that both these conditions are indispensable in the First figure:— (1) The major proposition must be universal, either affirmative or negative; (2) The minor proposition must be affirmative, either universal or particular or indefinite. Such must be the character of the premisses, in the first Figure, wherever the conclusion is valid and necessary; andvice versâ, the conclusion will be valid and necessary, when such is the character of the premisses.25
25Aristot. Analyt. Prior. I. iv. p. 26, b. 26, et sup.
25Aristot. Analyt. Prior. I. iv. p. 26, b. 26, et sup.
In regard to the four valid modes (Barbara,Celarent,Darii,Ferio, as we read in the scholastic Logic) Aristotle declares at once in general language that the conclusion follows necessarily; which he illustrates by setting down in alphabetical letters the skeleton of a syllogism inBarbara. If A is predicated of all B, and B of all C, A must necessarily be predicated of all C. But he does not justify it by any real example; he produces no special syllogism with real terms, and with a conclusion known beforehand to be true. He seems to think that the general doctrine will be accepted as evident without any such corroboration. He counts upon the learner’s memory and phantasy for supplying, out of the past discourse of common life, propositions conforming to the conditions in which the symbolical letters have been placed, and for not supplying any contradictory examples. This might suffice for a treatise; but we may reasonably believe that Aristotle, when teaching in his school, would superadd illustrative examples; for the doctrine was then novel, and he is not unmindful of the errors into which learners often fall spontaneously.26
26Analyt. Poster. I. xxiv. p. 85, b. 21.
26Analyt. Poster. I. xxiv. p. 85, b. 21.
When he deals with the remaining or invalid modes of the First figure, his manner of showing their invalidity is different, and in itself somewhat curious. “If (he says) the major term is affirmed of all the middle, while the middle is denied of all the minor, no necessary consequence follows from such being the fact, nor will there be any syllogism of the two extremes; for it is equally possible, either that the major term may be affirmed of all the minor, or that it may be denied of all the minor; so that no conclusion, either universal or particular, is necessary in all cases.�27Examples of such double possibility are then exhibited: first, of three terms arranged in two propositions (A and E), in which, from the terms specially chosen, the major happens to be truly affirmable of all the minor; so that the third proposition is an universal Affirmative:—
Major andMiddle.}Animal is predicable of every Man;Middle andMinor}Man is not predicable of any Horse;Major andMinor}Animal is predicable of every Horse.
Major andMiddle.}Animal is predicable of every Man;Middle andMinor}Man is not predicable of any Horse;Major andMinor}Animal is predicable of every Horse.
Next, a second example is set out with new terms, in which the major happens not to be truly predicable of any of the minor; thus exhibiting as third proposition an universal Negative:—
Major andMiddle.}Animal is predicable of every Man;Middle andMinor}Man is not predicable of any Stone;Major andMinor}Animal is not predicable of any Stone.
Major andMiddle.}Animal is predicable of every Man;Middle andMinor}Man is not predicable of any Stone;Major andMinor}Animal is not predicable of any Stone.
Here we see that the full exposition of a syllogism is indicated with real terms common and familiar to every one; alphabetical symbols would not have sufficed, for the learner must himself recognize the one conclusion as true, the other as false. Hence we are taught that, after two premisses thus conditioned, if we venture to join together the major and minor so as to form a pretended conclusion, we may in some cases obtain a true proposition universally Affirmative, in other cases a true proposition universally Negative. Therefore (Aristotle argues) there is no one necessary conclusion, the same in all cases, derivable from such premisses; in other words, this mode of syllogism is invalid and proves nothing. He applies the like reasoning to all the other invalid modes of the first Figure; setting them aside in the same way, and producing examples wherein double and opposite conclusions (improperly so called), both true, are obtained in different cases from the like arrangement of premisses.
27Analyt. Prior. I. iv. p. 26, a. 2, seq.
27Analyt. Prior. I. iv. p. 26, a. 2, seq.
This mode of reasoning plainly depends upon an appeal to prior experience. The validity or invalidity of each mode of the First figure is tested by applying it to different particular cases, each of which is familiar and known to the learneraliunde; in one case, the conjunction of the major and minor terms in the third proposition makes an universal Affirmative which he knows to be true; in another case, the like conjunction makes an universal Negative, which he also knows to be true; so that there is no onenecessary(i.e.no one uniform and trustworthy) conclusion derivable from such premisses.28In other words, these modes of the First figure are not valid or available in form; the negation being sufficiently proved by one single undisputed example.
28Though M. Barthélemy St. Hilaire (note, p. 19) declares Aristotle’s exposition to be a model of analysis, it appears to me that the grounds for disallowing this invalid mode of the First figure (A — E — A, or A — E — E) are not clearly set forth by Aristotle himself, while they are rendered still darker by some of his best commentators. Thus Waitz says (p. 381): “Per exempla allata probat (Aristoteles) quod demonstrare debebat ex ipsâ ratione quam singuli termini inter se habeant: est enim proprium artis logicæ, ut terminorum rationem cognoscat, dum res ignoret. Num de Caio prædicetur animal nescit, scit de Caio prædicari animal, si animal de homine et homo de Caio prædicetur.�This comment of Waitz appears to me founded in error. Aristotle had no means of shewing the invalidity of the mode A E in the First figure, except by an appeal to particular examples. The invalidity of the invalid modes, and the validity of the valid modes, rest alike upon this ultimate reference to examples of propositions known to be true or false, by prior experience of the learner. The valid modes are those which will stand this trial and verification; the invalid modes are those which will not stand it. Not till such verification has been made, is one warranted in generalizing the result, and enunciating a formula applicable to unknown particulars (rationem terminorum cognoscere, dum res ignoret). It was impossible for Aristotle to do what Waitz requires of him. I take the opposite ground, and regret that he did not set forth the fundamental test of appeal to example and experience, in a more emphatic and unmistakeable manner.M. Barthélemy St. Hilaire (in the note to his translation, p. 14) does not lend any additional clearness, when he talks of the “conclusion� from the propositions A and E in the First figure. Julius Pacius says (p. 134): “Si tamenconclusiodici debet, quæ non colligitur ex propositionibus,� &c. Moreover, M. St. Hilaire (p. 19) slurs over the legitimate foundation, the appeal to experience, much as Aristotle himself does: “Puis prenant des exemples où laconclusion est de toute évidence, Aristote les applique successivement à chacune de ces combinaisons; celles qui donnent laconclusion fournie d’ailleurs par le bon sens, sont concluantes ou syllogistiques, les autres sont asyllogistiques.�
28Though M. Barthélemy St. Hilaire (note, p. 19) declares Aristotle’s exposition to be a model of analysis, it appears to me that the grounds for disallowing this invalid mode of the First figure (A — E — A, or A — E — E) are not clearly set forth by Aristotle himself, while they are rendered still darker by some of his best commentators. Thus Waitz says (p. 381): “Per exempla allata probat (Aristoteles) quod demonstrare debebat ex ipsâ ratione quam singuli termini inter se habeant: est enim proprium artis logicæ, ut terminorum rationem cognoscat, dum res ignoret. Num de Caio prædicetur animal nescit, scit de Caio prædicari animal, si animal de homine et homo de Caio prædicetur.�
This comment of Waitz appears to me founded in error. Aristotle had no means of shewing the invalidity of the mode A E in the First figure, except by an appeal to particular examples. The invalidity of the invalid modes, and the validity of the valid modes, rest alike upon this ultimate reference to examples of propositions known to be true or false, by prior experience of the learner. The valid modes are those which will stand this trial and verification; the invalid modes are those which will not stand it. Not till such verification has been made, is one warranted in generalizing the result, and enunciating a formula applicable to unknown particulars (rationem terminorum cognoscere, dum res ignoret). It was impossible for Aristotle to do what Waitz requires of him. I take the opposite ground, and regret that he did not set forth the fundamental test of appeal to example and experience, in a more emphatic and unmistakeable manner.
M. Barthélemy St. Hilaire (in the note to his translation, p. 14) does not lend any additional clearness, when he talks of the “conclusion� from the propositions A and E in the First figure. Julius Pacius says (p. 134): “Si tamenconclusiodici debet, quæ non colligitur ex propositionibus,� &c. Moreover, M. St. Hilaire (p. 19) slurs over the legitimate foundation, the appeal to experience, much as Aristotle himself does: “Puis prenant des exemples où laconclusion est de toute évidence, Aristote les applique successivement à chacune de ces combinaisons; celles qui donnent laconclusion fournie d’ailleurs par le bon sens, sont concluantes ou syllogistiques, les autres sont asyllogistiques.�
We are now introduced to the Second figure, in which each of the two premisses has the middle term as Predicate.29To give a legitimate conclusion in this figure, one or other of the premisses must be negative, and the major premiss must be universal; moreover no affirmative conclusions can ever be obtained in it — none but negative conclusions, universal or particular. In this Second figure too, Aristotle recognizes four valid modes; settingaside the other possible modes as invalid30(in the same way as he had done in the First figure), because the third proposition or conjunction of the major term with the minor, might in some cases be a true universal affirmative, in other cases a true universal negative. As to the third and fourth of the valid modes, he demonstrates them by assuming the contradictory of the conclusion, together with the major premiss, and then showing that these two premisses form a new syllogism, which leads to a conclusion contradicting the minor premiss. This method, calledReductio ad Impossibile, is here employed for the first time; and employed without being ushered in or defined, as if it were familiarly known.31
29Analyt. Prior. I. v. p. 26, b. 34. As Aristotle enunciates a proposition by putting the predicate before the subject, he says that in this Second figure the middle term comes πρῶτον τῇ θέσει. In the Third figure, for the same reason, he calls it ἔσχατον τῇ θέσει, vi. p. 28, a. 15.
29Analyt. Prior. I. v. p. 26, b. 34. As Aristotle enunciates a proposition by putting the predicate before the subject, he says that in this Second figure the middle term comes πρῶτον τῇ θέσει. In the Third figure, for the same reason, he calls it ἔσχατον τῇ θέσει, vi. p. 28, a. 15.
30Analyt. Prior. I. v. p. 27, a. 18. In these invalid modes, Aristotle says there is nosyllogism; therefore we cannot properly speak of aconclusion, but only of a third proposition, conjoining the major with the minor.
30Analyt. Prior. I. v. p. 27, a. 18. In these invalid modes, Aristotle says there is nosyllogism; therefore we cannot properly speak of aconclusion, but only of a third proposition, conjoining the major with the minor.
31Ibid. p. 27, a. 15, 26, seq. It is said to involve ὑπόθεσις, p. 28, a. 7; to be ἐξ ὑποθέσεως xxiii. p. 41, a. 25; to be τοῦ ἐξ ὑποθέσεως, as opposed to δεικτικός, xxiii. p. 40, b. 25.M. B. St. Hilaire remarks justly, that Aristotle might be expected to define or explain what it is, on first mentioning it (note, p. 22).
31Ibid. p. 27, a. 15, 26, seq. It is said to involve ὑπόθεσις, p. 28, a. 7; to be ἐξ ὑποθέσεως xxiii. p. 41, a. 25; to be τοῦ ἐξ ὑποθέσεως, as opposed to δεικτικός, xxiii. p. 40, b. 25.
M. B. St. Hilaire remarks justly, that Aristotle might be expected to define or explain what it is, on first mentioning it (note, p. 22).
Lastly, we have the Third figure, wherein the middle term is the Subject in both premisses. Here one at least of the premisses must be universal, either affirmative or negative. But no universal conclusions can be obtained in this figure; all the conclusions are particular. Aristotle recognizes six legitimate modes; in all of which the conclusions are particular, four of them being affirmative, two negative. The other possible modes he sets aside as in the two preceding figures.32
32Ibid. I. vi. p. 28, a. 10-p. 29, a. 18.
32Ibid. I. vi. p. 28, a. 10-p. 29, a. 18.
But Aristotle assigns to the First figure a marked superiority as compared with the Second and Third. It is the only one that yields perfect syllogisms; those furnished by the other two are all imperfect. The cardinal principle of syllogistic proof, as he conceives it, is — That whatever can be affirmed or denied of a whole, can be affirmed or denied of any part thereof.33The major proposition affirms or denies something universally respecting a certain whole; the minor proposition declares a certain part to be included in that whole. To this principle the four modes of the First figure manifestly and unmistakably conform, without any transformation of their premisses. But in the other figures such conformity does not obviously appear, andmust be demonstrated by reducing their syllogisms to the First figure; either ostensively by exposition of a particular case, and conversion of the premisses, or byReductio ad Impossibile. Aristotle, accordingly, claims authority for the Second and Third figures only so far as they can be reduced to the First.34We must, however, observe that in this process of reduction no new evidence is taken in; the matter of evidence remains unchanged, and the form alone is altered, according to laws of logical conversion which Aristotle has already laid down and justified. Another ground of the superiority and perfection which he claims for the First figure, is, that it is the only one in which every variety of conclusion can be proved; and especially the only one in which the Universal Affirmative can be proved — the great aim of scientific research. Whereas, in the Second figure we can prove onlynegativeconclusions, universal or particular; and in the Third figure onlyparticularconclusions, affirmative or negative.35
33Ibid. I. xli. p. 49, b. 37: ὅλως γὰρ ὃ μή ἐστιν ὡς ὅλον πρὸς μέρος καὶ ἄλλο πρὸς τοῦτο ὡς μέρος πρὸς ὅλον, ἐξ οὐδενὸς τῶν τοιούτων δείκνυσιν ὁ δεικνύων, ὥστε οὐδὲ γίνεται συλλογισμός.He had before said this about the relation of the three terms in the Syllogism, I. iv. p. 25, b. 32: ὅταν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι τῷ μέσῳ καὶ τὸν μέσον ἐν ὅλῳ τῷ πρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι συλλογισμὸν τέλειον (Dictum de Omni et Nullo).
33Ibid. I. xli. p. 49, b. 37: ὅλως γὰρ ὃ μή ἐστιν ὡς ὅλον πρὸς μέρος καὶ ἄλλο πρὸς τοῦτο ὡς μέρος πρὸς ὅλον, ἐξ οὐδενὸς τῶν τοιούτων δείκνυσιν ὁ δεικνύων, ὥστε οὐδὲ γίνεται συλλογισμός.
He had before said this about the relation of the three terms in the Syllogism, I. iv. p. 25, b. 32: ὅταν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι τῷ μέσῳ καὶ τὸν μέσον ἐν ὅλῳ τῷ πρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι συλλογισμὸν τέλειον (Dictum de Omni et Nullo).
34Analyt. Prior. I. vii. p. 29, a. 30-b. 25.
34Analyt. Prior. I. vii. p. 29, a. 30-b. 25.
35Ibid. I. iv. p. 26, b. 30, p. 27, a. 1, p. 28, a. 9, p. 29, a. 15. An admissible syllogism in the Second or Third figure is sometimes called δυνατὸς as opposed to τέλειος, p. 41, b. 33. Compare Kampe, Die Erkenntniss-Theorie des Aristoteles, p. 245, Leipzig, 1870.
35Ibid. I. iv. p. 26, b. 30, p. 27, a. 1, p. 28, a. 9, p. 29, a. 15. An admissible syllogism in the Second or Third figure is sometimes called δυνατὸς as opposed to τέλειος, p. 41, b. 33. Compare Kampe, Die Erkenntniss-Theorie des Aristoteles, p. 245, Leipzig, 1870.
Such are the main principles of syllogistic inference and rules for syllogistic reasoning, as laid down by Aristotle. During the mediæval period, they were allowed to ramify into endless subtle technicalities, and to absorb the attention of teachers and studious men, long after the time when other useful branches of science and literature were pressing for attention. Through such prolonged monopoly — which Aristotle, among the most encyclopedical of all writers, never thought of claiming for them — they have become so discredited, that it is difficult to call back attention to them as they stood in the Aristotelian age. We have to remind the reader, again, that though language was then used with great ability for rhetorical and dialectical purposes, there existed as yet hardly any systematic or scientific study of it in either of these branches. The scheme and the terminology of any such science were alike unknown, and Aristotle was obliged to construct it himself from the foundation. The rhetorical and dialectical teaching as then given (he tells us) was mere unscientific routine, prescribing specimens of art to be committed to memory: respecting syllogism (or the conditions of legitimate deductive inference) absolutely nothing had been said.36Under these circumstances,his theory of names, notions, and propositions as employed for purposes of exposition and ratiocination, is a remarkable example of original inventive power. He had to work it out by patient and laborious research. No way was open to him except the diligent comparison and analysis of propositions. And though all students have now become familiar with the various classes of terms and propositions, together with their principal characteristics and relations, yet to frame and designate such classes for the first time without any precedent to follow, to determine for each the rules and conditions of logical convertibility, to put together the constituents of the Syllogism, with its graduation of Figures and difference of Modes, and with a selection, justified by reasons given, between the valid and the invalid modes — all this implies a high order of original systematizing genius, and must have required the most laborious and multiplied comparisons between propositions in detail.
36Aristot. Sophist. Elench. p. 184, a. 1, b. 2: διόπερ ταχεῖα μὲν ἄτεχνος δ’ ἦν ἡ διδασκαλία τοῖς μανθάνουσι παρ’ αὐτῶν· οὐ γὰρ τέχνην ἀλλὰ τὰ ἀπὸ τῆς τέχνης διδόντες παιδεύειν ὑπελάμβανον …περὶ δὲ τοῦ συλλογίζεσθαι παντελῶς οὐδὲν εἴχομεν πρότερον ἄλλο λέγειν,ἀλλ’ ἢ τριβῇ ζητοῦντες πολὺν χρόνον ἐπονοῦμεν.
36Aristot. Sophist. Elench. p. 184, a. 1, b. 2: διόπερ ταχεῖα μὲν ἄτεχνος δ’ ἦν ἡ διδασκαλία τοῖς μανθάνουσι παρ’ αὐτῶν· οὐ γὰρ τέχνην ἀλλὰ τὰ ἀπὸ τῆς τέχνης διδόντες παιδεύειν ὑπελάμβανον …περὶ δὲ τοῦ συλλογίζεσθαι παντελῶς οὐδὲν εἴχομεν πρότερον ἄλλο λέγειν,ἀλλ’ ἢ τριβῇ ζητοῦντες πολὺν χρόνον ἐπονοῦμεν.
The preceding abridgment of Aristotle’s exposition of the Syllogism applies only to propositions simply affirmative or simply negative. But Aristotle himself, as already remarked, complicates the exposition by putting the Modal propositions (Possible, Necessary) upon the same line as the above-mentioned Simple propositions. I have noticed, in dealing with the treatise De Interpretatione, the confusion that has arisen from thus elevating the Modals into a line of classification co-ordinate with propositions simply Assertory. In the Analytica, this confusion is still more sensibly felt, from the introduction of syllogisms in which one of the premisses is necessary, while the other is only possible. We may remark, however, that, in the Analytica, Aristotle is stricter in defining the Possible than he has been in the De Interpretatione; for he now disjoins the Possible altogether from the Necessary, making it equivalent to the Problematical (not merelymay be, butmay be or may not be).37In the middle, too, of his diffuse exposition of the Modals, he inserts one important remark, respecting universal propositions generally,which belongs quite as much to the preceding exposition about propositions simply assertory. He observes that universal propositions have nothing to do with time, present, past, or future; but are to be understood in a sense absolute and unqualified.38
37Analyt. Prior. I. viii. p. 29, a. 32; xiii. p. 32, a. 20-36: τὸ γὰρ ἀναγκαῖον ὁμωνύμως ἐνδέχεσθαι λέγομεν. In xiv. p. 33, b. 22, he excludes this equivocal meaning of τὸ ἐνδεχόμενον — δεῖ δὲ τὸ ἐνδέχεσθα λαμβάνειν μὴ ἐν τοῖς ἀναγκαίοις, ἀλλὰ κατὰ τὸν εἰρημένον διορισμόν. See xiii. p. 32, a. 33, where τὸ ἐνδέχεσθαι ὑπάρχειν is asserted to be equivalent to or convertible with τὸ ἐνδέχεσθαι μὴ ὑπάρχειν; and xix. p. 38, a. 35: τὸ ἐξ ἀνάγκης οὐκ ἦνἐνδεχόμενον. Theophrastus and Eudemus differed from Aristotle about his theory of the Modals in several points (Scholia ad Analyt. Priora, pp. 161, b. 30; 162, b. 23; 166, a. 12, b. 15, Brand.). Respecting the want of clearness in Aristotle about τὸ ἐνδεχόμενον, see Waitz’s noteadp. 32, b. 16. Moreover, he sometimes uses ὑπάρχον in the widest sense, including ἐνδεχόμενον and ἀναγκαῖον, xxiii. p. 40, b. 24.
37Analyt. Prior. I. viii. p. 29, a. 32; xiii. p. 32, a. 20-36: τὸ γὰρ ἀναγκαῖον ὁμωνύμως ἐνδέχεσθαι λέγομεν. In xiv. p. 33, b. 22, he excludes this equivocal meaning of τὸ ἐνδεχόμενον — δεῖ δὲ τὸ ἐνδέχεσθα λαμβάνειν μὴ ἐν τοῖς ἀναγκαίοις, ἀλλὰ κατὰ τὸν εἰρημένον διορισμόν. See xiii. p. 32, a. 33, where τὸ ἐνδέχεσθαι ὑπάρχειν is asserted to be equivalent to or convertible with τὸ ἐνδέχεσθαι μὴ ὑπάρχειν; and xix. p. 38, a. 35: τὸ ἐξ ἀνάγκης οὐκ ἦνἐνδεχόμενον. Theophrastus and Eudemus differed from Aristotle about his theory of the Modals in several points (Scholia ad Analyt. Priora, pp. 161, b. 30; 162, b. 23; 166, a. 12, b. 15, Brand.). Respecting the want of clearness in Aristotle about τὸ ἐνδεχόμενον, see Waitz’s noteadp. 32, b. 16. Moreover, he sometimes uses ὑπάρχον in the widest sense, including ἐνδεχόμενον and ἀναγκαῖον, xxiii. p. 40, b. 24.
38Analyt. Prior. I. xv. p. 34, b. 7.
38Analyt. Prior. I. xv. p. 34, b. 7.
Having finished with the Modals, Aristotle proceeds to lay it down, that all demonstration must fall under one or other of the three figures just described; and therefore that all may be reduced ultimately to the two first modes of the First figure. You cannot proceed a step with two terms only and one proposition only. You must have two propositions including three terms; the middle term occupying the place assigned to it in one or other of the three figures.39This is obviously true when you demonstrate by direct or ostensive syllogism; and it is no less true when you proceed byReductio ad Impossibile. This last is one mode of syllogizing from an hypothesis or assumption:40your conclusion being disputed, you prove it indirectly, by assuming its contradictory to be true, and constructing a new syllogism by means of that contradictory together with a second premiss admitted to be true; the conclusion of this new syllogism being a proposition obviously false or known beforehand to be false. Your demonstration must be conducted by a regular syllogism, as it is when you proceed directly and ostensively. The difference is, that the conclusion which you obtain is not that which you wish ultimately to arrive at, but something notoriously false. But as this false conclusion arises from your assumption or hypothesis that the contradictory of the conclusion originally disputed was true, you have indirectly made out your case that this contradictory must have been false, and therefore that the conclusion originally disputed was true. All this, however, has been demonstration by regular syllogism, but starting from an hypothesis assumed and admitted as one of the premisses.41
39Ibid. xxiii. p. 40, b. 20, p. 41, a. 4-20.
39Ibid. xxiii. p. 40, b. 20, p. 41, a. 4-20.
40Ibid. p. 40, b. 25: ἔτι ἢ δεικτικῶς ἢ ἐξ ὑποθέσεως· τοῦ δ’ἐξ ὑποθέσεωςμέρος τὸ διὰ τοῦ ἀδυνάτου.
40Ibid. p. 40, b. 25: ἔτι ἢ δεικτικῶς ἢ ἐξ ὑποθέσεως· τοῦ δ’ἐξ ὑποθέσεωςμέρος τὸ διὰ τοῦ ἀδυνάτου.
41Ibid. p. 41, b. 23: πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δ’ ἐξ ἀρχῆςἐξ ὑποθέσεωςδεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνῃ τῆς ἀντιφάσεως τεθείσης.It deserves to be remarked that Aristotle uses the phrase συλλογισμὸςἐξ ὑποθέσεως, not συλλογισμὸς ὑποθετικός. This bears upon the question as to his views upon what subsequently received the title ofhypothetical syllogisms; a subject to which I shall advert in a futurenote.
41Ibid. p. 41, b. 23: πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δ’ ἐξ ἀρχῆςἐξ ὑποθέσεωςδεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνῃ τῆς ἀντιφάσεως τεθείσης.
It deserves to be remarked that Aristotle uses the phrase συλλογισμὸςἐξ ὑποθέσεως, not συλλογισμὸς ὑποθετικός. This bears upon the question as to his views upon what subsequently received the title ofhypothetical syllogisms; a subject to which I shall advert in a futurenote.
Aristotle here again enforces what he had before urged — that in every valid syllogism, one premiss at least must be affirmative, and one premiss at least must be universal. If the conclusion be universal, both premisses must be so likewise;if it be particular, one of the premisses may not be universal. But without one universal premiss at least, there can be no syllogistic proof. If you have a thesis to support, you cannot assume (or ask to be conceded to you) that very thesis, without committingpetitio principii,(i.e.quæsitiorprobandi); you must assume (or ask to have conceded to you) some universal proposition containing it and more besides; under which universal you may bring the subject of your thesis as a minor, and thus the premisses necessary for supporting it will be completed. Aristotle illustrates this by giving a demonstration that the angles at the base of an isosceles triangle are equal; justifying every step in the reasoning by an appeal to some universal proposition.42
42Analyt. Prior. I. xxiv. p. 41, b. 6-31. The demonstration given (b. 13-22) is different from that which we read in Euclid, and is not easy to follow. It is more clearly explained by Waitz (p. 434) than either by Julius Pacius or by M. Barth. St. Hilaire (p. 108).
42Analyt. Prior. I. xxiv. p. 41, b. 6-31. The demonstration given (b. 13-22) is different from that which we read in Euclid, and is not easy to follow. It is more clearly explained by Waitz (p. 434) than either by Julius Pacius or by M. Barth. St. Hilaire (p. 108).
Again, every demonstration is effected by two propositions (anevennumber) and by three terms (anoddnumber); though the same proposition may perhaps be demonstrable by more than one pair of premisses, or through more than one middle term;43that is, by two or more distinct syllogisms. If there be more than three terms and two propositions, either the syllogism will no longer be one but several; or there must be particulars introduced for the purpose of obtaining an universal by induction; or something will be included, superfluous and not essential to the demonstration, perhaps for the purpose of concealing from the respondent the real inference meant.44In the case (afterwards calledSorites) where the ultimate conclusion is obtained through several mean terms in continuous series, the number of terms will always exceed by one the number of propositions; but the numbers may be odd or even, according to circumstances. As terms are added, the total of intermediate conclusions, if drawn out in form, will come to be far greater than that of the terms or propositions, multiplying as it will do in an increasing ratio to them.45
43Ibid. I. xxv. p. 41, b. 36, seq.
43Ibid. I. xxv. p. 41, b. 36, seq.
44Ibid. xxv. p. 42, a. 23: μάτην ἔσται εἰλημμένα, εἰ μὴ ἐπαγωγῆς ἢ κρύψεως ἤ τινος ἄλλου τῶν τοιούτων χάριν. Ib. a. 38: οὗτος ὁ λόγος ἢ οὐ συλλελόγισται ἢ πλείω τῶν ἀναγκαίων ἠρώτηκε πρὸς τὴν θέσιν.
44Ibid. xxv. p. 42, a. 23: μάτην ἔσται εἰλημμένα, εἰ μὴ ἐπαγωγῆς ἢ κρύψεως ἤ τινος ἄλλου τῶν τοιούτων χάριν. Ib. a. 38: οὗτος ὁ λόγος ἢ οὐ συλλελόγισται ἢ πλείω τῶν ἀναγκαίων ἠρώτηκε πρὸς τὴν θέσιν.
45Ibid. p. 42, b. 5-26.
45Ibid. p. 42, b. 5-26.
It will be seen clearly from the foregoing remarks that there is a great difference between one thesis and another as to facility of attack or defence in Dialectic. If the thesis be an Universal Affirmative proposition, it can be demonstrated only in the First figure, and only by one combination of premisses; while, on theother hand, it can be impugned either by an universal negative, which can be demonstrated both in the First and Second figures, or by a particular negative, which can be demonstrated in all the three figures. Hence an Universal Affirmative thesis is at once the hardest to defend and the easiest to oppugn: more so than either a Particular Affirmative, which can be proved both in the First and Third figures; or a Universal Negative, which can be proved either in First or Second.46To the opponent, an universal thesis affords an easier victory than a particular thesis; in fact, speaking generally, his task is easier than that of the defendant.
46Analyt. Prior. I. xxvi. p. 42, b. 27, p. 43, a. 15.
46Analyt. Prior. I. xxvi. p. 42, b. 27, p. 43, a. 15.
In the Analytica Priora, Aristotle proceeds to tell us that he contemplates not only theory, but also practice and art. The reader must be taught, not merely to understand the principles of Syllogism, but likewise where he can find the matter for constructing syllogisms readily, and how he can obtain the principles of demonstration pertinent to each thesis propounded.47
47Ibid. I. xxvii. p. 43, a. 20: πῶς δ’ εὐπορήσομεν αὐτοὶ πρὸς τὸ τιθέμενον ἀεὶ συλλογισμῶν, καὶ διὰ ποίας ὁδοῦ ληψόμεθα τὰς περὶ ἕκαστον ἀρχάς, νῦν ἤδη λεκτέον· οὐ γὰρ μόνον ἴσως δεῖ τὴν γένεσιν θεωρεῖν τῶν συλλογισμῶν, ἀλλὰ καὶ τὴν δύναμιν ἔχειν τοῦ ποιεῖν. The second section of Book I. here begins.
47Ibid. I. xxvii. p. 43, a. 20: πῶς δ’ εὐπορήσομεν αὐτοὶ πρὸς τὸ τιθέμενον ἀεὶ συλλογισμῶν, καὶ διὰ ποίας ὁδοῦ ληψόμεθα τὰς περὶ ἕκαστον ἀρχάς, νῦν ἤδη λεκτέον· οὐ γὰρ μόνον ἴσως δεῖ τὴν γένεσιν θεωρεῖν τῶν συλλογισμῶν, ἀλλὰ καὶ τὴν δύναμιν ἔχειν τοῦ ποιεῖν. The second section of Book I. here begins.
A thesis being propounded in appropriate terms, with subject and predicate, how are you the propounder to seek out arguments for its defence? In the first place, Aristotle reverts to the distinction already laid down at the beginning of the Categoriæ.48Individual things or persons are subjects only, never appearing as predicates — this is the lowest extremity of the logical scale: at the opposite extremity of the scale, there are the highest generalities, predicates only, and not subjects of any predication, though sometimes supposed to be such, as matters of dialectic discussion.49Between the lowest and highest we have intermediate or graduate generalities, appearing sometimes as subjects, sometimes as predicates; and it is among these that the materials both of problems for debate, and of premisses for proof, are usually found.50
48Ibid. I. xxvii. p. 43, a. 25, seq.
48Ibid. I. xxvii. p. 43, a. 25, seq.
49Ibid. p. 43, a. 39: πλὴν εἰ μὴ κατὰ δόξαν. Cf. Schol. of Alexander, p. 175, a. 44, Br.: ἐνδόξως καὶ διαλεκτικῶς, ὥσπερ εἶπεν ἐν τοῖς Τοπικοῖς, that even theprincipiaof science may be debated; for example, in bookB. of the Metaphysica. Aristotle does not recognize either τὸ ὄν or τὸ ἕν as true genera, but only as predicates.
49Ibid. p. 43, a. 39: πλὴν εἰ μὴ κατὰ δόξαν. Cf. Schol. of Alexander, p. 175, a. 44, Br.: ἐνδόξως καὶ διαλεκτικῶς, ὥσπερ εἶπεν ἐν τοῖς Τοπικοῖς, that even theprincipiaof science may be debated; for example, in bookB. of the Metaphysica. Aristotle does not recognize either τὸ ὄν or τὸ ἕν as true genera, but only as predicates.
50Ibid. a. 40-43.
50Ibid. a. 40-43.
You must begin by putting down, along with the matter in hand itself, its definition and itspropria; after that, its other predicates; next, those predicates whichcannotbelong to it;lastly, those other subjects, of which it may itself be predicated. You must classify its various predicates distinguishing the essential, thepropria, and the accidental; also distinguishing the true and unquestionable, from the problematical and hypothetical.51You must look out for those predicates which belong to it as subject universally, and not to certain portions of it only; since universal propositions are indispensable in syllogistic proof, and indefinite propositions can only be reckoned as particular. When a subject is included in some larger genus — as, for example, man in animal — you must not look for the affirmative or negative predicates which belong to animal universally (since all these will of course belong to man also) but for those which distinguish man from other animals; nor must you, in searching for those lower subjects of which man is the predicate, fix your attention on the higher genus animal; for animal will of course be predicable of all those of which man is predicable. You must collect what pertains to man specially, either as predicate or subject; nor merely that which pertains to him necessarily and universally, but also usually and in the majority of cases; for most of the problems debated belong to this latter class, and the worth of the conclusion will be co-ordinate with that of the premisses.52Do not select predicates that are predicable53both of the predicate and subject; for no valid affirmative conclusion can be obtained from them.