12Aristot. Analyt. Post. I. ii. p. 71, b. 9-17. Julius Pacius says in a note, ad c. ii. p. 394: “Propositio demonstrativa est prima, immediata, et indemonstrabilis. His tribus verbis significatur una et eadem conditio; nam propositio prima est, quæ, quod medio caret, demonstrari nequit.�So also Zabarella (In lib. I. Post. Anal. Comm., p. 340, Op. ed. Venet. 1617): “Duæ illæ dictiones (primisetimmediatis) unam tantum significant conditionem ordine secundam, non duas; idem namque est, principia esse medio carentia, ac esse prima.�
12Aristot. Analyt. Post. I. ii. p. 71, b. 9-17. Julius Pacius says in a note, ad c. ii. p. 394: “Propositio demonstrativa est prima, immediata, et indemonstrabilis. His tribus verbis significatur una et eadem conditio; nam propositio prima est, quæ, quod medio caret, demonstrari nequit.�
So also Zabarella (In lib. I. Post. Anal. Comm., p. 340, Op. ed. Venet. 1617): “Duæ illæ dictiones (primisetimmediatis) unam tantum significant conditionem ordine secundam, non duas; idem namque est, principia esse medio carentia, ac esse prima.�
13Aristot. Analyt. Post. I. ii. p. 72, a. 1-24; Themistius, Paraphr. I. ii. p. 10, ed. Spengel; Schol. p. 199, b. 44. Themistius quotes the definition of an Axiom as given by Theophrastus: Ἀξίωμά ἐστιδόξατις, &c. This shows the difficulty of adhering precisely to a scientific terminology. Theophrastus explains an axiom to be a sort of δόξα, thus lapsing into the common loose use of the word. Yet still both he and Aristotle declare δόξα to be of inferior intellectual worth as compared with ἐπιστήμη (Anal. Post. I. xxiii.), while at the same time they declare the Axiom to be the very maximum of scientific truth. Theophrastus gave, as examples of Axioms, themaximof Contradiction, universally applicable, and, “If equals be taken from equals the remainders will be equal,â€� applicable to homogeneous quantities. Even Aristotle himself sometimes falls into the same vague employment of δόξα, as including the Axioms. See Metaphys.B.ii. p. 996, b. 28;Γ.iii. p. 1005, b. 33.
13Aristot. Analyt. Post. I. ii. p. 72, a. 1-24; Themistius, Paraphr. I. ii. p. 10, ed. Spengel; Schol. p. 199, b. 44. Themistius quotes the definition of an Axiom as given by Theophrastus: Ἀξίωμά ἐστιδόξατις, &c. This shows the difficulty of adhering precisely to a scientific terminology. Theophrastus explains an axiom to be a sort of δόξα, thus lapsing into the common loose use of the word. Yet still both he and Aristotle declare δόξα to be of inferior intellectual worth as compared with ἐπιστήμη (Anal. Post. I. xxiii.), while at the same time they declare the Axiom to be the very maximum of scientific truth. Theophrastus gave, as examples of Axioms, themaximof Contradiction, universally applicable, and, “If equals be taken from equals the remainders will be equal,â€� applicable to homogeneous quantities. Even Aristotle himself sometimes falls into the same vague employment of δόξα, as including the Axioms. See Metaphys.B.ii. p. 996, b. 28;Γ.iii. p. 1005, b. 33.
14Aristot. Anal. Post. I. ii. p. 72, a. 25, b. 4. I translate these words in conformity with Themistius, pp. 12-13, and with Mr. Poste’s translation, p. 43. Julius Pacius and M. Barthélemy St. Hilaire render them somewhat differently. They also read ἀμετάπτωτος, while Waitz and Firmin Didot read ἀμετάπειστος, which last seems preferable.
14Aristot. Anal. Post. I. ii. p. 72, a. 25, b. 4. I translate these words in conformity with Themistius, pp. 12-13, and with Mr. Poste’s translation, p. 43. Julius Pacius and M. Barthélemy St. Hilaire render them somewhat differently. They also read ἀμετάπτωτος, while Waitz and Firmin Didot read ἀμετάπειστος, which last seems preferable.
In Aristotle’s time two doctrines had been advanced, in opposition to the preceding theory: (1) Some denied the necessity of any indemonstrableprincipia, and affirmed the possibility of, demonstrating backwardsad infinitum; (2) Others agreed in denying the necessity of any indemonstrableprincipia, but contended that demonstration in a circle is valid and legitimate —e.g.that A may be demonstrated by means of B, and B by means of A. Against both these doctrines Aristotle enters his protest. The first of them — the supposition of an interminable regress — he pronounces to be obviously absurd: the second he declares tantamount to proving a thing by itself; the circular demonstration, besides, having been shown to be impossible, except in the First figure, with propositions in which the predicate reciprocates or is co-extensive with the subject — a very small proportion among propositions generally used in demonstrating.15
15Aristot. Analyt. Post. I. iii. p. 72, b. 5-p. 73, a. 20: ὥστ’ ἐπειδὴὀλίγα τοιαῦταἐν ταῖς ἀποδείξεσιν, &c.
15Aristot. Analyt. Post. I. iii. p. 72, b. 5-p. 73, a. 20: ὥστ’ ἐπειδὴὀλίγα τοιαῦταἐν ταῖς ἀποδείξεσιν, &c.
Demonstrative Science is attained only by syllogizing from necessary premisses, such as cannot possibly be other than they are. The predicate must be (1)de omni, (2)per se, (3)quatenus ipsum, so that it is aPrimum Universale; this third characteristic not being realized without the preceding two. First, the predicate must belong, and belong at all times, to everything called by the name of the subject. Next, it must belong thereuntoper se, or essentially; that is, either the predicate must be stated in the definition declaring the essence of the subject, or the subject must be stated in the definition declaring the essence of the predicate. The predicate must not be extra-essential to the subject, nor attached to it as an adjunct from without, simply concomitant or accidental. The like distinction holds in regard to events: some are accidentally concomitant sequences which may or may not be realized (e.g., a flash of lightning occurring when a man is on his journey); in others, the conjunction is necessary or causal (as when an animal dies under the sacrificial knife).16Both these two characteristics (de omniandper se) are presupposed in the third (quatenus ipsum); but this last implies farther, that the predicate is attached to the subject in the highest universality consistent with truth;i.e., that it is a First Universal, a primary predicate and not a derivative predicate. Thus, the predicate of having its three angles equal to two right angles, is a characteristic not merelyde omniandper se, but also a First Universal, applied to a triangle. It is applied to a triangle,quatenustriangle, as a primary predicate. If applied to a subject of higher universality (e.g., to every geometrical figure), it would not be always true. If applied to a subject of lower universality (e.g., to a right-angled triangle or an isosceles triangle), it would be universally true and would be trueper se, but it would be a derivative predicate and not a First Universal; it would not be applied to the isoscelesquatenusisosceles, for there is a still higher Universal of which it is predicable, being true respecting any triangle you please. Thus, the properties with which Demonstration, or full and absoluteScience, is conversant, arede omni,per se, andquatenus ipsum, orUniversalia Prima;17all of them necessary, such as cannot but be true.
16Aristot. Analyt. Post. I. iv. p. 73, a. 21, b. 16.Τὰ ἄρα λεγόμενα ἐπὶ τῶν ἁπλῶς ἐπιστητῶν καθ’ αὑτὰ οὕτως ὡς ἐνυπάρχειν τοῖς κατηγορουμένοις ἢ ἐνυπάρχεσθαι δι’ αὑτά τέ ἐστι καὶ ἐξ ἀνάγκης (b. 16, seq.).Linemust be included in the definition of the oppositesstraightorcurve. Also it is essential to every line that it is either straight or curve.Numbermust be included in the definition of the oppositesoddoreven; and to be either odd or even is essentially predicable of every number. You cannot understand what is meant bystraightorcurveunless you have the notion of aline.The example given by Aristotle ofcausalconjunction (the death of an animal under the sacrificial knife) shows that he had in his mind the perfection of Inductive Observation, including full application of the Method of Difference.
16Aristot. Analyt. Post. I. iv. p. 73, a. 21, b. 16.
Τὰ ἄρα λεγόμενα ἐπὶ τῶν ἁπλῶς ἐπιστητῶν καθ’ αὑτὰ οὕτως ὡς ἐνυπάρχειν τοῖς κατηγορουμένοις ἢ ἐνυπάρχεσθαι δι’ αὑτά τέ ἐστι καὶ ἐξ ἀνάγκης (b. 16, seq.).Linemust be included in the definition of the oppositesstraightorcurve. Also it is essential to every line that it is either straight or curve.Numbermust be included in the definition of the oppositesoddoreven; and to be either odd or even is essentially predicable of every number. You cannot understand what is meant bystraightorcurveunless you have the notion of aline.
The example given by Aristotle ofcausalconjunction (the death of an animal under the sacrificial knife) shows that he had in his mind the perfection of Inductive Observation, including full application of the Method of Difference.
17Aristot. Analyt. Post. I. iv. p. 73, b. 25-p. 74, a. 3. ὃ τοίνυντὸ τυχὸν πρῶτονδείκνυται δύο ὀρθὰς ἔχον ἢ ὁτιοῦν ἄλλο, τούτῳ πρώτῳ ὑπάρχει καθόλου, καὶ ἡἀπόδειξις καθ’ αὑτὸτούτου καθόλου ἐστὶ, τῶν δ’ ἄλλων τρόπον τινὰ οὐ καθ’ αὑτό· οὐδὲ τοῦ ἰσοσκέλους οὐκ ἔστι καθόλου ἀλλ’ ἐπὶ πλέον.About the precise signification of καθόλου in Aristotle, see a valuable note of Bonitz (ad Metaphys.Z.iii.) p. 299; also Waitz (ad Aristot. De Interpr. c. vii.) I. p. 334. Aristotle gives it here, b. 26: καθόλου δὲ λέγω ὃ ἂν κατὰ παντός τε ὑπάρχῃ καὶ καθ’ αὑτὸ καὶ ᾗ αὐτό. Compare Themistius, Paraphr. p. 19, Spengel. Τὸ καθ’ αὑτό is described by Aristotle confusedly. Τὸ καθόλου, is that which is predicable of the subject as a whole orsummum genus: τὸ κατὰ παντός, that which is predicable of every individual, either of thesummum genusor of any inferior species contained therein. Cf. Analyt. Post. I. xxiv. p. 85, b. 24: ᾧ γὰρ καθ’ αὑτὸ ὑπάρχει τι, τοῦτο αὐτὸ αὑτῷ αἴτιον — the subject is itself the cause orfundamentumof the propertiesper se. See the explanation and references in Kampe, Die Erkenntniss-theorie des Aristoteles, ch. v. pp. 160-165.
17Aristot. Analyt. Post. I. iv. p. 73, b. 25-p. 74, a. 3. ὃ τοίνυντὸ τυχὸν πρῶτονδείκνυται δύο ὀρθὰς ἔχον ἢ ὁτιοῦν ἄλλο, τούτῳ πρώτῳ ὑπάρχει καθόλου, καὶ ἡἀπόδειξις καθ’ αὑτὸτούτου καθόλου ἐστὶ, τῶν δ’ ἄλλων τρόπον τινὰ οὐ καθ’ αὑτό· οὐδὲ τοῦ ἰσοσκέλους οὐκ ἔστι καθόλου ἀλλ’ ἐπὶ πλέον.
About the precise signification of καθόλου in Aristotle, see a valuable note of Bonitz (ad Metaphys.Z.iii.) p. 299; also Waitz (ad Aristot. De Interpr. c. vii.) I. p. 334. Aristotle gives it here, b. 26: καθόλου δὲ λέγω ὃ ἂν κατὰ παντός τε ὑπάρχῃ καὶ καθ’ αὑτὸ καὶ ᾗ αὐτό. Compare Themistius, Paraphr. p. 19, Spengel. Τὸ καθ’ αὑτό is described by Aristotle confusedly. Τὸ καθόλου, is that which is predicable of the subject as a whole orsummum genus: τὸ κατὰ παντός, that which is predicable of every individual, either of thesummum genusor of any inferior species contained therein. Cf. Analyt. Post. I. xxiv. p. 85, b. 24: ᾧ γὰρ καθ’ αὑτὸ ὑπάρχει τι, τοῦτο αὐτὸ αὑτῷ αἴτιον — the subject is itself the cause orfundamentumof the propertiesper se. See the explanation and references in Kampe, Die Erkenntniss-theorie des Aristoteles, ch. v. pp. 160-165.
Aristotle remarks that there is great liability to error about theseUniversalia Prima. We sometimes demonstrate a predicate to be true, universally andper se, of a lower species, without being aware that it might also be demonstrated to be true, universally andper se, of the higher genus to which that species belongs; perhaps, indeed, that higher genus may not yet have obtained a current name. That proportions hold by permutation, was demonstrated severally for numbers, lines, solids, and intervals of time; but this belongs to each of them, not from any separate property of each, but from what is common to all: that, however, which is common to all had received no name, so that it was not known that one demonstration might comprise all the four.18In like manner, a man may know that an equilateral and an isosceles triangle have their three angles equal to two right angles, and also that a scalene triangle has its three angles equal to two right angles; yet he may not know (except sophistically and by accident19) that a trianglein generehas its three angles equal to two right angles, though there be no other triangles except equilateral, isosceles, and scalene. He does not know that this may be demonstrated of every trianglequatenustriangle. The only way to obtain acertain recognition ofPrimum Universale, is, to abstract successively from the several conditions of a demonstration respecting the concrete and particular, until the proposition ceases to be true. Thus, you have before you a brazen isosceles triangle, the three angles whereof are equal to two right angles. You may eliminate the condition brazen, and the proposition will still remain true. You may also eliminate the condition isosceles; still the proposition is true. But you cannot eliminate the condition triangle, so as to retain only the higher genus, geometrical figure; for the proposition then ceases to be always true. Triangle is in this case thePrimum Universale.20
18Aristot. Analyt. Post. I. v. p. 74, a. 4-23. ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι πάντα ταῦτα ἕν, ἀριθμοί, μήκη, χρόνος, στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμβάνετο. What these four have in common is that which he himself expresses by Ποσόν —Quantum— in the Categoriæ and elsewhere. (Categor. p. 4, b. 20, seq.; Metaph.Δ. p. 1020, a. 7, seq.)
18Aristot. Analyt. Post. I. v. p. 74, a. 4-23. ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι πάντα ταῦτα ἕν, ἀριθμοί, μήκη, χρόνος, στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμβάνετο. What these four have in common is that which he himself expresses by Ποσόν —Quantum— in the Categoriæ and elsewhere. (Categor. p. 4, b. 20, seq.; Metaph.Δ. p. 1020, a. 7, seq.)
19Aristot. Analyt. Post. I. v. p. 74, a. 27: οὔπω οἶδε τὸ τρίγωνον ὅτι δύο ὀρθαῖς, εἰ μὴτὸν σοφιστικὸν τρόπονοὐδὲ καθόλου τρίγωνον, οὔδ’ εἰ μηδέν ἐστι παρὰ ταῦτα τρίγωνον ἕτερον. The phrase τὸν σοφιστικὸν τρόπον is equivalent to τὸν σοφιστικὸντρόποντὸν κατὰ συμβεβηκός, p. 71, b. 10. I see nothing in it connected with Aristotle’s characteristic of a Sophist (special professional life purpose — τοῦ βίου τῇ προαιρέσει, Metaphys.Γ.p. 1004, b. 24): the phrase means nothing more thanunscientific.
19Aristot. Analyt. Post. I. v. p. 74, a. 27: οὔπω οἶδε τὸ τρίγωνον ὅτι δύο ὀρθαῖς, εἰ μὴτὸν σοφιστικὸν τρόπονοὐδὲ καθόλου τρίγωνον, οὔδ’ εἰ μηδέν ἐστι παρὰ ταῦτα τρίγωνον ἕτερον. The phrase τὸν σοφιστικὸν τρόπον is equivalent to τὸν σοφιστικὸντρόποντὸν κατὰ συμβεβηκός, p. 71, b. 10. I see nothing in it connected with Aristotle’s characteristic of a Sophist (special professional life purpose — τοῦ βίου τῇ προαιρέσει, Metaphys.Γ.p. 1004, b. 24): the phrase means nothing more thanunscientific.
20Aristot. Analyt. Post. I. v. p. 74, a. 32-b. 4.
20Aristot. Analyt. Post. I. v. p. 74, a. 32-b. 4.
In every demonstration theprincipiaor premisses must be not only true, but necessarily true; the conclusion also will then be necessarily true, by reason of the premisses, and this constitutes Demonstration. Wherever the premisses are necessarily true, the conclusion will be necessarily true; but you cannot say,vice versâ, that wherever the conclusion is necessarily true, the syllogistic premisses from which it follows must always be necessarily true. They may be true without being necessarily true, or they may even be false: if, then, the conclusion be necessarily true, it is not so by reason of these premisses; and the syllogistic proof is in this case no demonstration. Your syllogism may have true premisses and may lead to a conclusion which is true by reason of them; but still you have not demonstrated, since neither premisses nor conclusion arenecessarilytrue.21When an opponent contests your demonstration, he succeeds if he can disprove thenecessityof your conclusion; if he can show any single case in which it either is or may be false.22It is not enough to proceed upon a premiss which is either probable or simply true: it may be true, yet not appropriate to the case: you must take your departure from the first or highest universal of the genus about which you attempt to demonstrate.23Again, unless you can state thewhyof your conclusion; that is to say, unless the middle term, by reason of which the conclusion is necessarily true, be itself necessarily true, — you have not demonstrated it, nor do you know it absolutely. Yourmiddle term not being necessary may vanish, while the conclusion to which it was supposed to lead abides: in truth no conclusion was known through that middle.24In the complete demonstrative or scientific syllogism, the major term must be predicable essentially orper seof the middle, and the middle term must be predicable essentially orper seof the minor; thus alone can you be sure that the conclusion also isper seor necessary. The demonstration cannot take effect through a middle term which is merely a Sign; the sign, even though it be a constant concomitant, yet being not, or at least not known to be,per se, will not bring out thewhyof the conclusion, nor make the conclusion necessary. Of non-essential concomitants altogether there is no demonstration; wherefore it might seem to be useless to put questions about such; yet, though the questions cannot yield necessary premisses for a demonstrative conclusion, they may yield premisses from which a conclusion will necessarily follow.25
21Ibid. vi. p. 74, b. 5-18. ἐξ ἀληθῶν μὲν γὰρ ἔστι καὶ μὴ ἀποδεικνύντα συλλογίσθαι, ἐξ ἀναγκαίων δ’ οὐκ ἔστιν ἀλλ’ ἢ ἀποδεικνύντα· τοῦτο γὰρ ἤδη ἀποδείξεώς ἐστιν. Compare Analyt. Prior. I. ii. p. 53, b. 7-25.
21Ibid. vi. p. 74, b. 5-18. ἐξ ἀληθῶν μὲν γὰρ ἔστι καὶ μὴ ἀποδεικνύντα συλλογίσθαι, ἐξ ἀναγκαίων δ’ οὐκ ἔστιν ἀλλ’ ἢ ἀποδεικνύντα· τοῦτο γὰρ ἤδη ἀποδείξεώς ἐστιν. Compare Analyt. Prior. I. ii. p. 53, b. 7-25.
22Aristot. Analyt. Post. I. vi. p. 74, b. 18: σημεῖον δ’ ὅτι ἡ ἀπόδειξις ἐξ ἀναγκαίων, ὅτι καὶ τὰς ἐνστάσεις οὕτω φέρομεν πρὸς τοὺς οἰομένους ἀποδεικνύναι, ὅτι οὐκ ἀνάγκη, &c.
22Aristot. Analyt. Post. I. vi. p. 74, b. 18: σημεῖον δ’ ὅτι ἡ ἀπόδειξις ἐξ ἀναγκαίων, ὅτι καὶ τὰς ἐνστάσεις οὕτω φέρομεν πρὸς τοὺς οἰομένους ἀποδεικνύναι, ὅτι οὐκ ἀνάγκη, &c.
23Ibid. vi. p. 74, b. 21-26: δῆλον δ’ ἐκ τούτων καὶ ὅτι εὐήθεις οἱ λαμβάνειν οἰόμενοι καλῶς τὰς ἀρχάς, ἐὰν ἔνδοξος ᾖ ἡ πρότασις καὶ ἀληθής, οἷον οἱ σοφισταὶ ὅτι τὸ ἐπίστασθαι τὸ ἐπιστήμην ἔχειν·, &c.
23Ibid. vi. p. 74, b. 21-26: δῆλον δ’ ἐκ τούτων καὶ ὅτι εὐήθεις οἱ λαμβάνειν οἰόμενοι καλῶς τὰς ἀρχάς, ἐὰν ἔνδοξος ᾖ ἡ πρότασις καὶ ἀληθής, οἷον οἱ σοφισταὶ ὅτι τὸ ἐπίστασθαι τὸ ἐπιστήμην ἔχειν·, &c.
24Aristot. Analyt. Post. I. vi. p. 74, b. 26-p. 75, a. 17.
24Aristot. Analyt. Post. I. vi. p. 74, b. 26-p. 75, a. 17.
25Ibid. vi. p. 75, a. 8-37.On the point last mentioned, M. Barthélemy St. Hilaire observes in his note, p. 41: “Dans les questions de dialectique, la conclusion est nécessaire en ce sens, qu’elle suit nécessairement des prémisses; elle n’est pas du tout nécessaire en ce sens, que la chose qu’elle exprime soit nécessaire. Ainsi il faut distinguer la nécessité de la forme et la nécessité de la matière: ou comme disent les scholastiques,necessitas illationis et necessitas materiæ. La dialectique se contente de la première, mais la demonstration a essentiellement besoin des deux.�
25Ibid. vi. p. 75, a. 8-37.
On the point last mentioned, M. Barthélemy St. Hilaire observes in his note, p. 41: “Dans les questions de dialectique, la conclusion est nécessaire en ce sens, qu’elle suit nécessairement des prémisses; elle n’est pas du tout nécessaire en ce sens, que la chose qu’elle exprime soit nécessaire. Ainsi il faut distinguer la nécessité de la forme et la nécessité de la matière: ou comme disent les scholastiques,necessitas illationis et necessitas materiæ. La dialectique se contente de la première, mais la demonstration a essentiellement besoin des deux.�
In every demonstration three things may be distinguished: (1) The demonstrated conclusion, or Attribute essential to a certain genus; (2) The Genus, of which the attributesper seare the matter of demonstration; (3) The Axioms, out of which, or through which, the demonstration is obtained. These Axioms may be and are common to several genera: but the demonstration cannot be transferred from one genus to another; both the extremes as well as the middle term must belong to the same genus. An arithmetical demonstration cannot be transferred to magnitudes and their properties, except in so far as magnitudes are numbers, which is partially true of some among them. The demonstrations in arithmetic may indeed be transferred to harmonics, because harmonics is subordinate to arithmetic; and, for the like reason, demonstrations in geometry may be transferred to mechanics and optics. But we cannot introduce into geometry any property of lines, which does not belong to themquâlines; such, for example, as that a straight line is the most beautiful of all lines, or is the contrary of a circular line; for these predicates belong to it, notquâline, butquâmember of a different or more extensive genus.26There can be nocomplete demonstration about perishable things, or about any individual line, except in regard to its attributes as member of the genus line. Where the conclusion is not eternally true, but true at one time and not true at another, this can only be because one of its premisses is not universal or essential. Where both premisses are universal and essential, the conclusion must be eternal or eternally true. As there is no demonstration, so also there can be no definition, of perishable attributes.27
26Ibid. vii. p. 75, a. 38-b. 20. Mr. Poste, in his translation, here cites (p. 50) a good illustrative passage from Dr. Whewell’s Philosophy of the Inductive Sciences, Book II. ii.:— “But, in order that we may make any real advance in the discovery of truth, our ideas must not only be clear; they must also beappropriate. Each science has for its basis a different class of ideas; and the steps which constitute the progress of one science can never be made by employing the ideas of another kind of science. No genuine advance could ever be obtained in Mechanics by applying to the subject the ideas of space and time merely; no advance in Chemistry by the use of mere mechanical conceptions; no discovery in Physiology by referring facts to mere chemical and mechanical principles.� &c.
26Ibid. vii. p. 75, a. 38-b. 20. Mr. Poste, in his translation, here cites (p. 50) a good illustrative passage from Dr. Whewell’s Philosophy of the Inductive Sciences, Book II. ii.:— “But, in order that we may make any real advance in the discovery of truth, our ideas must not only be clear; they must also beappropriate. Each science has for its basis a different class of ideas; and the steps which constitute the progress of one science can never be made by employing the ideas of another kind of science. No genuine advance could ever be obtained in Mechanics by applying to the subject the ideas of space and time merely; no advance in Chemistry by the use of mere mechanical conceptions; no discovery in Physiology by referring facts to mere chemical and mechanical principles.� &c.
27Aristot. Analyt. Post. I. viii. p. 75, b. 21-36. Compare Metaphys.Z.p. 1040, a. 1: δῆλον ὅτι οὐκ ἂν εἴη αὐτῶν (τῶν φθαρτῶν) οὔθ’ ὁρισμὸς οὔτ’ ἀπόδειξις. Also Biese, Die Philosophie des Aristoteles, ch. iv. p. 249.
27Aristot. Analyt. Post. I. viii. p. 75, b. 21-36. Compare Metaphys.Z.p. 1040, a. 1: δῆλον ὅτι οὐκ ἂν εἴη αὐτῶν (τῶν φθαρτῶν) οὔθ’ ὁρισμὸς οὔτ’ ἀπόδειξις. Also Biese, Die Philosophie des Aristoteles, ch. iv. p. 249.
For complete demonstration, it is not sufficient that the premisses be true, immediate, and undemonstrable; they must, furthermore, be essential and appropriate to the class in hand. Unless they be such, you cannot be said to know the conclusionabsolutely; you know it only by accident. You can only know a conclusion when demonstrated from its own appropriate premisses; and you know it best when it is demonstrated from its highest premisses. It is sometimes difficult to determine whether we really know or not; for we fancy that we know, when we demonstrate from true and universalprincipia, without being aware whether they are, or are not, theprincipiaappropriate to the case.28But theseprincipiamust always be assumed without demonstration — the class whose essential constituent properties are in question, the universal Axioms, and the Definition or meaning of the attributes to be demonstrated. If these definitions and axioms are not always formally enunciated, it is because we tacitly presume them to be already known and admitted by the learner.29He may indeed always refuse to grant them in express words, but they are such that he cannot help granting them by internal assent in his mind, to which every syllogism must address itself. When you assume a premiss without demonstrating it, though it be really demonstrable, this, if the learner is favourable and willing to grant it, is an assumption or Hypothesis, valid relatively to him alone, but not valid absolutely: if he is reluctant or adverse, it is a Postulate, whichyou claim whether he is satisfied or not.30The Definition by itself is not an hypothesis; for it neither affirms nor denies the existence of anything. The pupil must indeed understand the terms of it; but this alone is not an hypothesis, unless you call the fact that the pupil comes to learn, an hypothesis.31The Hypothesis or assumption is contained in the premisses, being that by which the reason of the conclusion comes to be true. Some object that the geometer makes a false hypothesis or assumption, when he declares a given line drawn to be straight, or to be a foot long, though it is neither one nor the other. But this objection has no pertinence, since the geometer does not derive his conclusions from what is true of the visible lines drawn before his eyes, but from what is true of the lines conceived in his own mind, and signified or illustrated by the visible diagrams.32
28Ibid. ix. p. 75, b. 37-p. 76, a. 30.
28Ibid. ix. p. 75, b. 37-p. 76, a. 30.
29Ibid. x. p. 76, a. 31-b. 22.
29Ibid. x. p. 76, a. 31-b. 22.
30Aristot. Analyt. Post. I. x. p. 76, b. 29-34: ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὔχ ἁπλῶς ὑπόθεσις, ἀλλὰ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμίᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτό, αἰτεῖται. καὶ τούτῳ διαφέρειὑπόθεσιςκαὶαἴτημα, &c. Themistius, Paraphras. p. 37, Spengel.
30Aristot. Analyt. Post. I. x. p. 76, b. 29-34: ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὔχ ἁπλῶς ὑπόθεσις, ἀλλὰ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμίᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτό, αἰτεῖται. καὶ τούτῳ διαφέρειὑπόθεσιςκαὶαἴτημα, &c. Themistius, Paraphras. p. 37, Spengel.
31Ibid. p. 76, b. 36: τοῦτο δ’ οὐχ ὑπόθεσις, εἰ μὴ καὶτὸ ἀκούεινὑπόθεσίν τις εἶναι φήσει. For the meaning ofτὸ ἀκούειν, compare ὁ ἀκούων, infra, Analyt. Post. I. xxiv. p. 85, b. 22.
31Ibid. p. 76, b. 36: τοῦτο δ’ οὐχ ὑπόθεσις, εἰ μὴ καὶτὸ ἀκούεινὑπόθεσίν τις εἶναι φήσει. For the meaning ofτὸ ἀκούειν, compare ὁ ἀκούων, infra, Analyt. Post. I. xxiv. p. 85, b. 22.
32Ibid. p. 77, a. 1: ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῷ τήνδε εἶναι τὴν γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα.Themistius, Paraphr. p. 37: ὥσπερ οὐδ’ οἱ γεωμέτραι κέχρηνται ταῖς γραμμαῖς ὑπὲρ ὧν διαλέγονται καὶ δεικνύουσιν, ἀλλ’ ἃς ἔχουσιν ἐν τῇ ψυχῇ, ὧν εἰσὶ σύμβολα αἱ γραφόμεναι.A similar doctrine is asserted, Analyt. Prior. I. xli. p. 49, b. 35, and still more clearly in De Memoria et Reminiscentia, p. 450, a. 2-12.
32Ibid. p. 77, a. 1: ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῷ τήνδε εἶναι τὴν γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα.
Themistius, Paraphr. p. 37: ὥσπερ οὐδ’ οἱ γεωμέτραι κέχρηνται ταῖς γραμμαῖς ὑπὲρ ὧν διαλέγονται καὶ δεικνύουσιν, ἀλλ’ ἃς ἔχουσιν ἐν τῇ ψυχῇ, ὧν εἰσὶ σύμβολα αἱ γραφόμεναι.
A similar doctrine is asserted, Analyt. Prior. I. xli. p. 49, b. 35, and still more clearly in De Memoria et Reminiscentia, p. 450, a. 2-12.
The process of Demonstration neither requires, nor countenances, the Platonic theory of Ideas — universal substances beyond and apart from particulars. But it does require that we should admit universal predications; that is, one and the same predicate truly applicable in the same sense to many different particulars. Unless this be so, there can be no universal major premiss, nor appropriate middle term, nor valid demonstrative syllogism.33
33Aristot. Analyt. Post. I. xi. p. 77, a. 5-9.
33Aristot. Analyt. Post. I. xi. p. 77, a. 5-9.
The Maxim or Axiom of Contradiction, in its most general enunciation, is never formally enunciated by any special science; but each of them assumes the Maxim so far as applicable to its own purpose, whenever theReductio ad Absurdumis introduced.34It is in this and the other common principles or Axioms that all the sciences find their point of contact and communion; and that Dialectic also comes into communion with all of them, as also the science (First Philosophy) that scrutinizes the validity or demonstrability of the Axioms.35The dialectician is not confinedto any one science, or to any definite subject-matter. His liberty of interrogation is unlimited; but his procedure is essentially interrogatory, and he is bound to accept the answer of the respondent — whatever it be, affirmative or negative — as premiss for any syllogism that he may construct. In this way he can never be sure of demonstrating any thing; for the affirmative and the negative will not be equally serviceable for that purpose. There is indeed also, in discussions on the separate sciences, a legitimate practice of scientific interrogation. Here the questions proper to be put are limited in number, and the answers proper to be made are determined beforehand by the truths of the science — say Geometry; still, an answer thus correctly made will serve to the interrogator as premiss for syllogistic demonstration.36The respondent must submit to have such answer tested by appeal to geometricalprincipiaand to other geometrical propositions already proved as legitimate conclusions from theprincipia; if he finds himself involved in contradictions, he is confutedquâgeometer, and must correct or modify his answer. But he is not bound,quâgeometer, to undergo scrutiny as to the geometricalprincipiathemselves; this would carry the dialogue out of the province of Geometry into that of First Philosophy and Dialectic. Care, indeed, must be taken to keep both questions and answers within the limits of the science. Now there can be no security for this restriction, except in the scientific competence of the auditors. Refrain, accordingly, from all geometrical discussions among men ignorant of geometry and confine yourself to geometrical auditors, who alone can distinguish what questions and answers are really appropriate. And what is here said about geometry, is equally true about the other special sciences.37Answers may be improper either as foreign to the science under debate, or as appertaining to the science, yet false as to the matter, or as equivocal in middle term; though this last is less likely to occur in Geometry, since the demonstrations are accompanied by diagrams, which helpto render conspicuous any such ambiguity.38To an inductive proposition, bringing forward a single case as contributory to an ultimate generalization, no general objection should be offered; the objection should be reserved until the generalization itself is tendered.39Sometimes the mistake is made of drawing an affirmative conclusion from premisses in the Second figure; this is formally wrong, but the conclusion may in some cases be true, if the major premiss happens to be a reciprocating proposition, having its predicate co-extensive with its subject. This, however, cannot be presumed; nor can a conclusion be made to yield up its principles by necessary reciprocation; for we have already observed that, though the truth of the premisses certifies the truth of the conclusion, we cannot sayvice versâthat the truth of the conclusion certifies the truth of the premisses. Yet propositions are more frequently found to reciprocate in scientific discussion than in Dialectic; because, in the former, we take no account of accidental properties, but only of definitions and what follows from them.40
34Ibid. a. 10, seq.
34Ibid. a. 10, seq.
35Ibid. a. 26-30: καὶ εἴ τις καθόλου πειρῷτο δεικνύναι τὰ κοινά, οἷον ὅτι ἅπαν φάναι ἢ ἀποφάναι, ἢ ὅτι ἴσα ἀπὸ ἴσων, ἢ τῶν τοιούτων ἄττα. Compare Metaph.K.p. 1061,b. 18.
35Ibid. a. 26-30: καὶ εἴ τις καθόλου πειρῷτο δεικνύναι τὰ κοινά, οἷον ὅτι ἅπαν φάναι ἢ ἀποφάναι, ἢ ὅτι ἴσα ἀπὸ ἴσων, ἢ τῶν τοιούτων ἄττα. Compare Metaph.K.p. 1061,b. 18.
36Aristot. Analyt. Post. I. xii, p. 77, a. 36-40; Themistius, p. 40.The text is here very obscure. He proceeds to distinguish Geometry especially (also other sciences, though less emphatically) from τὰ ἐν τοῖς διαλόγοις (I. xii. p. 78, a. 12).Julius Pacius, ad Analyt. Post. I. viii. (he divides the chapters differently), p. 417, says:— “Differentia interrogationis dialecticæ et demonstrativæ hæc est. Dialecticus ita interrogat, ut optionem det adversario, utrum malit affirmare an negare. Demonstrator vero interrogat ut rem evidentiorem faciat; id est, ut doceat ex principiis auditori notis.â€�
36Aristot. Analyt. Post. I. xii, p. 77, a. 36-40; Themistius, p. 40.
The text is here very obscure. He proceeds to distinguish Geometry especially (also other sciences, though less emphatically) from τὰ ἐν τοῖς διαλόγοις (I. xii. p. 78, a. 12).
Julius Pacius, ad Analyt. Post. I. viii. (he divides the chapters differently), p. 417, says:— “Differentia interrogationis dialecticæ et demonstrativæ hæc est. Dialecticus ita interrogat, ut optionem det adversario, utrum malit affirmare an negare. Demonstrator vero interrogat ut rem evidentiorem faciat; id est, ut doceat ex principiis auditori notis.�
37Ibid. I. xii. p. 77, b. 1-15; Themistius, p. 41: οὐ γὰρ ὥσπερ τῶν ἐνδόξων οἱ πολλοὶ κριταί, οὕτω καὶ τῶν κατ’ ἐπιστήμην οἱ ἀνεπιστήμονες.
37Ibid. I. xii. p. 77, b. 1-15; Themistius, p. 41: οὐ γὰρ ὥσπερ τῶν ἐνδόξων οἱ πολλοὶ κριταί, οὕτω καὶ τῶν κατ’ ἐπιστήμην οἱ ἀνεπιστήμονες.
38Analyt. Post. I. xii. p. 77, b. 16-33. Propositions within the limits of the science, but false as to matter, are styled by Aristotle ψευδογραφήματα. See Aristot. Sophist. Elench. xi. p. 171, b. 14; p. 172, a. 1.“L’interrogation syllogistique se confondant avec la proposition, il s’ensuit que l’interrogation doit être, comme la proposition, propre à la science dont il s’agit.â€� (Barthélemy St Hilaire, note, p. 70). Interrogation here has a different meaning from that which it bears in Dialectic.
38Analyt. Post. I. xii. p. 77, b. 16-33. Propositions within the limits of the science, but false as to matter, are styled by Aristotle ψευδογραφήματα. See Aristot. Sophist. Elench. xi. p. 171, b. 14; p. 172, a. 1.
“L’interrogation syllogistique se confondant avec la proposition, il s’ensuit que l’interrogation doit être, comme la proposition, propre à la science dont il s’agit.� (Barthélemy St Hilaire, note, p. 70). Interrogation here has a different meaning from that which it bears in Dialectic.
39Ibid. I. xii. p. 77,b. 34 seq. This passage is to me hardly intelligible. It is differently understood by commentators and translators. John Philoponus in the Scholia (p. 217, b. 17-32, Brandis), cites the explanation of it given by Ammonius, but rejects that explanation, and waits for others to supply him with a better. Zabarella (Comm. in Analyt. Post. pp. 426, 456, ed. Venet 1617) admits that as it stands, and where it stands, it is unintelligible, but transposes it to another part of the book (to the end of cap. xvii., immediately before the words φανερὸν δὲ καὶ ὅτι, &c., of c. xviii.), and gives an explanation of it in this altered position. But I do not think he has succeeded in clearing it up.
39Ibid. I. xii. p. 77,b. 34 seq. This passage is to me hardly intelligible. It is differently understood by commentators and translators. John Philoponus in the Scholia (p. 217, b. 17-32, Brandis), cites the explanation of it given by Ammonius, but rejects that explanation, and waits for others to supply him with a better. Zabarella (Comm. in Analyt. Post. pp. 426, 456, ed. Venet 1617) admits that as it stands, and where it stands, it is unintelligible, but transposes it to another part of the book (to the end of cap. xvii., immediately before the words φανερὸν δὲ καὶ ὅτι, &c., of c. xviii.), and gives an explanation of it in this altered position. But I do not think he has succeeded in clearing it up.
40Ibid. I. xii. p. 77, b. 40-p. 78, a. 13.
40Ibid. I. xii. p. 77, b. 40-p. 78, a. 13.
Knowledge of Fact and knowledge of the Cause must be distinguished, and even within the same Science.41In some syllogisms the conclusion only brings out τὸ ὅτι — the reality of certain facts; in others, it ends in τὸ διότι — the affirmation of a cause, or of theWhy. The syllogism of theWhyis, where the middle term is not merely the cause, but the proximate cause, of the conclusion. Often, however, the effect is more notorious, so that we employ it as middle term, and conclude from it to its reciprocating cause; in which case our syllogism is only of the ὅτι; and so it is also when we employ as middle term a cause not proximate but remote, concluding from that to the effect.42Sometimesthe syllogisms of the ὅτι may fall under one science, those of the διότι under another, namely, in the case where one science is subordinate to another, as optics to geometry, and harmonics to arithmetic; the facts of optics and harmonics belonging to sense and observation, the causes thereof to mathematical reasoning. It may happen, then, that a man knows τὸ διότι well, but is comparatively ignorant τοῦ ὅτι: the geometer may have paid little attention to optical facts.43Cognition of the διότι is the maximum, the perfection, of all cognition; and this, comprising arithmetical and geometrical theorems, is almost always attained by syllogisms in the First figure. This figure is the most truly scientific of the three; the other two figures depend upon it for expansion and condensation. It is, besides, the only one in which universal affirmative conclusions can be obtained; for in the Second figure we get only negative conclusions; in the Third, only particular. Accordingly, propositions declaring Essence or Definition, obtained only through universal affirmative conclusions, are yielded in none but the First figure.44
41Ibid. I. xiii. p. 77, a. 22 seq.
41Ibid. I. xiii. p. 77, a. 22 seq.
42Themistius, p. 45: πολλάκις συμβαίνει καὶ ἀντιστρέφειν ἀλλήλοις τὸ αἰτιον καὶ τὸ σημεῖον καὶ ἄμφω δείκνυσθαι δι’ ἀλλήλων, διὰ τοῦ σημείου μὲν ὡς τὸ ὅτι, διὰ θατέρου δὲ ὡς τὸ διότι.“Cum enim vera demonstratio, id est τοῦ διότι, fiat per causam proximam, consequens est, ut demonstratio vel per effectum proximum, vel per causam remotam, sit demonstratio τοῦ ὅτιâ€� (Julius Pacius, Comm. p. 422).M. Barthélemy St. Hilaire observes (Note, p. 82):— “La cause éloignée non immédiate, donne un syllogisme dans la seconde figure. — Il est vrai qu’Aristote n’appelle cause que la cause immédiate; et que la cause éloignée n’est pas pour lui une véritable cause.â€�See in Schol. p. 188, a. 19, the explanation given by Alexander of the syllogism τοῦ διότι.
42Themistius, p. 45: πολλάκις συμβαίνει καὶ ἀντιστρέφειν ἀλλήλοις τὸ αἰτιον καὶ τὸ σημεῖον καὶ ἄμφω δείκνυσθαι δι’ ἀλλήλων, διὰ τοῦ σημείου μὲν ὡς τὸ ὅτι, διὰ θατέρου δὲ ὡς τὸ διότι.
“Cum enim vera demonstratio, id est τοῦ διότι, fiat per causam proximam, consequens est, ut demonstratio vel per effectum proximum, vel per causam remotam, sit demonstratio τοῦ ὅτιâ€� (Julius Pacius, Comm. p. 422).
M. Barthélemy St. Hilaire observes (Note, p. 82):— “La cause éloignée non immédiate, donne un syllogisme dans la seconde figure. — Il est vrai qu’Aristote n’appelle cause que la cause immédiate; et que la cause éloignée n’est pas pour lui une véritable cause.�
See in Schol. p. 188, a. 19, the explanation given by Alexander of the syllogism τοῦ διότι.
43Analyt. Post. I. xiii. p. 79, a. 2, seq.: ἐνταῦθα γὰρ τὸ μὲν ὅτι τῶν αἰσθητικῶν εἰδέναι, τὸ δὲ διότι τῶν μαθηματικῶν, &c. Compare Analyt. Prior. II. xxi. p. 67, a. 11; and Metaphys.A.p. 981, a. 15.
43Analyt. Post. I. xiii. p. 79, a. 2, seq.: ἐνταῦθα γὰρ τὸ μὲν ὅτι τῶν αἰσθητικῶν εἰδέναι, τὸ δὲ διότι τῶν μαθηματικῶν, &c. Compare Analyt. Prior. II. xxi. p. 67, a. 11; and Metaphys.A.p. 981, a. 15.
44Analyt. Post. I. xiv. p. 79, a. 17-32.
44Analyt. Post. I. xiv. p. 79, a. 17-32.
As there are some affirmative propositions that are indivisible,i.e., having affirmative predicates which belong to a subject at once, directly, immediately, indivisibly, — so there are also some indivisible negative propositions,i.e., with predicates that belong negatively to a subject at once, directly, &c. In all such there is no intermediate step to justify either the affirmation of the predicate, or the negation of the predicate, respecting the given subject. This will be the case where neither the predicate nor the subject is contained in any higher genus.45
45Ibid. I. xv. p. 79, a. 33-b. 22. The point which Aristotle here especially insists upon is, that there may be and are immediate, undemonstrable,negative(as well as affirmative) predicates: φανερὸν οὖν ὅτι ἐνδέχεταί τε ἄλλο ἄλλῳμὴ ὑπάρχεινἀτόμως. (Themistius, Paraphr. p. 48, Spengel: ἄμεσοι δὲ προτάσεις οὐ καταφάσεις μόνον εἰσίν, ἀλλὰ καὶ ἀποφάσεις ὁμοίως αἳ μὴ δύνανται διὰ συλλογισμοῦ δειχθῆναι, αὗται δ’ εἰσὶν ἐφ’ ὧν οὐδετέρου τῶν ὅρων ἄλλος τις ὅλου κατηγορεῖται.) It had been already shown, in an earlier chapter of this treatise (p. 72, b. 19), that there wereaffirmativepredicates immediate and undemonstrable. This may be compared with that which Plato declares in the Sophistes (pp. 253-254, seq.) about the intercommunion τῶν γενῶν καὶ τῶν εἰδῶν with each other. Some of them admit such intercommunion, others repudiate it.
45Ibid. I. xv. p. 79, a. 33-b. 22. The point which Aristotle here especially insists upon is, that there may be and are immediate, undemonstrable,negative(as well as affirmative) predicates: φανερὸν οὖν ὅτι ἐνδέχεταί τε ἄλλο ἄλλῳμὴ ὑπάρχεινἀτόμως. (Themistius, Paraphr. p. 48, Spengel: ἄμεσοι δὲ προτάσεις οὐ καταφάσεις μόνον εἰσίν, ἀλλὰ καὶ ἀποφάσεις ὁμοίως αἳ μὴ δύνανται διὰ συλλογισμοῦ δειχθῆναι, αὗται δ’ εἰσὶν ἐφ’ ὧν οὐδετέρου τῶν ὅρων ἄλλος τις ὅλου κατηγορεῖται.) It had been already shown, in an earlier chapter of this treatise (p. 72, b. 19), that there wereaffirmativepredicates immediate and undemonstrable. This may be compared with that which Plato declares in the Sophistes (pp. 253-254, seq.) about the intercommunion τῶν γενῶν καὶ τῶν εἰδῶν with each other. Some of them admit such intercommunion, others repudiate it.
In regard both to these propositions immediate and indivisible, and to propositions mediate and deducible, there are two varieties of error.46You may err simply, from ignorance, not knowing better, and not supposing yourself to know at all; or your error may be a false conclusion, deduced by syllogism through a middle term, and accompanied by a belief on your part that you do know. This may happen in different ways. Suppose the negative proposition, No B is A, to be true immediately or indivisibly. Then, if you conclude the contrary of this47(All B is A) to be true, by syllogism through the middle term C, your syllogism must be in the First figure; it must have the minor premiss false (since B is brought under C, when it is not contained in any higher genus), and it may have both premisses false. Again, suppose the affirmative proposition, All B is A, to be true immediately or indivisibly. Then if you conclude the contrary of this (No B is A) to be true, by syllogism through the middle term C, your syllogism may be in the First figure, but it may also be in the Second figure, your false conclusion being negative. If it be in the First figure, both its premisses may be false, or one of them only may be false, either indifferently.48If it be in the Second figure, either premiss singly may be wholly false, or both may be partly false.49