Again, there are certain genera in which negation carries with it the affirmation of an opposite; such as odd and even, in numbers. In such genera, if we are to admit any medium apart from and between the two members of the Antiphasis, we should be forced to admit some number which is neither odd nor even (p. 1012, a. 11). This is impossible: the definition excludes it. (Alexander gives this as the definition of number: πᾶς γὰρ ἀριθμὸς ἢ ἄρτιός ἐστιν ἢ περιττός, καὶ ἀριθμός ἐστιν ὃς ἢ ἄρτιός ἐστιν ἢ περιττός — Schol. p. 682, a. 16.)
Again, if the Antiphasis could be divided, and a half or intermediate position found, as this theory contends, the division of it must be admissible farther and farther, ad infinitum. After bisecting the Antiphasis, you can proceed to bisect each of the sections; and so on. Each section will afford an intermediate term which may be denied with reference to each of the two members of the original Antiphasis. Two new Antiphases will thus be formed, each of which may be bisected in the same manner; and so bisection, with the formation of successive new Antiphases, may proceed without end (p. 1012, a. 13).
Again, suppose a questioner to ask you, Is this subject white? You answer, No. Now you have denied nothing else than the being-white: this is the ἀπόφασις, or negative member of the Antiphasis. But you have neither denied nor affirmed the intermediate stage between the affirmative and the negative; nor is there any answer possible by which you could do so. Therefore there is no real intermediate stage between them (ἔτι ὅταν ἐρομένου εἰ λευκόν ἐστιν εἴπῃ ὅτι οὔ, οὐθὲν ἄλλο ἀποπέφηκεν ἢ τὸ εἶναι· ἀπόφασις δὲ τὸ μὴ εἶναι — p. 1012, a. 15; see Alex. Schol. p. 682, b. 15-38, and Bonitz’s note. Bonitz suggests, though timidly, ἀποπέφηκεν instead of the common reading ἀποπέφυκεν, which none of the commentators explain, and which seems unintelligible. I think Bonitz is right, though ἀποπέφηκεν is an unknown tense from ἀπόφημι: it is quite as regular as ἀποφήσω or ἀπέφησα.).
The doctrines which we have been just controverting (Aristotle says) arise, like other paradoxes, either from the embarrassment in which men find themselves when they cannot solve a sophistical difficulty; or from their fancying that an explanation may be demanded of every thing. In replying to them, you must take your start from the definition, which assigns to each word one fixed and constant signification. The doctrine of Herakleitus — That all things are and all things are not — makes all propositions true; that of Anaxagoras — That every thing is intermingled with every thing — makes all propositions false: such mixture is neither good, nor not good; neither of the members of the Antiphasis is true (a. 17-28). Our preceding reasonings have refuted both these doctrines, and have shown that neither of the two one-sided extremes can be universally true: neither the doctrine — Every proposition is true; nor that — Every proposition is false; still less that which comprehends them both — Every proposition is both true and false. Among these three doctrines, the second might seem the most plausible, yet it is inadmissible, like the other two (b. 4).
In debating with all these reasoners, you must require them (as we have already laid down), not to admit either existence or non-existence but, to admit a constant signification for each word. You must begin by defining truth and falsehood; each of them belongs only to affirmation in a certain way. Where the affirmation is true the denial is false; all propositions cannot be false; one member of each Antiphasis must be true, and the other member must be false. Each of these doctrines labours under the often-exposed defect — that it destroys itself (p. 1012, b. 14, τὸ θρυλλούμενον — allusion to the Theætetus, according to Alexander). For whoever declares all propositions to be true, declares the contradictory of this declaration to be true as well as the rest, and therefore his own declaration not to be true. Whoever declares all propositions to be false, declares his own declaration to be false as well as all other propositions (b. 17). And, even if we suppose each of these persons to make a special exception in regard to the particular propositions here respectively indicated, still this will not serve. The man who declares all propositions to be false, will be compelled to admit an infinite number of true propositions; because the proposition declaring the true proposition to be true, must itself be true; a second proposition declaring this last to be true, will itself be true; and so on to a third, a fourth, &c., in endless scale of ascent. The like may be said about the man who declares all propositions to be true: he too will be obliged to admit an infinite number of false propositions; for that which declares a true proposition to be false, must itself be false; and so on through a second, a third, &c., in endless scale of ascent as in the former case (b. 22).
It follows from what has been just proved, that those who affirm every thing to be at rest, and those who affirm every thing to be in motion, are both alike wrong. For, if every thing were at rest, the same propositions would be always true and always false. But this is plainly contrary to evidence; for the very reasoner who affirms it was once non-existent, and will again be non-existent. On the other hand, if every thing were in motion, no proposition would be true, and all would be false: but we have proved above that this is not so. Nor is it true that all things are alternately in motion or at rest; for there must be something ever-moving and other things ever-moved — and this prime movent must be itself immovable (p. 1012, b. 22-30).
Book E.
The First Philosophy investigates the causes and principles of Entia quatenus Entia (p. 1025, b. 3). It is distinguished from other sciences, by applying to all Entia, and in so far as they are Entia; for each of the other sciences applies itself to some separate branch of Entia, and investigates the causes and principles of that branch exclusively. Each assumes either from data of perception, or avowedly by way of hypothesis, the portion or genus of Entia to which it applies; not investigating the entity thereof, but pre-supposing this process to have been already performed by Ontology: each then investigates the properties belonging per se to that genus (b. 13). It is plain that by such an induction not one of these sciences can demonstrate either the essence of its own separate genus, nor whether that genus has any real existence. Both these questions — both εἰ ἔστιν and τί ἐστιν — belong to Ontology (b. 18). (The belief derived from perception and induction never amounts to demonstration, as has been shown in the Analytica; you may always contest the universality of the conclusion—Alex. p. 734, b. 16, Br.)
Apart from Ontology, each of these separate sciences is either theoretical, or practical, or constructive (p. 1025, b. 21). Two of the separate sciences are theoretical — Physics and Mathematics; and, as Ontology (or Theology) is also theoretical, there are three varieties of theoretical science (p. 1026, a. 18).