Though it be true, therefore, that neither Ens nor Unum is a true genus, nor separable, but both of them aggregates of analogical derivatives, yet since all these derivatives have their root in one and the same fundamentum, the study of all of them belongs to one and the same science (p. 1005, a. 6-11). It is not the province of the geometer to examine what is The Opposite, The Perfect, Ens, Unum, Idem, Diversum, except in their application to his own problems. The general enquiry devolves upon the First Philosopher; who will investigate Ens quatenus Ens, together with the belongings or appendages (τὰ ὑπάρχοντα) of Ens quatenus Ens, including Prius, Posterius, Genus, Species, Totum, Pars, and such like (a. 11-18).
It falls to the First Philosopher also to investigate and explain what mathematicians call their Axioms: the mathematician ought not to do this himself, but to leave it to the First Philosopher. These Axioms are, in their highest generality, affirmations respecting Ens quatenus Ens, all of which belong to the First Philosopher; from whom the mathematician accepts them, and applies them as far as his own department requires (p. 1005, a. 20, seq.).
In First Philosophy, the firmest, best known, and most unquestionable of all principles is this: It is impossible for the same predicate at the same time and in the same sense to belong and not to belong to the same subject (p. 1005, b. 20). No one can at the same time believe that the same thing both is and is not; though Herakleitus professed to believe this, we must not suppose that he really did believe it (b. 25). No man can hold two contrary opinions at the same time (b. 31). This is by nature the first principle of all other axioms; to which principle all demonstrations are in the last resort brought back (b. 33: φύσει γὰρ ἀρχὴ καὶ τῶν ἄλλων ἀξιωμάτων αὕτη πάντων).
Aristotle then proceeds to explain and vindicate at length this ἀρχή â€” the Principle of Contradiction, which many at that time denied. This principle is at once the most knowable, and noway assumed as hypothesis (γνωριμωτάτην καὶ ἀνυπόθετον — p. 1005, b. 13). You cannot indeed demonstrate it to be true; the very attempt to demonstrate it would be unphilosophical: demonstration of every thing, is an impossibility. You cannot march upwards in an infinite progression of demonstrations; you must arrive ultimately at some first truth which is not demonstrable; and, if any such first truth is to be recognized, no one can point out any truth better entitled to such privilege than the Principle of Contradiction (p. 1006, a. 11). But you can convict an opponent of self-contradiction (ἀποδεῖξαι ἐλεγκτικῶς, a. 12, 15), if he will only consent to affirm any proposition in significant terms — that is, in terms which he admits to be significant to himself and which he intends as such to others; in other words, if he will enter into dialogue with you, for without significant speech there can be no dialogue with him at all (a. 21).
When the opponent has shown his willingness to comply with the conditions of dialogue, by advancing a proposition in terms each having one definite signification, it is plain, by his own admission, that the proposition does not both signify and not signify the same. First, the copula of the proposition (est) does not signify what would be signified if the copula were non est; so that here is one case wherein the affirmative and the negative cannot be both of them true (p. 1006, a. 30; see Alex. Schol. and Bonitz’s note). Next, let the subject of the proposition be homo; a term having only one single definite signification, or perhaps having two or three (or any definite number of) distinct significations, each definite. If the number of distinct significations be indefinite, the term is unfit for the purpose of dialogue (a. 30-b. 10). The term homo will signify one thing only; it will have one determinate essence and definition — say animal bipes: that is, if any thing be a man, the same will be animal bipes. But this last cannot be the essence and definition of non-homo also: non-homo, as a different name, must have different definition; homo and non-homo cannot be like λώπιον and ἱμάτιον, two terms having the same signification, essence and definition; for homo signifies one subject of constant and defined nature, not simply one among many predicates applicable by accident to this same constant subject; it signifies μίαν φύσιν and not ἄλλην τινὰ φύσιν (Scholia, p. 656, b. 21). Since each name indeed is applied by convention to what it denominates, the name non-homo may be applied elsewhere to that which we term homo; but this is a mere difference of naming; what bears the name homo, and what bears the name non-homo, must always be different, if homo is defined to signify one determinate nature (b. 22). The one single nature and essence defined as belonging to homo, cannot be the same as that belonging to non-homo. If any thing be homo, the same cannot be non-homo: if any thing be non-homo, the same cannot be homo (b. 25-34). Whoever says that homo and non-homo have the same meaning, must say à fortiori that homo, fortis, musicus, simus, pulcher, &c., have the same meaning; for not one of these terms is so directly and emphatically opposite to homo, as non-homo is. He must therefore admit that the meaning, not merely of all these words but also, of a host besides is the same; in other words, that not merely Opposites are one, but all other things besides, under different names (ὅτι ἓν πάντα ἔσται καὶ οὐ μόνον τὰ ἀντικείμενα — p. 1007, a. 6).
This argument is directed against those who maintain that affirmative and negative are both true at once, but who still desire to keep up dialogue (Alex. Schol. p. 658, a. 26, Br.: τῷ τήν τε ἀντίφασιν συναληθεύειν λέγοντι, καὶ σώζειν βουλομένῳ τὸ διαλέγεσθαι). No man who maintains this opinion, can keep his consistency in dialogue, if he will only give direct answers to the questions put to him, without annexing provisoes and gratuitous additions to his answers. If you ask him, Whether it is true that Sokrates is homo? he ought to answer plainly Yes, or No. He ought not to answer: “Yes, but Sokrates is also non-homo,â€� meaning that Sokrates is also the subject of many other accidental predicates — fair, flat-nosed, brave, accomplished, &c. He ought to answer simply to the question, whether the one essence or definition signified by the word man, belongs to Sokrates or not; he ought not to introduce the mention of these accidental predicates, to which the question did not refer. These accidental predicates are infinite in number; he cannot enumerate them all, and therefore he ought not to introduce the mention of any of them. Sokrates is homo, by the essence and definition of the word; he is non-homo, ten thousand times over, by accidental predicates; that is, he is fair, brave, musical, flat-nosed, &c., all of which are varieties of the general word non-homo (p. 1007, a. 7-19).
Those who contend that both members of the Antiphasis are at once true disallow Essentia altogether, and the distinction between it and Accidens (p. 1007, a. 21). When we say that the word homo signifies a certain Essentia, we mean that its Essentia is nothing different from this, and that the being homo cannot be the same as the being non-homo, or the not being homo. Those against whom we are reasoning discard Essentia as distinguished from Accidens, and consider all predicates as Accidentia. Albus belongs to homo as an accident; but the essence of albus does not coincide with that of homo, and cannot be predicated of homo (a. 32). Upon the theory of these opponents, there would be no Prima Essentia to which all accidents are attached; but this theory is untenable. Accidents cannot be attached one to another in an infinite ascending series (b. 1). You cannot proceed more than two steps upward: first one accident, then a second; the two being joined by belonging to one and the same subject. No accident can be the accident of another accident. Τὸ λευκόν may have the accident μουσικόν, or τὸ μουσικόν may have the accident λευκόν; each of these may be called indifferently the accident of the other; but the truth is, that λευκός and μουσικός are both of them accidents belonging to the common Essentia — homo. But, when we affirm homo est musicus, we implicate the accident with the Essentia to which it belongs; that Essentia is signified by the subject homo. There must thus be one word which has signification as Essentia; and, when such is the case, we have already shown that both members of the Antiphasis cannot be predicated at once (b. 5-18).
(Alexander, in Scholia, p. 658, b. 40-p. 659, b. 14, Br., remarks on this argument of Aristotle: Those who held the opinion here controverted by Aristotle — τὴν ἀντίφασιν συναληθεύειν — had in their minds accidental propositions, in regard to which they were right, except that both members of the Antiphasis cannot be true at the same time. Sokrates est musicus — Sokrates non est musicus: these two propositions are both true, in the sense that one or other of them is true only potentially, and that both cannot be actually true at the same time. One of them is true, and the other false, at the present moment; but that which is now false has been true in the past, and may become true in the future. Aristotle does not controvert this theory so far as regards accidental propositions; but he maintains that it is untenable about essential propositions, and that the theorists overlooked this distinction.)
Moreover, if you say that both members of the Antiphasis are alike true respecting every predicate of a given subject, you must admit that all things are one (p. 1007, b. 20). The same thing will be at once a wall, a trireme, a man. Respecting every subject, you may always either affirm or deny any given predicate; but, according to this theory, whenever it is true to affirm, it is always equally true to deny. If you can say truly, Homo non est triremis, you may say with equal truth, according to the theory before us, Homo est triremis. And, of course, Homo non est triremis may be said truly; since (still according to this theory) the much more special negative, Homo non est homo, may be said truly (b. 32).
Again, if this theory be admitted, the doctrine that every predicate may be either affirmed or denied of any given subject, will no longer hold true. For, if it be true to say of Sokrates both Est homo and Est non-homo: it must also be true to say of him both Non est homo and Non est non-homo. If both affirmative and negative may be alike affirmed, both may be alike denied (p. 1008, a. 2-7). If both members of the Antiphasis are alike true, both must be alike false (Alex. Schol. p. 663, a. 14-34).