46Analyt. Post. I. xvi. p. 79, b. 23: ἄγνοια κατ’ ἀπόφασιν — ἄγνοια κατὰ διάθεσιν. See Themistius, p. 49, Spengel. In regard to simple and uncombined ideas, ignorance is not possible as an erroneous combination, but only as a mental blank. You either have the idea and thus know so much truth, or you have not the idea and are thus ignorant to that extent; this is the only alternative. Cf. Aristot. Metaph.Θ. p. 1051, a. 34; De Animâ, III. vi. p. 430, a. 26.
46Analyt. Post. I. xvi. p. 79, b. 23: ἄγνοια κατ’ ἀπόφασιν — ἄγνοια κατὰ διάθεσιν. See Themistius, p. 49, Spengel. In regard to simple and uncombined ideas, ignorance is not possible as an erroneous combination, but only as a mental blank. You either have the idea and thus know so much truth, or you have not the idea and are thus ignorant to that extent; this is the only alternative. Cf. Aristot. Metaph.Θ. p. 1051, a. 34; De Animâ, III. vi. p. 430, a. 26.
47Analyt. Post. I. xvi. p. 79, b. 29. M. Barthélemy St. Hilaire remarks (p. 95, n.):— “Il faut remarquer qu’Aristote ne s’occupe que des modes universels dans la première et dans la seconde figure, parceque, la démonstration étant toujours universelle, les propositions qui expriment l’erreur opposée doivent l’être comme elle. Ainsi ce sont les propositions contraires, et non les contradictoires, dont il sera question ici.�For the like reason the Third figure is not mentioned here, but only the First and Second: because in the Third figure no universal conclusion can be proved (Julius Pacius, p. 431).
47Analyt. Post. I. xvi. p. 79, b. 29. M. Barthélemy St. Hilaire remarks (p. 95, n.):— “Il faut remarquer qu’Aristote ne s’occupe que des modes universels dans la première et dans la seconde figure, parceque, la démonstration étant toujours universelle, les propositions qui expriment l’erreur opposée doivent l’être comme elle. Ainsi ce sont les propositions contraires, et non les contradictoires, dont il sera question ici.�
For the like reason the Third figure is not mentioned here, but only the First and Second: because in the Third figure no universal conclusion can be proved (Julius Pacius, p. 431).
48Analyt. Post. I. xvi. p. 80, a. 6-26.
48Analyt. Post. I. xvi. p. 80, a. 6-26.
49Ibid. a. 27-b. 14: ἐν δὲ τῷ μέσῳ σχήματι ὅλας μὲν εἶναι τὰς προτάσεις ἀμφοτέρας ψευδεῖς οὐκ ἐνδέχεται — ἐπί τι δ’ ἑκατέραν οὐδὲν κωλύει ψευδῆ εἶναι.
49Ibid. a. 27-b. 14: ἐν δὲ τῷ μέσῳ σχήματι ὅλας μὲν εἶναι τὰς προτάσεις ἀμφοτέρας ψευδεῖς οὐκ ἐνδέχεται — ἐπί τι δ’ ἑκατέραν οὐδὲν κωλύει ψευδῆ εἶναι.
Let us next assume the affirmative proposition, All B is A, to be true, but mediate and deducible through the middle term C. If you conclude the contrary of this (No B is A) through the same middle term C, in the First figure, your error cannot arise from falsity in the minor premiss, because your minor (by the laws of the figure) must be affirmative; your error must arise from a false major, because a negative major is not inconsistent with the laws of the First figure. On the other hand, if you conclude the contrary in the First figure through a differentmiddle term, D, either both your premisses will be false, or your minor premiss will be false.50If you employ the Second figure to conclude your contrary, both your premisses cannot be false, though either one of them singly may be false.51
50Analyt. Post. I. xvi. p. 80, b. 17-p. 81, a. 4.
50Analyt. Post. I. xvi. p. 80, b. 17-p. 81, a. 4.
51Ibid. p. 81, a. 5-14.
51Ibid. p. 81, a. 5-14.
Such will be the case when the deducible proposition assumed to be true is affirmative, and when therefore the contrary conclusion which you profess to have proved is negative. But if the deducible proposition assumed to be true is negative, and if consequently the contrary conclusion must be affirmative, — then, if you try to prove this contrary through the same middle term, your premisses cannot both be false, but your major premiss must always be false.52If, however, you try to prove the contrary through a different and inappropriate middle term, you cannot convert the minor premiss to its contrary (because the minor premiss must continue affirmative, in order that you may arrive at any conclusion at all), but the major can be so converted. Should the major premiss thus converted be true, the minor will be false; should the major premiss thus converted be false, the minor may be either true or false. Either one of the premisses, or both the premisses, may thus be false.53
52Ibid. xvii. p. 81, a. 15-20.
52Ibid. xvii. p. 81, a. 15-20.
53Ibid. a. 20-34. Mr. Poste’s translation (pp. 65-70) is very perspicuous and instructive in regard to these two difficult chapters.
53Ibid. a. 20-34. Mr. Poste’s translation (pp. 65-70) is very perspicuous and instructive in regard to these two difficult chapters.
Errors of simple ignorance (not concluded from false syllogism) may proceed from defect or failure of sensible perception, in one or other of its branches. For without sensation there can be no induction; and it is from induction only that the premisses for demonstration by syllogism are obtained. We cannot arrive at universal propositions, even in what are called abstract sciences, except through induction of particulars; nor can we demonstrate except from universals. Induction and Demonstration are the only two ways of learning; and the particulars composing our inductions can only be known through sense.54
54Analyt. Post. I. xviii. p. 81, a. 38-b. 9. In this important chapter (the doctrines of which are more fully expanded in the last chapter of the Second Book of the Analyt. Post.), the text of Waitz does not fully agree with that of Julius Pacius. In Firmin Didot’s edition the text is the same as in Waitz; but his Latin translation remains adapted to that of Julius Pacius. Waitz gives the substance of the chapter as follows (ad Organ. II. p. 347):— “Universales propositiones omnes inductione comparantur, quum etiam in iis, quæ a sensibus maxime aliena videntur et quæ, ut mathematica (τὰ ἐξ ἀφαιρέσεως), cogitatione separantur à materia quacum conjuncta sunt, inductione probentur ea quæ de genero (e.g., de linea vel de corpore mathematico), ad quod demonstratio pertineat, prædicentur καθ’ αὑτά et cum ejus natura conjuncta sint. Inductio autem iis nititur quæ sensibus percipiuntur; nam res singulares sentiuntur, scientia vero rerum singularium non datur sine inductione, non datur inductio sine sensu.â€�
54Analyt. Post. I. xviii. p. 81, a. 38-b. 9. In this important chapter (the doctrines of which are more fully expanded in the last chapter of the Second Book of the Analyt. Post.), the text of Waitz does not fully agree with that of Julius Pacius. In Firmin Didot’s edition the text is the same as in Waitz; but his Latin translation remains adapted to that of Julius Pacius. Waitz gives the substance of the chapter as follows (ad Organ. II. p. 347):— “Universales propositiones omnes inductione comparantur, quum etiam in iis, quæ a sensibus maxime aliena videntur et quæ, ut mathematica (τὰ ἐξ ἀφαιρέσεως), cogitatione separantur à materia quacum conjuncta sunt, inductione probentur ea quæ de genero (e.g., de linea vel de corpore mathematico), ad quod demonstratio pertineat, prædicentur καθ’ αὑτά et cum ejus natura conjuncta sint. Inductio autem iis nititur quæ sensibus percipiuntur; nam res singulares sentiuntur, scientia vero rerum singularium non datur sine inductione, non datur inductio sine sensu.â€�
Aristotle next proceeds to show (what in previous passages hehad assumed)55that, if Demonstration or the syllogistic process be possible — if there be any truths supposed demonstrable, this implies that there must be primary or ultimate truths. It has been explained that the constituent elements assumed in the Syllogism are three terms and two propositions or premisses; in the major premiss, A is affirmed (or denied) of all B; in the minor, B is affirmed of all C; in the conclusion, A is affirmed (or denied) of all C.56Now it is possible that there may be some one or more predicates higher than A, but it is impossible that there can be an infinite series of such higher predicates. So also there may be one or more subjects lower than C, and of which C will be the predicate; but it is impossible that there can be an infinite series of such lower subjects. In like manner there may perhaps be one or more middle terms between A and B, and between B and C; but it is impossible that there can be an infinite series of such intervening middle terms. There must be a limit to the series ascending, descending, or intervening.57These remarks have no application to reciprocating propositions, in which the predicate is co-extensive with the subject.58But they apply alike to demonstrations negative and affirmative, and alike to all the three figures of Syllogism.59
55Analyt. Prior. I. xxvii. p. 43, a. 38; Analyt. Post. I. ii. p. 71, b. 21.
55Analyt. Prior. I. xxvii. p. 43, a. 38; Analyt. Post. I. ii. p. 71, b. 21.
56Analyt. Post. I. xix. p. 81, b. 10-17.
56Analyt. Post. I. xix. p. 81, b. 10-17.
57Ibid. p. 81, b. 30-p. 82,a. 14.
57Ibid. p. 81, b. 30-p. 82,a. 14.
58Ibid. p. 82, a. 15-20. M. Barthélemy St. Hilaire, p. 117:— “Ceci ne saurait s’appliquer aux termes réciproques, parce que dans les termes qui peuvent être attribués réciproquement l’un à l’autre, on ne peut pas dire qu’il y ait ni premier ni dernier rélativement à l’attribution.�
58Ibid. p. 82, a. 15-20. M. Barthélemy St. Hilaire, p. 117:— “Ceci ne saurait s’appliquer aux termes réciproques, parce que dans les termes qui peuvent être attribués réciproquement l’un à l’autre, on ne peut pas dire qu’il y ait ni premier ni dernier rélativement à l’attribution.�
59Analyt. Post. I. xx., xxi. p. 82, a. 21-b. 36.
59Analyt. Post. I. xx., xxi. p. 82, a. 21-b. 36.
In Dialectical Syllogism it is enough if the premisses be admitted or reputed as propositions immediately true, whether they are so in reality or not; but in Scientific or Demonstrative Syllogism they must be so in reality: the demonstration is not complete unless it can be traced up to premisses that are thus immediately or directly true (without any intervening middle term).60That there are and must be such primary or immediate premisses, Aristotle now undertakes to prove, by some dialectical reasons, and other analytical or scientific reasons.61He himselfthus distinguishes them; but the distinction is faintly marked, and amounts, at most, to this, that the analytical reasons advert only to essential predication, and to the conditions of scientific demonstration, while the dialectical reasons dwell upon these, but include something else besides, viz., accidental predication. The proof consists mainly in the declaration that, unless we assume some propositions to be true immediately, indivisibly, undemonstrably, — Definition, Demonstration, and Science would be alike impossible. If the ascending series of predicates is endless, so that we never arrive at a highest generic predicate; if the descending series of subjects is endless, so that we never reach a lowest subject, — no definition can ever be attained. The essential properties included in the definition, must be finite in number; and the accidental predicates must also be finite in number, since they have no existence except as attached to some essential subject, and since they must come under one or other of the nine later Categories.62If, then, the two extremes are thus fixed and finite — the highest predicate and the lowest subject — it is impossible that there can be an infinite series of terms between the two. The intervening terms must be finite in number. The Aristotelian theory therefore is, that there are certain propositions directly and immediately true, and others derived from them by demonstration through middle terms.63It is alike an error to assert that every thing can be demonstrated, and that nothing can be demonstrated.
60Ibid. xix. p. 81, b. 18-29.
60Ibid. xix. p. 81, b. 18-29.
61Ibid. xxi. p. 82, b. 35; xxii. p. 84, a. 7:λογικῶςμὲν οὖν ἐκ τούτων ἄν τις πιστεύσειε περὶ τοῦ λεχθέντος,ἀναλυτικῶςδὲ διὰ τῶνδε φανερὸν συντομώτερον. In Scholia, p. 227, a. 42, the same distinction is expressed by Philoponus in the terms λογικώτερα and πραγματωδέστερα. Compare Biese, Die Philosophie des Aristoteles, pp. 134, 261; Bassow, De Notionis Definitione, pp. 19, 20; Heyder, Aristot. u. Hegel. Dialektik, pp. 316, 317.Aristotle, however, does not always adhere closely to the distinction. Thus, if we compare thelogicalordialecticalreasons given, p. 82, b. 37, seq., with theanalytical, announced as beginning p. 84, a. 8, seq., we find the same main topic dwelt upon in both, namely, that to admit an infinite series excludes the possibility of Definition. Both Alexander and Ammonius agree in announcing this as the capital topic on which the proof turned; but Alexander inferred from hence that the argument was purelydialectical(λογικὸν ἐπιχείρημα), while Ammonius regarded it as a reason thoroughly convincing and evident: ὁ μέντοι φιλόσοφος (Ammonius) ἔλεγε μὴ διὰ τοῦτο λέγεινλογικὰτὰ ἐπιχειρήματα· ἐναργὲς γὰρ ὅτι εἰσὶν ὁρισμοί, εἰ μὴ ἀκαταληψίαν εἰσαγάγωμεν (Schol. p. 227, a. 40, seq., Brand.).
61Ibid. xxi. p. 82, b. 35; xxii. p. 84, a. 7:λογικῶςμὲν οὖν ἐκ τούτων ἄν τις πιστεύσειε περὶ τοῦ λεχθέντος,ἀναλυτικῶςδὲ διὰ τῶνδε φανερὸν συντομώτερον. In Scholia, p. 227, a. 42, the same distinction is expressed by Philoponus in the terms λογικώτερα and πραγματωδέστερα. Compare Biese, Die Philosophie des Aristoteles, pp. 134, 261; Bassow, De Notionis Definitione, pp. 19, 20; Heyder, Aristot. u. Hegel. Dialektik, pp. 316, 317.
Aristotle, however, does not always adhere closely to the distinction. Thus, if we compare thelogicalordialecticalreasons given, p. 82, b. 37, seq., with theanalytical, announced as beginning p. 84, a. 8, seq., we find the same main topic dwelt upon in both, namely, that to admit an infinite series excludes the possibility of Definition. Both Alexander and Ammonius agree in announcing this as the capital topic on which the proof turned; but Alexander inferred from hence that the argument was purelydialectical(λογικὸν ἐπιχείρημα), while Ammonius regarded it as a reason thoroughly convincing and evident: ὁ μέντοι φιλόσοφος (Ammonius) ἔλεγε μὴ διὰ τοῦτο λέγεινλογικὰτὰ ἐπιχειρήματα· ἐναργὲς γὰρ ὅτι εἰσὶν ὁρισμοί, εἰ μὴ ἀκαταληψίαν εἰσαγάγωμεν (Schol. p. 227, a. 40, seq., Brand.).
62Analyt. Post. I. xxii. p. 83, a. 20, b. 14. Only eight of the ten Categories are here enumerated.
62Analyt. Post. I. xxii. p. 83, a. 20, b. 14. Only eight of the ten Categories are here enumerated.
63Ibid. I. xxii. p. 84, a. 30-35. The paraphrase of Themistius (pp. 55-58, Spengel) states the Aristotelian reasoning in clearer language than Aristotle himself. Zabarella (Comm. in Analyt. Post. I. xviii.; context. 148, 150, 154) repeats that Aristotle’s proof is founded upon the undeniable fact that therearedefinitions, and that without them there could be no demonstration and no science. This excludes the supposition of an infinite series of predicates and of middle terms:— “Sumit rationem à definitione; si inpredicatis in quidprocederetur ad infinitum, sequeretur auferri definitionem et omnino essentiæ cognitionem; sed hoc dicendum non est, quum omnium consensioni adversetur� (p. 466, Ven. 1617).
63Ibid. I. xxii. p. 84, a. 30-35. The paraphrase of Themistius (pp. 55-58, Spengel) states the Aristotelian reasoning in clearer language than Aristotle himself. Zabarella (Comm. in Analyt. Post. I. xviii.; context. 148, 150, 154) repeats that Aristotle’s proof is founded upon the undeniable fact that therearedefinitions, and that without them there could be no demonstration and no science. This excludes the supposition of an infinite series of predicates and of middle terms:— “Sumit rationem à definitione; si inpredicatis in quidprocederetur ad infinitum, sequeretur auferri definitionem et omnino essentiæ cognitionem; sed hoc dicendum non est, quum omnium consensioni adversetur� (p. 466, Ven. 1617).
It is plain from Aristotle’s own words64that he intended these four chapters (xix.-xxii.) as a confirmation of what he had already asserted in chapter iii. of the present treatise, and as farther refutation of the two distinct classes of opponents there indicated: (1) those who said that everything was demonstrable, demonstration in a circle being admissible; (2) those who said that nothing was demonstrable, inasmuch as the train of predicationupwards, downwards, and intermediate, was infinite. Both these two classes of opponents agreed in saying, that there were no truths immediate and indemonstrable; and it is upon this point that Aristotle here takes issue with them, seeking to prove that there are and must be such truths. But I cannot think the proof satisfactory; nor has it appeared so to able commentators either of ancient or modern times — from Alexander of Aphrodisias down to Mr. Poste.65The elaborate amplificationadded in these last chapters adds no force to the statement already given at the earlier stage; and it is in one respect a change for the worse, inasmuch as it does not advert to the important distinction announced in chapter iii., between universal truths known by Induction (from sense and particulars), and universal truths known by Deduction from these. The truths immediate and indemonstrable (not known through a middle term) are the inductive truths, as Aristotle declares in many places, and most emphatically at the close of the Analytica Posteriora. But in these chapters, he hardly alludes to Induction. Moreover, while trying to prove that there must be immediate universal truths, he neither gives any complete list of them, nor assigns any positive characteristic whereby to identify them. Opponents might ask him whether these immediate universal truths were not ready-made inspirations of the mind; and if so, what better authority they had than the Platonic Ideas, which are contemptuously dismissed.
64Analyt. Post. I. xxii. p. 84, a. 32: ὅπερ ἔφαμέν τινας λέγειν κατ’ ἀρχάς, &c.
64Analyt. Post. I. xxii. p. 84, a. 32: ὅπερ ἔφαμέν τινας λέγειν κατ’ ἀρχάς, &c.
65See Mr. Poste’s note, p. 77, of his translation of this treatise. After saying that the first of Aristotle’sdialecticalproofs is faulty, and that the second is apetitio principii, Mr. Poste adds, respecting the so-calledanalyticalproof given by Aristotle:— “It is not so much a proof, as a more accurate determination of the principle to be postulated. This postulate, the existence of first principles, as concerning the constitution of the world, appears to belong properly to Metaphysics, and is merely borrowed by Logic. See Metaph. ii. 2, and Introduction.â€� In the passage of the Metaphysica (α. p. 994) here cited the main argument of Aristotle is open to the same objection ofpetitio principiiwhich Mr. Poste urges against Aristotle’s seconddialecticalargument in this place.Mr. John Stuart Mill, in his System of Logic, takes for granted that theremustbe immediate, indemonstrable truths, to serve as a basis for deduction; “that there cannot be a chain of proof suspended from nothing;â€� that there must be ultimate laws of nature, though we cannot be sure that the laws now known to us are ultimate.On the other hand, we read in the recent work of an acute contemporary philosopher, Professor DelbÅ“uf (Essai de Logique Scientifique, Liège, 1865, Pref. pp. v, vii, viii, pp. 46, 47:) — “Il est des points sur lesquels je crains de ne m’être pas expliqué assez nettement, entre autres la question du fondement de la certitude. Je suis de ceux qui repoussent de toutes leurs forces l’axiome si spécieux qu’on ne peut tout démontrer; cette proposition aurait, à mes yeux, plus besoin que toute autre d’une démonstration. Cette démonstration ne sera en partie donnée que quand on aura une bonne fois énuméré toutes les propositions indémontrables; et quand on aura bien défini le caractère auquel on les reconnait. Nulle part on ne trouve ni une semblable énumération, ni une semblable définition. On reste à cet égard dans une position vague, et par cela même facile à défendre.â€�It would seem, by these words, that M. DelbÅ“uf stands in the most direct opposition to Aristotle, who teaches us that the ἀρχαὶ orprincipiafrom which demonstration starts cannot be themselves demonstrated. But when we compare other passages of M. DelbÅ“uf’s work, we find that, in rejecting all undemonstrable propositions, what he really means is to reject allself-evident universal truths, “C’est donc une véritable illusion d’admettre des vérités évidentes par elles-mêmes. Il n’y a pas de proposition fausse que nous ne soyons disposés d’admettre comme axiome, quand rien ne nous a encore autorisés à la repousserâ€� (p. ix.). This is quite true in my opinion; but the immediate indemonstrable truths for which Aristotle contends as ἀρχαὶ of demonstration, are not announced by him asself-evident, they are declared to be results of sense and induction, to be raised from observation of particulars multiplied, compared, and permanently formularized under the intellectualhabituscalled Noûs. By Demonstration Aristotle means deduction in its most perfect form, beginning from these ἀρχαὶ which are inductively known but not demonstrable (i. e.not knowable deductively). And in this view the very able and instructive treatise of M. DelbÅ“uf mainly coincides, assigning even greater preponderance to the inductive process, and approximating in this respect to the important improvements in logical theory advanced by Mr. John Stuart Mill.Among the universal propositions which are not derived from Induction, but which serve as ἀρχαὶ for Deduction and Demonstration, we may reckon the religious, ethical, æsthetical, social, political, &c., beliefs received in each different community, and impressed upon all newcomers born into it by the force of precept, example, authority. Here the major premiss is felt by each individual as carrying an authority of its own, stamped and enforced by the sanction of society, and by the disgrace or other penalties in store for those who disobey it. It is ready to be interpreted and diversified by suitable minor premisses in all inferential applications. But these ἀρχαὶ for deduction, differing widely at different times and places, though generated in the same manner and enforced by the same sanction, would belong more properly to the class which Aristotle terms τὰ ἔνδοξα.
65See Mr. Poste’s note, p. 77, of his translation of this treatise. After saying that the first of Aristotle’sdialecticalproofs is faulty, and that the second is apetitio principii, Mr. Poste adds, respecting the so-calledanalyticalproof given by Aristotle:— “It is not so much a proof, as a more accurate determination of the principle to be postulated. This postulate, the existence of first principles, as concerning the constitution of the world, appears to belong properly to Metaphysics, and is merely borrowed by Logic. See Metaph. ii. 2, and Introduction.â€� In the passage of the Metaphysica (α. p. 994) here cited the main argument of Aristotle is open to the same objection ofpetitio principiiwhich Mr. Poste urges against Aristotle’s seconddialecticalargument in this place.
Mr. John Stuart Mill, in his System of Logic, takes for granted that theremustbe immediate, indemonstrable truths, to serve as a basis for deduction; “that there cannot be a chain of proof suspended from nothing;� that there must be ultimate laws of nature, though we cannot be sure that the laws now known to us are ultimate.
On the other hand, we read in the recent work of an acute contemporary philosopher, Professor Delbœuf (Essai de Logique Scientifique, Liège, 1865, Pref. pp. v, vii, viii, pp. 46, 47:) — “Il est des points sur lesquels je crains de ne m’être pas expliqué assez nettement, entre autres la question du fondement de la certitude. Je suis de ceux qui repoussent de toutes leurs forces l’axiome si spécieux qu’on ne peut tout démontrer; cette proposition aurait, à mes yeux, plus besoin que toute autre d’une démonstration. Cette démonstration ne sera en partie donnée que quand on aura une bonne fois énuméré toutes les propositions indémontrables; et quand on aura bien défini le caractère auquel on les reconnait. Nulle part on ne trouve ni une semblable énumération, ni une semblable définition. On reste à cet égard dans une position vague, et par cela même facile à défendre.�
It would seem, by these words, that M. DelbÅ“uf stands in the most direct opposition to Aristotle, who teaches us that the ἀρχαὶ orprincipiafrom which demonstration starts cannot be themselves demonstrated. But when we compare other passages of M. DelbÅ“uf’s work, we find that, in rejecting all undemonstrable propositions, what he really means is to reject allself-evident universal truths, “C’est donc une véritable illusion d’admettre des vérités évidentes par elles-mêmes. Il n’y a pas de proposition fausse que nous ne soyons disposés d’admettre comme axiome, quand rien ne nous a encore autorisés à la repousserâ€� (p. ix.). This is quite true in my opinion; but the immediate indemonstrable truths for which Aristotle contends as ἀρχαὶ of demonstration, are not announced by him asself-evident, they are declared to be results of sense and induction, to be raised from observation of particulars multiplied, compared, and permanently formularized under the intellectualhabituscalled Noûs. By Demonstration Aristotle means deduction in its most perfect form, beginning from these ἀρχαὶ which are inductively known but not demonstrable (i. e.not knowable deductively). And in this view the very able and instructive treatise of M. DelbÅ“uf mainly coincides, assigning even greater preponderance to the inductive process, and approximating in this respect to the important improvements in logical theory advanced by Mr. John Stuart Mill.
Among the universal propositions which are not derived from Induction, but which serve as ἀρχαὶ for Deduction and Demonstration, we may reckon the religious, ethical, æsthetical, social, political, &c., beliefs received in each different community, and impressed upon all newcomers born into it by the force of precept, example, authority. Here the major premiss is felt by each individual as carrying an authority of its own, stamped and enforced by the sanction of society, and by the disgrace or other penalties in store for those who disobey it. It is ready to be interpreted and diversified by suitable minor premisses in all inferential applications. But these ἀρχαὶ for deduction, differing widely at different times and places, though generated in the same manner and enforced by the same sanction, would belong more properly to the class which Aristotle terms τὰ ἔνδοξα.
We have thus recognized that there exist immediate (ultimate or primary) propositions, wherein the conjunction between predicate and subject is such that no intermediate term can be assigned between them. When A is predicated both of B and C, this may perhaps be in consequence of some common property possessed by B and C, and such common property will form a middle term. For example, equality of angles to two right angles belongs both to an isosceles and to a scalene triangle, and it belongs to them by reason of their common property — triangular figure; which last is thus the middle term. But this need not be always the case.66It is possible that the two propositions — A predicated of B, A predicated of C — may both of them be immediate propositions; and that there may be no community of nature between B and C. Whenever a middle term can be found, demonstration is possible; but where no middle term can be found, demonstration is impossible. The proposition, whether affirmative or negative, is then an immediate or indivisible one. Such propositions, and the terms of which they are composed, are the ultimate elements orprincipiaof Demonstration. Predicate and subject are brought constantly into closer and closer conjunction, until at last they become one and indivisible.67Here we reach the unit or elementof the syllogizing process. In all scientific calculations there is assumed an unit to start from, though in each branch of science it is a different unit;e.g.in barology, the pound-weight; in harmonics, the quarter-tone; in other branches of science, other units.68Analytical research teaches us that the corresponding unit in Syllogism is the affirmative or negative proposition which is primary, immediate, indivisible. In Demonstration and Science it is the Noûs or Intellect.69
66Analyt. Post. I. xxiii. p. 84, b. 3-18. τοῦτο δ’ οὐκ ἀεὶ οὕτως ἔχει.
66Analyt. Post. I. xxiii. p. 84, b. 3-18. τοῦτο δ’ οὐκ ἀεὶ οὕτως ἔχει.
67Ibid. b. 25-37. ἀεὶ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γένηται καὶ ἕν. ἔστι δ’ ἕν, ὅταν ἄμεσον γένηται καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος.
67Ibid. b. 25-37. ἀεὶ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γένηται καὶ ἕν. ἔστι δ’ ἕν, ὅταν ἄμεσον γένηται καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος.
68Analyt. Post. I. xxiii. p. 84, b. 37: καὶ ὥσπερ ἐν τοῖς ἄλλοις ἡ ἀρχὴ ἁπλοῦν, τοῦτο δ’ οὐ ταὐτὸ πανταχοῦ, ἀλλ’ ἐν βαρεῖ μὲν μνᾶ, ἐν δὲ μέλει δίεσις, ἄλλο δ’ ἐν ἄλλῳ, οὕτως ἐν συλλογισμῷ τὸ ἓν πρότασις ἄμεσος, ἐν δ’ ἀποδείξει καὶ ἐπιστήμῃ ὁ νοῦς.
68Analyt. Post. I. xxiii. p. 84, b. 37: καὶ ὥσπερ ἐν τοῖς ἄλλοις ἡ ἀρχὴ ἁπλοῦν, τοῦτο δ’ οὐ ταὐτὸ πανταχοῦ, ἀλλ’ ἐν βαρεῖ μὲν μνᾶ, ἐν δὲ μέλει δίεσις, ἄλλο δ’ ἐν ἄλλῳ, οὕτως ἐν συλλογισμῷ τὸ ἓν πρότασις ἄμεσος, ἐν δ’ ἀποδείξει καὶ ἐπιστήμῃ ὁ νοῦς.
69Ibid. b. 35-p. 85, a. 1.
69Ibid. b. 35-p. 85, a. 1.
Having thus, in the long preceding reasoning, sought to prove that all demonstration must take its departure from primary undemonstrableprincipia— from some premisses, affirmative and negative, which are directly true in themselves, and not demonstrable through any middle term or intervening propositions, Aristotle now passes to a different enquiry. We have some demonstrations in which the conclusion is Particular, others in which it is Universal: again, some Affirmative, some Negative, Which of the two, in each of these alternatives, is the best? We have also demonstrations Direct or Ostensive, and demonstrations Indirect or by way ofReductio ad Absurdum. Which of these two is the best? Both questions appear to have been subjected to debate by contemporary philosophers.70
70Ibid. xxiv. p. 85, a. 13-18. ἀμφισβητεῖται ποτέρα βελτίων· ὡς δ’ αὕτως καὶ περὶ τῆς ἀποδεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης ἀποδείξεως.
70Ibid. xxiv. p. 85, a. 13-18. ἀμφισβητεῖται ποτέρα βελτίων· ὡς δ’ αὕτως καὶ περὶ τῆς ἀποδεικνύναι λεγομένης καὶ τῆς εἰς τὸ ἀδύνατον ἀγούσης ἀποδείξεως.
Aristotle discusses these points dialectically (as indeed he points out in the Topica that the comparison of two things generally, as to better and worse, falls under the varieties ofdialecticalenquiry71), first stating and next refuting the arguments on the weaker side. Some persons may think (he says) that demonstration of the Particular is better than demonstration of the Universal: first, because it conducts to fuller cognition of that which the thing is in itself, and not merely that which it isquatenusmember of a class; secondly, because demonstrations of the Universal are apt to generate an illusory belief, that the Universal is a distinct reality apart from and independent of all its particulars (i.e., that figure in general has a real existence apart from all particular figures, and number in general apart from all particular numbers, &c.), while demonstrations of the Particular do not lead to any such illusion.72
71Aristot. Topic. III. i. p. 116, a. 1, seq.
71Aristot. Topic. III. i. p. 116, a. 1, seq.
72Analyt. Post. I. xxiv. p. 85, a. 20-b. 3. Themistius, pp. 58-59, Spengel: οὐ γὰρ ὁμώνυμον τὸ καθόλου ἐστίν, οὐδὲ φωνὴ μόνον, ἀλλ’ ὑπόστασις, οὐ χωριστὴ μὲν ὥσπερ οὐδὲ τὰ συμβεβηκότα, ἐναργῶς δ’ οὖν ἐμφαινομένη τοῖς πράγμασιν. The Scholastic doctrine ofUniversalia in reis here expressed very clearly by Themistius.
72Analyt. Post. I. xxiv. p. 85, a. 20-b. 3. Themistius, pp. 58-59, Spengel: οὐ γὰρ ὁμώνυμον τὸ καθόλου ἐστίν, οὐδὲ φωνὴ μόνον, ἀλλ’ ὑπόστασις, οὐ χωριστὴ μὲν ὥσπερ οὐδὲ τὰ συμβεβηκότα, ἐναργῶς δ’ οὖν ἐμφαινομένη τοῖς πράγμασιν. The Scholastic doctrine ofUniversalia in reis here expressed very clearly by Themistius.
To these arguments Aristotle replies:— 1. It is not correct to say that cognition of the Particular is more complete, or bears more upon real existence, than cognition of the Universal. The reverse would be nearer to the truth. To know that the isosceles,quatenustriangle, has its three angles equal to two right angles, is more complete cognition than knowing simply that the isosceles has its three angles equal to two right angles. 2. If the Universal be not an equivocal term — if it represents one property and one definition common to many particulars, it then has a real existence as much or more than any one or any number of the particulars. For all these particulars are perishable, but the class is imperishable. 3. He who believes that the universal term has one meaning in all the particulars, need not necessarily believe that it has any meaningapartfrom all particulars; he need not believe this about Quiddity, any more than he believes it about Quality or Quantity. Or if he does believe so, it is his own individual mistake, not imputable to the demonstration. 4. We have shown that a complete demonstration is one in which the middle term is the cause or reason of the conclusion. Now the Universal is most of the nature of Cause; for it represents the First Essence or thePer Se, and is therefore its own cause, or has no other cause behind it. The demonstration of the Universal has thus more of the Cause or theWhy, and is therefore better than the demonstration of the Particular. 5. In the Final Cause or End of action, there is always some ultimate end for the sake of which the intermediate ends are pursued, and which, as it is better than they, yields, when it is known, the only complete explanation of the action. So it is also with the Formal Cause: there is one highest form which contains theWhyof the subordinate forms, and the knowledge of which is therefore better; as when, for example, the exterior angles of a given isosceles triangle are seen to be equal to four right angles, not because it is isosceles or triangle, but because it is a rectilineal figure. 6. Particulars, as such, fall into infinity of number, and are thus unknowable; the Universal tends towards oneness and simplicity, and is thus essentially knowable, more fully demonstrable than the infinity of particulars. The demonstration thereof is therefore better. 7. It is also better, on another ground; for he that knows the Universal does in a certain sense know also the Particular;73but he that knows the Particular cannot be said in any sense toknow the Universal. 8. Theprincipiumor perfection of cognition is to be found in the immediate proposition, trueper se. When we demonstrate, and thus employ a middle term, the nearer the middle term approaches to thatprincipium, the better the demonstration is. The demonstration of the Universal is thus better and more accurate than that of the Particular.74
73Compare Analyt. Post. I. i. p. 71, a. 25; also Metaphys.A. p. 981, a. 12.
73Compare Analyt. Post. I. i. p. 71, a. 25; also Metaphys.A. p. 981, a. 12.
74Analyt. Post. I. xxiv. p. 85, b. 4-p. 86, a. 21. Schol. p. 233, b. 6: ὁμοίως δὲ ὄντων γνωρίμων, ἡ δι’ ἐλαττόνων μέσων αἱρετωτέρα· μᾶλλον γὰρ ἐγγυτέρω τῆς τοῦ νοῦ ἐνεργείας.
74Analyt. Post. I. xxiv. p. 85, b. 4-p. 86, a. 21. Schol. p. 233, b. 6: ὁμοίως δὲ ὄντων γνωρίμων, ἡ δι’ ἐλαττόνων μέσων αἱρετωτέρα· μᾶλλον γὰρ ἐγγυτέρω τῆς τοῦ νοῦ ἐνεργείας.
Such are the several reasons enumerated by Aristotle in refutation of the previous opinion stated in favour of the Particular. Evidently he does not account them all of equal value: he intimates that some are purely dialectical (λογικά); and he insists most upon the two following:— 1. He that knows the Universal knows in a certain sense the Particular; if he knows that every triangle has its three angles equal to two right angles, he knows potentially that the isosceles has its three angles equal to the same, though he may not know as yet that the isoscelesisa triangle. But he that knows the Particular does not in any way know the Universal, either actually or potentially.752. The Universal is apprehended by Intellect or Noûs, the highest of all cognitive powers; the Particular terminates in sensation. Here, I presume, he means, that, in demonstration of the Particular, the conclusion teaches you nothing more than you might have learnt from a direct observation of sense; whereas in that of the Universal the conclusion teaches you more than you could have learnt from direct sensation, and comes into correlation with the highest form of our intellectual nature.76
75Analyt. Post. I. xxiv. p. 86,a. 22: ἀλλὰ τῶν μὲν εἰρημένων ἔνια λογικά ἐστι·μάλισταδὲ δῆλον ὅτι ἡ καθόλου κυριωτέρα, ὅτι — ὁ δὲ ταύτην ἔχων τὴν πρότασιν (the Particular)τὸ καθόλου οὐδαμῶς οἶδεν,οὔτε δυνάμει οὔτ’ ἐνεργείᾳ.
75Analyt. Post. I. xxiv. p. 86,a. 22: ἀλλὰ τῶν μὲν εἰρημένων ἔνια λογικά ἐστι·μάλισταδὲ δῆλον ὅτι ἡ καθόλου κυριωτέρα, ὅτι — ὁ δὲ ταύτην ἔχων τὴν πρότασιν (the Particular)τὸ καθόλου οὐδαμῶς οἶδεν,οὔτε δυνάμει οὔτ’ ἐνεργείᾳ.
76Ibid. a. 29: καὶ ἡ μὲν καθόλου νοητή, ἡ δὲ κατὰ μέρος εἰς αἴσθησιν τελευτᾷ. Compare xxiii. p. 84, b. 39, where we noticed the doctrine that Νοῦς is theunitof scientific demonstration.
76Ibid. a. 29: καὶ ἡ μὲν καθόλου νοητή, ἡ δὲ κατὰ μέρος εἰς αἴσθησιν τελευτᾷ. Compare xxiii. p. 84, b. 39, where we noticed the doctrine that Νοῦς is theunitof scientific demonstration.
Next, Aristotle compares the Affirmative with the Negative demonstration, and shows that the Affirmative is the better. Of two demonstrations (he lays it down) that one which proceeds upon a smaller number of postulates, assumptions, or propositions, is better than the other; for, to say nothing of other reasons, it conducts you more speedily to knowledge than the other, and that is an advantage. Now, both in the affirmative and in the negative syllogism, you must have three terms and two propositions; but in the affirmative you assume only that somethingis; while in the negative you assume both that somethingis, and that somethingis not. Here is a double assumption instead of a single; therefore the negative is the worse orinferior of the two.77Moreover, for the demonstration of a negative conclusion, you require one affirmative premiss (since from two negative premisses nothing whatever can be concluded); while for the demonstration of an affirmative conclusion, you must have two affirmative premisses, and you cannot admit a negative. This, again, shows that the affirmative is logically prior, more trustworthy, and better than the negative.78The negative is only intelligible and knowable through the affirmative, just asNon-Ensis knowable only throughEns. The affirmative demonstration therefore, as involving better principles, is, on this ground also, better than the negative.79A fortiori, it is also better than the demonstration by way ofReductio ad Absurdum, which was the last case to be considered. This, as concluding only indirectly and from impossibility of the contradictory, is worse even than the negative; much more therefore is it worse than the direct affirmative.80
77Analyt. Post. I. xxv. p. 86, a. 31-b. 9.
77Analyt. Post. I. xxv. p. 86, a. 31-b. 9.
78Ibid. b. 10-30.
78Ibid. b. 10-30.
79Ibid. b. 30-39.
79Ibid. b. 30-39.
80Ibid. I. xxvi. p. 87, a. 2-30. Waitz (II. p. 370), says: “deductio (ad absurdum), quippe quæ per ambages cogat, post ponenda, est demonstrationi rectæ.â€�Philoponus says (Schol. pp. 234-235,Brand.) that the Commentators all censured Aristotle for the manner in which he here laid out the Syllogism δι’ ἀδυνάτου. I do not, however, find any such censure in Themistius. Philoponus defends Aristotle from the censure.
80Ibid. I. xxvi. p. 87, a. 2-30. Waitz (II. p. 370), says: “deductio (ad absurdum), quippe quæ per ambages cogat, post ponenda, est demonstrationi rectæ.�
Philoponus says (Schol. pp. 234-235,Brand.) that the Commentators all censured Aristotle for the manner in which he here laid out the Syllogism δι’ ἀδυνάτου. I do not, however, find any such censure in Themistius. Philoponus defends Aristotle from the censure.
If we next compare one Science with another, the prior and more accurate of the two is, (1) That which combines at once the ὅτι and the διότι; (2) That which is abstracted from material conditions, as compared with that which is immersed therein — for example, arithmetic is more accurate than harmonics; (3) The more simple as compared with the more complex: thus, arithmetic is more accurate than geometry, a monad or unit is a substance without position, whereas a point (more concrete) is a substance with position.81One and the same science is that which belongs to one and the same generic subject-matter. The premisses of a demonstration must be included in the same genus with the conclusion; and where the ultimate premisses are heterogeneous, the cognition derived from them must be considered as not one but a compound of several.82You may find two or more distinct middle terms for demonstrating the same conclusion; sometimes out of the same logical series or table, sometimes out of different tables.83
81Analyt. Post. I. xxvii. p. 87, a. 31-37. Themistius, Paraphras. p. 60, ed. Speng.: κατ’ ἄλλον δὲ (τρόπον), ἐὰν ἡ μὲν περὶ ὑποκείμενά τινα καὶ αἰσθητὰ πραγματεύηται, ἡ δὲ περὶ νοητὰ καὶ καθόλου.Philoponus illustrates this (Schol. p. 235, b. 41, Br.): οἷον τὰ Θεοδοσίου σφαιρικὰ ἀκριβέστερά ἐστιν ἐπιστήμῃ τῆς τῶν Αὐτολύκου περὶ κινουμένης σφαίρας. &c.
81Analyt. Post. I. xxvii. p. 87, a. 31-37. Themistius, Paraphras. p. 60, ed. Speng.: κατ’ ἄλλον δὲ (τρόπον), ἐὰν ἡ μὲν περὶ ὑποκείμενά τινα καὶ αἰσθητὰ πραγματεύηται, ἡ δὲ περὶ νοητὰ καὶ καθόλου.
Philoponus illustrates this (Schol. p. 235, b. 41, Br.): οἷον τὰ Θεοδοσίου σφαιρικὰ ἀκριβέστερά ἐστιν ἐπιστήμῃ τῆς τῶν Αὐτολύκου περὶ κινουμένης σφαίρας. &c.
82Analyt. Post. I. xxviii. p. 87, a. 38-b. 5. Themistius, p. 61: δῆλον δὲ τοῦτο γίνεται προϊοῦσιν ἐπὶ τὰς ἀναποδείκτους ἀρχάς· αὗται γὰρ εἰ μηδεμίαν ἔχοιεν συγγένειαν, ἕτεραι αἱ ἐπιστῆμαι.
82Analyt. Post. I. xxviii. p. 87, a. 38-b. 5. Themistius, p. 61: δῆλον δὲ τοῦτο γίνεται προϊοῦσιν ἐπὶ τὰς ἀναποδείκτους ἀρχάς· αὗται γὰρ εἰ μηδεμίαν ἔχοιεν συγγένειαν, ἕτεραι αἱ ἐπιστῆμαι.