We must pronounce whether it pertains to the same science to study both the so-called axioms in mathematics and substance, or to different sciences. It is obvious that the investigation of these axioms too pertains to one science, namely the science of the philosopher; for they apply to all existing things, and not to a particular class separate and distinct from the rest. Moreover all thinkers employ them—because they are axioms of Being qua Being, and every genus possesses Being— but employ them only in so far as their purposes require; i.e., so far as the genus extends about which they are carrying out their proofs. Hence since these axioms apply to all things qua Being (for this is what is common to them), it is the function of him who studies Being qua Being to investigate them as well. For this reason no one who is pursuing a particular inquiry—neither a geometrician nor an arithmetician—attempts to state whether they are true or false; but some of the physicists did so, quite naturally; for they alone professed to investigate nature as a whole, and Being. But inasmuch as there is a more ultimate type of thinker than the natural philosopher (for nature is only a genus of Being), the investigation of these axioms too will belong to the universal thinker who studies the primary reality. 1005bNatural philosophy is a kind of Wisdom, but not the primary kind. As for the attempts of some of those who discuss how the truth should be received, they are due to lack of training in logic; for they should understand these things before they approach their task, and not investigate while they are still learning. Clearly then it is the function of the philosopher, i.e. the student of the whole of reality in its essential nature, to investigate also the principles of syllogistic reasoning. And it is proper for him who best understands each class of subject to be able to state the most certain principles of that subject; so that he who understands the modes of Being qua Being should be able to state the most certain principles of all things. Now this person is the philosopher, and the most certain principle of all is that about which one cannot be mistaken; for such a principle must be both the most familiar (for it is about the unfamiliar that errors are always made), and not based on hypothesis. For the principle which the student of any form of Being must grasp is no hypothesis; and that which a man must know if he knows anything he must bring with him to his task. Clearly, then, it is a principle of this kind that is the most certain of all principles. Let us next state what this principle is. “It is impossible for the same attribute at once to belong and not to belong to the same thing and in the same relation”; and we must add any further qualifications that may be necessary to meet logical objections. This is the most certain of all principles, since it possesses the required definition; for it is impossible for anyone to suppose that the same thing is and is not, as some imagine that Heraclitus says¹—for what a man says does not necessarily represent what he believes. And if it is impossible for contrary attributes to belong at the same time to the same subject (the usual qualifications must be added to this premiss also), and an opinion which contradicts another is contrary to it, then clearly it is impossible for the same man to suppose at the same time that the same thing is and is not; for the man who made this error would entertain two contrary opinions at the same time. Hence all men who are demonstrating anything refer back to this as an ultimate belief; for it is by nature the starting-point of all the other axioms as well.

There are some, however, as we have said, who both state themselves that the same thing can be and not be, 1006aand say that it is possible to hold this view. Many even of the physicists adopt this theory. But we have just assumed that it is impossible at once to be and not to be, and by this means we have proved that this is the most certain of all principles. Some, indeed, demand to have the law proved, but this is because they lack education²; for it shows lack of education not to know of what we should require proof, and of what we should not. For it is quite impossible that everything should have a proof; the process would go on to infinity, so that even so there would be no proof.³ If on the other hand there are some things of which no proof need be sought, they cannot say what principle they think to be more self-evident. Even in the case of this law, however, we can demonstrate the impossibility by refutation, if only our opponent makes some statement. If he makes none, it is absurd to seek for an argument against one who has no arguments of his own about anything, in so far as he has none; for such a person, in so far as he is such, is really no better than a vegetable. And I say that proof by refutation differs from simple proof in that he who attempts to prove might seem to beg the fundamental question, whereas if the discussion is provoked thus by someone else, refutation and not proof will result. The starting-point for all such discussions is not the claim that he should state that something is or is not so (because this might be supposed to be a begging of the question), but that he should say something significant both to himself and to another (this is essential if any argument is to follow; for otherwise such a person cannot reason either with himself or with another); and if this is granted, demonstration will be possible, for there will be something already defined. But the person responsible is not he who demonstrates but he who acquiesces; for though he disowns reason he acquiesces to reason. Moreover, he who makes such an admission as this has admitted the truth of something apart from demonstration [so that not everything will be “so and not so”]. Thus in the first place it is obvious that this at any rate is true: that the term “to be” or “not to be” has a definite meaning; so that not everything can be “so and not so.” Again, if “man” has one meaning, let this be “two-footed animal.” By “has one meaning” I mean this: if X means “man,” then if anything is a man, its humanity will consist in being X. And it makes no difference even if it be said that “man” has several meanings, provided that they are limited in number; 1006bfor one could assign a different name to each formula. For instance, it might be said that “man” has not one meaning but several, one of which has the formula “two-footed animal,” and there might be many other formulae as well, if they were limited in number; for a particular name could be assigned to each for formula. If on the other hand it be said that “man” has an infinite number of meanings, obviously there can be no discourse; for not to have one meaning is to have no meaning, and if words have no meaning there is an end of discourse with others, and even, strictly speaking, with oneself; because it is impossible to think of anything if we do not think of one thing; and even if this were possible, one name might be assigned to that of which we think. Now let this name, as we said at the beginning, have a meaning; and let it have one meaning. Now it is impossible that “being man” should have the same meaning as “not being man,” that is, if “man” is not merely predicable of one subject but has one meaning (for we do not identify “having one meaning” with “being predicable of one subject,” since in this case “cultured” and “white” and “man” would have one meaning, and so all things would be one; for they would all have the same meaning). And it will be impossible for the same thing to be and not to be, except by equivocation, as e.g. one whom we call “man” others might call “not-man”; but the problem is whether the same thing can at once be and not be “man,” not in name , but in fact . If “man” and “not-man” have not different meanings, clearly “not being a man” will mean nothing different from “being a man”; and so “being a man” will be “not being a man”; they will be one. For “to be one” means, as in the case of “garment” and “coat,” that the formula is one. And if “being man” and “being not-man” are to be one, they will have the same meaning; but it has been proved above that they have different meanings. If then anything can be truly said to be “man,” it must be “two-footed animal”; for this is what “man” was intended to mean. And if this is necessarily so, it is impossible that at the same time the same thing should not be “two-footed animal.” For “to be necessarily so” means this: that it is impossible not to be so. Thus it cannot be true to say at the same time that the same thing is and is not man. And the same argument holds also in the case of not being man; 1007abecause “being man” and “being not-man” have different meanings if “being white” and “being man” have different meanings (for the opposition is much stronger in the former case so as to produce different meanings). And if we are told that “white” too means one and the same thing,⁴ we shall say again just what we said before,⁵ that in that case all things, and not merely the opposites, will be one. But if this is impossible, what we have stated follows; that is, if our opponent answers our question; but if when asked the simple question he includes in his answer the negations, he is not answering our question. There is nothing to prevent the same thing from being “man” and “white” and a multitude of other things; but nevertheless when asked whether it is true to say that X is man, or not, one should return an answer that means one thing, and not add that X is white and large. It is indeed impossible to enumerate all the infinity of accidents; and so let him enumerate either all or none. Similarly therefore, even if the same thing is ten thousand times “man” and “not-man,” one should not include in one’s answer to the question whether it is “man” that it is at the same time also “not-man,” unless one is also bound to include in one’s answer all the other accidental things that the subject is or is not. And if one does this, he is not arguing properly. In general those who talk like this do away with substance and essence, for they are compelled to assert that all things are accidents, and that there is no such thing as “being essentially man” or “animal.” For if there is to be such a thing as “being essentially man,” this will not be “being not-man” nor “not-being man” (and yet these are negations of it); for it was intended to have one meaning, i.e. the substance of something. But to denote a substance means that the essence is that and nothing else; and if for it “being essentially man” is the same as either “being essentially not-man” or “essentially not-being man,” the essence will be something else. Thus they are compelled to say that nothing can have such a definition as this, but that all things are accidental; for this is the distinction between substance and accident: “white” is an accident of “man,” because although he is white, he is not white in essence. And since the accidental always implies a predication about some subject, if all statements are accidental, there will be nothing primary about which they are made; 1007bso the predication must proceed to infinity. But this is impossible, for not even more than two accidents can be combined in predication. An accident cannot be an accident of an accident unless both are accidents of the same thing. I mean, e.g., that “white” is “cultured” and “cultured” “white” merely because both are accidents of a man. But it is not in this sense—that both terms are accidents of something else—that Socrates is cultured. Therefore since some accidents are predicated in the latter and some in the former sense, such as are predicated in the way that “white” is of Socrates cannot be an infinite series in the upper direction; e.g. there cannot be another accident of “white Socrates,” for the sum of these predications does not make a single statement. Nor can “white” have a further accident, such as “cultured”; for the former is no more an accident of the latter than vice versa; and besides we have distinguished that although some predicates are accidental in this sense, others are accidental in the sense that “cultured” is to Socrates; and whereas in the former case the accident is an accident of an accident, it is not so in the latter; and thus not all predications will be of accidents. Therefore even so there will be something which denotes substance. And if this is so, we have proved that contradictory statements cannot be predicated at the same time. Again, if all contradictory predications of the same subject at the same time are true, clearly all things will be one. For if it is equally possible either to affirm or deny anything of anything, the same thing will be a trireme and a wall and a man; which is what necessarily follows for those who hold the theory of Protagoras.⁶ For if anyone thinks that a man is not a trireme, he is clearly not a trireme; and so he also is a trireme if the contradictory statement is true. And the result is the dictum of Anaxagoras, “all things mixed together”⁷; so that nothing truly exists. It seems, then, that they are speaking of the Indeterminate; and while they think that they are speaking of what exists, they are really speaking of what does not; for the Indeterminate is that which exists potentially but not actually. But indeed they must admit the affirmation or negation of any predicate of any subject, for it is absurd that in the case of each term its own negation should be true, and the negation of some other term which is not true of it should not be true. I mean, e.g., that if it is true to say that a man is not a man, it is obviously also true to say that he is or is not a trireme. Then if the affirmation is true, so must the negation be true; but if the affirmation is not true the negation will be even truer than the negation of the original term itself. 1008aTherefore if the latter negation is true, the negation of “trireme” will also be true; and if this is true, the affirmation will be true too. And not only does this follow for those who hold this theory, but also that it is not necessary either to affirm or to deny a statement. For if it is true that X is both man and not-man, clearly he will be neither man nor not-man; for to the two statements there correspond two negations, and if the former is taken as a single statement compounded out of two, the latter is also a single statement and opposite to it. Again, either this applies to all terms, and the same thing is both white and not-white, and existent and non-existent, and similarly with all other assertions and negations; or it does not apply to all, but only to some and not to others. And if it does not apply to all, the exceptions will be admitted⁸; but if it does apply to all, again either (a) the negation will be true wherever the affirmation is true, and the affirmation will be true wherever the negation is true, or (d) the negation will be true wherever the assertion is true, but the assertion will not always be true where the negation is true. And in the latter case there will be something which definitely is not, and this will be a certain belief; and if that it is not is certain and knowable, the opposite assertion will be still more knowable. But if what is denied can be equally truly asserted, it must be either true or false to state the predicates separately and say, e.g., that a thing is white, and again that it is not-white. And if it is not-true to state them separately, our opponent does not say what he professes to say, and nothing exists; and how can that which does not exist speak or walk?⁹ And again all things will be one, as we said before,¹⁰ and the same thing will be “man” and “God” and “trireme” and the negations of these terms. For if it is equally possible to assert or deny anything of anything, one thing will not differ from another; for if anything does differ, it will be true and unique. And similarly even if it is possible to make a true statement while separating the predicates, what we have stated follows. Moreover it follows that all statements would be true and all false; and that our opponent himself admits that what he says is false. Besides, it is obvious that discussion with him is pointless, because he makes no real statement. For he says neither “yes” nor “no,” but “yes and no”; and again he denies both of these and says “neither yes nor no”; otherwise there would be already some definite statement. Again, if when the assertion is true the negation is false, and when the latter is true the affirmation is false, it will be impossible to assert and deny with truth the same thing at the same time. 1008bBut perhaps it will be said that this is the point at issue. Again, is the man wrong who supposes that a thing is so or not so, and he who supposes both right? If he is right, what is the meaning of saying that “such is the nature of reality”?¹¹ And if he is not right, but is more right than the holder of the first view, reality will at once have a definite nature, and this will be true, and not at the same time not-true. And if all men are equally right and wrong, an exponent of this view can neither speak nor mean anything, since at the same time he says both “yes” and “no.” And if he forms no judgement, but “thinks” and “thinks not” indifferently, what difference will there be between him and the vegetables? Hence it is quite evident that no one, either of those who profess this theory or of any other school, is really in this position. Otherwise, why does a man walk to Megara and not stay at home, when he thinks he ought to make the journey? Why does he not walk early one morning into a well or ravine, if he comes to it, instead of clearly guarding against doing so, thus showing that he does not think that it is equally good and not good to fall in? Obviously then he judges that the one course is better and the other worse. And if this is so, he must judge that one thing is man and another not man, and that one thing is sweet and another not sweet. For when, thinking that it is desirable to drink water and see a man, he goes to look for them, he does not look for and judge all things indifferently; and yet he should, if the same thing were equally man and not-man. But as we have said, there is no one who does not evidently avoid some things and not others. Hence, as it seems, all men form unqualified judgements, if not about all things, at least about what is better or worse. And if they do this by guesswork and without knowledge, they should be all the more eager for truth; just as a sick man should be more eager for health than a healthy man; for indeed the man who guesses, as contrasted with him who knows, is not in a healthy relation to the truth. Again, however much things may be “so and not so,” yet differences of degree are inherent in the nature of things. For we should not say that 2 and 3 are equally even; nor are he who thinks that 4 is 5, and he who thinks it is 1000, equally wrong: hence if they are not equally wrong, the one is clearly less wrong, and so more right. If then that which has more the nature of something is nearer to that something, 1009athere will be some truth to which the more true is nearer. And even if there is not, still there is now something more certain and true, and we shall be freed from the undiluted doctrine which precludes any mental determination.