LESSON 5

The Principle of Non-Contradiction

  Chapters 5 & 6: 1061b 34-1062b 19

             934. There is a principle in existing things about which it is impossible to make a mistake, but of which one must always do the contrary, I mean acknowledge it as true, namely, that the same thing cannot both be and not be at one and the same time; and the same is also true of other things which are opposed in this way (326-328).

             935. And while there is no demonstration in the strict sense of such principles, one may employ an argument ad hominem; for it is impossible to construct a syllogism from a more certain principle than this one. But this would be necessary if there were demonstration of it in the strict sense (329-331).

             936. Now anyone who wants to prove to an opponent making statements opposite to one's own that he is wrong must take some such principle which is the same as this one--that the same thing cannot both be and not be at the same time--but apparently is not the same. For this will be the only method of demonstration that can be used against one who says that opposite statements can be truly made about the same subject.

             937. Accordingly, those who are to join in some discussion must understand each other to some extent. And if this does not happen, how will they join in a common discussion? Therefore each of the terms used must be understood and must signify something, and not many things but only one. But if a term does signify many things, it must be made clear to which of these it refers. Hence, one who says that this is and is not, totally denies what he affirms, and thus denies that the term signifies what it signifies. But this is impossible. Hence, if to be this has some meaning, the contradictory cannot be said to be true of the same subject (332-340).

             938. Again, if a term signifies something and this is affirmed truly, it must necessarily be so; and what is necessarily so cannot not be. Hence opposite affirmations and negations cannot be true of the same subject (337-338).

             939. Again, if the affirmation is in no way truer than the negation, it will not be truer to say that something is a man than to say that it is not a man. And it would also seem that it is either more or not less true to say that a man is not a horse than to say that he is not a man. Hence one will also be right in saying that the same thing is a horse; for it was assumed that opposite statements are equally true. Therefore it follows that the same thing is a man and a horse, or any other animal (343-345). Hence, while there is no demonstration in the strict sense of these principles, there is still a demonstration ad hominem against one who makes these assumptions.

             940. And perhaps if one had questioned Heraclitus himself in this way, he would quickly have forced him to admit that opposite statements can never be true of the same subjects. But he adopted this view without understanding his own statement (328). And in general if what he said is true, not even this statement will be true--I mean that the same thing can both be and not be at one and the same time. For just as when they are separated the affirmation will not be truer than the negation (346), in a similar way when both are combined and taken together as though they were one affirmation, the negation will not be truer than the whole statement regarded as an affirmation.

             941. Again, if it is possible to affirm nothing truly, even this statement--that no affirmation is true--will be false (396-397). But if there is a true affirmation, this will refute what is said by those who raise such objections and completely destroy discussion.

Chapter 6

             942. The statement made by Protagoras is similar to those mentioned; for he said that man is the measure of all things, meaning simply that whatever appears so to anyone is just as it appears to him. But if this is true, it follows that the same thing is and is not, and is good and evil, and that other statements involving opposites are true; because often a particular thing appears to be good to some and just the opposite to others, and that which appears to each man is the measure.

COMMENTARY

             2211. Having shown that a study of the common principles of demonstration belongs chiefly to the consideration of this philosophical science, the Philosopher now deals with the first of these principles (934:C 2212). For just as all beings must be referred to one first being, in a similar fashion all principles of demonstration must be referred to some principle which pertains in a more basic way to the consideration of this philosophical science. This principle is that the same thing cannot both be and not be at the same time. It is the first principle because its terms, being and non-being, are the first to be apprehended by the intellect.

             2212. This part is divided into two members. In the first (934:C 2211) he establishes the truth of this principle. In the second (936:C 2214) he rejects an error ("Now anyone who").

             In reference to the first part he does two things regarding this principle. First, he says that in regard to beings there is a principle of demonstration "about which it is impossible to make a mistake" (i.e., so far as its meaning is concerned), but of which we "must always do the contrary," namely, acknowledge it as true. This principle is that the same thing cannot both be and not be at one and the same time, granted of course that the other conditions which it is customary to give in the case of a contradiction are fulfilled, namely, in the same respect, in an unqualified sense, and the like. For no one can think that this principle is false, because, if someone were to think that contradictories may be true at the same time, he would then have contrary opinions at the same time; for opinions about contradictories are contrary. For example, the opinion that "Socrates is sitting" is contrary to the opinion that "Socrates is not sitting."

             2213. And while (935).

             Second, he says that, while there cannot be demonstration in the strict sense of the above-mentioned principle and other similar ones, one may offer an argument ad hominem in support of it. That it cannot be demonstrated in the strict sense he proves thus: no one can prove this principle by constructing a syllogism from some principle which is better known. But such would be necessary if that principle were to be demonstrated in the strict sense. However, this principle can be demonstrated by using an argument ad hominem against one who admits some other statement, though less known, and denies this one.

             2214. Now anyone who (936).

             Then he rejects the opinion of those who deny this principle; and this is divided into two parts. First (936:C 2214), he argues against those who deny this principle. Second (943:C 2225), he shows how one can meet this opinion ("Now this difficulty").

             In regard to the first he does two things. First (936:C 2214), he argues against those who unqualifiedly deny this principle. Second (940:C 2221), he turns his attention to certain particular opinions ("And perhaps").

             In regard to the first he does two things. First, he gives the method of arguing against this error. He says that in arguing against an opponent who claims that contradictory propositions may be true, anyone who wants to show that this opinion is false ought to take some such principle which is the same as this one--that the same thing cannot both be and not be at the same time--but apparently is not the same. For, if it were evidently the same, it would not be admitted by an opponent. Yet if it were not the same, he could not prove his thesis, because a principle of this kind cannot be demonstrated from some principle which is better known. Hence, it is only in this way that a demonstration can be made against those who say that contradictories may be true of the same subject, namely, by assuming as a premise what is in fact the same as the conclusion but apparently is not.

             2215. Accordingly (937).

             Second, he begins to argue dialectically against the above-mentioned error; and in regard to this he gives three arguments. First, he argues as follows: if two men are to join in a discussion in such a way that one may communicate his view to the other in a dispute, each must understand something that the other is saying. For if this were not the case, no statement would be understood by both of them; and thus an argument with an opponent would be pointless.

             2216. However, if one of them is to understand what the other is saying, each of the terms used must be understood according to its proper meaning and must therefore signify some one thing and not many things. And if it should signify many, it will be necessary to make clear which of the many things it signifies; otherwise one would not know what the other person means.

             2217. Now granted that a term signifies one thing, it is evident that one who says both that this is and that this is not, for example, that Socrates is a man and that he is not a man, denies the one thing which he attributed to Socrates, namely, that he is a man, when he adds that he is not a man; and thus he denies what he first signified. Hence it follows that a word does not signify what it signifies. But this is impossible. Consequently, if a term signifies some definite thing, the contradictory cannot be truly affirmed of the same subject.

             2218. Again, if a term (938).

             Then he gives the second argument, which runs as follows: if a term signifies some attribute, and the attribute signified by the term is truly affirmed of the same subject of which the term is first predicated, this attribute must belong to the subject of which the term is predicated so long as the proposition is true. For this conditional proposition, "If Socrates is a man, Socrates is a man," is clearly true. Now every true conditional proposition is a necessary one. Hence, if the consequent is true, the antecedent must be true. But what is, cannot sometimes not be, because to be necessary and to be incapable of not being are equivalent. Therefore so long as the proposition "Socrates is a man" is true, the proposition "Socrates is not a man" cannot be true. Thus it is evident that opposite affirmations and negations cannot be true of the same subject at the same time.

             2219. Again, if the affirmation (939).

             Then he gives the third argument, which is as follows: if an affirmation is not truer than the negation opposed to it, one who says that Socrates is a man does not speak with greater truth than one who says that Socrates is not a man. But it is evident that one who says that a man is not a horse speaks either with greater or with no less truth than one who says that a man is not a man. Hence, according to this argument, he who says that a man is not a horse will speak with equal or no less truth. But if contradictory opposites are true at the same time, for example, if the proposition "Man is not a horse" is true, and the proposition "Man is a horse" is also true, then it follows that a man is a horse and also any other animal.

             2220. But because someone could criticize the foregoing arguments on the grounds that the things assumed in them are less known than the intended conclusion, he therefore answers this by saying that no one of the foregoing arguments is demonstrative in the strict sense, although there can be an argument ad hominem against an opponent who gives this argument, because the things assumed must be admitted to be true even though they are less known absolutely than what he denies.

             2221. And perhaps (940).

             Then he rejects the above error by considering certain particular thinkers. He does this, first (940:C 2221), with regard to Heraclitus; and second (942:C 2224), with regard to Protagoras ("The statement").

             Now Heraclitus posited two things: first, that an affirmation and a negation may be true at the same time (and from this it would follow that every proposition, affirmative as well as negative, is true); and second, that there may be an intermediate between affirmation and negation (and from this it would follow that neither an affirmation nor a negation can be true). Consequently every proposition is false.

             2222. First (940:C 2222), he raises an argument against Heraclitus' first position; and second (941:C 2223), against his second position ("Again, if it is possible").

             He accordingly says, first (940), that by giving an argument ad hominem in this way one may easily bring even Heraclitus, who was the author of this statement, to admit that opposite propositions may not be true of the same subject. For he seems to have accepted the opinion that they may be true of the same subject because he did not understand his own statement. And he would be forced to deny his statement in the following way: if what he said is true, namely, that one and the same thing can both be and not be at one and the same time, it follows that this very statement will not be true; for if an affirmation and a negation are taken separately, an affirmation is not truer than a negation; and if an affirmation and a negation are taken together in such a way that one affirmation results from them, the negation will not be less true of the whole statement made up of the affirmation and the negation than of the opposite affirmation. For it is clearly possible for some copulative proposition to be true, just as for some simple proposition; and it is possible to take its negation. And whether the copulative proposition be composed of two affirmative propositions, as when we say "Socrates is sitting and arguing," or of two negative propositions, as when we say "It is true that Socrates is not a stone or an ass," or of an affirmative proposition and a negative proposition, as when we say "It is true that Socrates is sitting and not arguing," nevertheless a copulative proposition is always taken to be true because one affirmative proposition is true. And he who says that it is false takes the negation as applying to the whole copulative proposition. Hence he who says that it is true that man is and is not at the same time, takes this as a kind of affirmation; and that this is not true is the negation of this. Hence, if an affirmation and a negation are true at the same time, it follows that the negation which states that this is not true, i.e., that an affirmation and a negation are true at the same time, is equally true. For if any negation is true at the same time as the affirmation opposed to it, every negation must be true at the same time as the affirmation opposed to it; for the reasoning is the same in all cases.

             2223. Again, if it is possible (941).

             Then he introduces an argument against the second position of Heraclitus: that no affirmation is true. For if it is possible to affirm that nothing is true, and if one who says that no affirmation is true does affirm something, namely, that it is true that no affirmation is true, then this statement will be false. And if some affirmative statement is true, the opinion of people such as those who oppose all statements will be rejected. And those who adopt this position destroy the whole debate, because if nothing is true, nothing can be conceded on which an argument may be based. And if an affirmation and a negation are true at the same time, it will be impossible to signify anything by a word, as was said above (937:C 2215), and then the argument will cease.

             2224. The statement (942).

             Here he considers the opinion of Protagoras. He says that the statement made by Protagoras is similar to the one made by Heraclitus and by others who claim that an affirmation and a negation are true at the same time. For Protagoras says that man is the measure of all things, i.e., according to the intellect and the senses, as has been explained in Book IX (753:C 1800), as if the being of a thing depended upon intellectual and sensory apprehension. And one who says that man is the measure of all things merely says that whatever appears so to anyone is true. But if this is maintained, it follows that the same thing both is and is not and is both good and evil at the same time. The same thing is also true of other opposites, because often something seems to be good to some and just the opposite to others, and the way in which things seem or appear is the measure of all things according to the opinion of Protagoras; so that, inasmuch as a thing appears, to that extent it is true.