Since the science of the philosopher is concerned with Being qua Being universally,¹ and not with some part of it, and since the term Being has several meanings and is not used only in one sense, if it is merely equivocal and has no common significance it cannot fall under one science (for there is no one class in things of this kind); but if it has a common significance it must fall under one science. Now it would seem that it is used in the sense which we have described, like “medical” and “healthy,” for we use each of these terms in several senses; 1061aand each is used in this way because it has a reference, one to the science of medicine, and another to health, and another to something else; but each refers always to the same concept. A diagnosis and a scalpel are both called medical, because the one proceeds from medical science and the other is useful to it. The same is true of “healthy”; one thing is so called because it is indicative, and another because it is productive, of health; and the same applies to all other cases. Now it is in this same way that everything which exists is said to be ; each thing is said to be because it is a modification or permanent or temporary state or motion or some other such affection of Being qua Being. And since everything that is can be referred to some one common concept, each of the contrarieties too can be referred to the primary differentiae and contrarieties of Being—whether the primary differentiae of Being are plurality and unity, or similarity and dissimilarity, or something else; for we may take them as already discussed.² It makes no difference whether that which is is referred to Being or Unity; for even if they are not the same but different, they are in any case convertible, since that which is one also in a sense is , and that which is is one. Now since the study of contraries pertains to one and the same science, and each contrary is so called in virtue of privation (although indeed one might wonder in what sense they can be called contraries in virtue of privation when they admit of a middle term—e.g. “unjust” and “just”), in all such cases we must regard the privation as being not of the whole definition but of the ultimate species. E.g., if the just man is “one who is obedient to the laws in virtue of some volitional state,” the unjust man will not be entirely deprived of the whole definition, but will be “one who is in some respect deficient in obedience to the laws”; and it is in this respect that the privation of justice will apply to him (and the same holds good in all other cases). And just as the mathematician makes a study of abstractions (for in his investigations he first abstracts everything that is sensible, such as weight and lightness, hardness and its contrary, and also heat and cold and all other sensible contrarieties, leaving only quantity and continuity—sometimes in one, sometimes in two and sometimes in three dimensions—and their affections qua quantitative and continuous, and does not study them with respect to any other thing; and in some cases investigates the relative positions of things and the properties of these, 1061band in others their commensurability or incommensurability, and in others their ratios; yet nevertheless we hold that there is one and the same science of all these things, viz. geometry), so it is the same with regard to Being. For the study of its attributes in so far as it is Being, and of its contrarieties³ qua Being, belongs to no other science than Philosophy; for to physics one would assign the study of things not qua Being but qua participating in motion, while dialectics and sophistry deal with the attributes of existing things, but not of things qua Being, nor do they treat of Being itself in so far as it is Being. Therefore it remains that the philosopher is the man who studies the things which we have described, in so far as they are Being. And since everything that is , although the term has several meanings, is so described in virtue of some one common concept, and the same is true of the contraries (since they can be referred to the primary contrarieties and differences of Being), and since things of this kind can fall under one science, the difficulty which we stated at the beginning⁴ may be regarded as solved⁵—I mean the problem as to how there can be one science of several things which are different in genus.

Since even the mathematician uses the common axioms only in a particular application, it will be the province of Primary Philosophy to study the principles of these as well.⁶ That when equals are taken from equals the remainders are equal is an axiom common to all quantities; but mathematics isolates a particular part of its proper subject matter and studies it separately; e.g. lines or angles or numbers or some other kind of quantity, but not qua Being, but only in so far as each of them is continuous in one, two or three dimensions. But philosophy does not investigate particular things in so far as each of them has some definite attribute, but studies that which is , in so far as each particular thing is . The same applies to the science of physics as to mathematics, for physics studies the attributes and first principles of things qua in motion, and not qua Being; but Primary Science, as we have said, deals with these things only in so far as the subjects which underlie them are existent, and not in respect of anything else. Hence we should regard both physics and mathematics as subdivisions of Wisdom.

There is a principle in existing things about which we cannot make a mistake⁷; of which, on the contrary, we must always realize the truth—viz. that the same thing cannot at one and the same time be and not be, 1062anor admit of any other similar pair of opposites. Of such axioms although there is a proof ad hominem, there is no absolute proof; because there is no principle more convincing than the axiom itself on which to base an argument, whereas there must be such a principle if there is to be absolute proof. But he who wants to convince an opponent who makes opposite statements that he is wrong must obtain from him an admission which shall be identical with the proposition that the same thing cannot at one and the same time be and not be, but shall seem not to be identical with it. This is the only method of proof which can be used against one who maintains that opposite statements can be truly made about the same subject. Now those who intend to join in discussion must understand one another to some extent; for without this how can there be any common discussion between them? Therefore each of the terms which they use must be intelligible and signify something; not several things, but one only; or if it signifies more than one thing, it must be made clear to which of these the term is applied. Now he who says that A is and is not denies what he asserts, and therefore denies that the term signifies what it does signify. But this is impossible. Therefore if “to be so-and-so” has a definite meaning, the opposite statement about the same subject cannot be true. Again, if the term has a definite significance and this is truly stated, it must of necessity be so.⁸ But that which of necessity is can never not be. Hence opposite statements about the same subject cannot be true. Again, if the assertion is no more true than the negation, it will be no more true to say “A is man” than to say “A is not man.” ⁹ But it would also be admitted that it is more or at least not less true to say that a man is not a horse than to say that he is not a man; and therefore, since it was assumed that opposite statements are equally true, it will be true to say that the same person is also a horse. It follows therefore, that the same person is a man and a horse, or any other animal. Thus, although there is no absolute proof of these axioms, there is an ad hominem proof where one’s opponent makes these assumptions.¹⁰ Perhaps even Heraclitus himself, if he had been questioned on these lines, would have been compelled to admit that opposite statements can never be true of the same subjects; as it is, he adopted this theory through ignorance of what his doctrine implied. In general,¹¹ if what he says is true, not even this statement itself 1062b(I mean “that the same thing can at one and the same time be and not be”) will be true; because just as, when they are separated, the affirmation is no more true than the negation, so in the same way, if the complex statement is taken as a single affirmation, the negation will be just as true as the whole statement regarded as an affirmation. And further, if nothing can be truly affirmed, then this very statement—that there is no such thing as a true affirmation—will be false. But if there is such a thing, the contentions of those who raise objections of this kind and utterly destroy rational discourse may be considered to be refuted.¹²