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 isosceles is a triangle. But he that knows the Particular does not in any way know the Universal, either actually or potentially.[75] 2. 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]
[75] Analyt. Post. I. xxiv. p. 86, a. 22: ἀλλὰ τῶν μὲν εἰρημένων ἔνια λογικά ἐστι· μάλιστα δὲ δῆλον ὅτι ἡ καθόλου κυριωτέρα, ὅτι — ὁ δὲ ταύτην ἔχων τὴν πρότασιν (the Particular) τὸ καθόλου οὐδαμῶς οἶδεν, οὔτε δυνάμει οὔτ’ ἐνεργείᾳ.
[76] Ibid. 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 the unit of 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 something is; while in the negative you assume both that something is, and that something is not. Here is a double assumption instead of a single; therefore the negative is the worse or inferior of the two.[77] Moreover, 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.[78] The negative is only intelligible and knowable through the affirmative, just as Non-Ens is knowable only through Ens. The affirmative demonstration therefore, as involving better principles, is, on this ground also, better than the negative.[79] A fortiori, it is also better than the demonstration by way of Reductio 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]
[77] Analyt. Post. I. xxv. p. 86, a. 31-b. 9.
[78] Ibid. b. 10-30.
[79] Ibid. b. 30-39.
[80] Ibid. 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.[81] One 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.[82] You 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]