Book IV. Section V.
How people come to doubt the law of contradiction:
(i.) The belief that contradictions and contraries can be true at the same time comes to them from seeing the contraries generated from the same thing. Then if what is not cannot be generated, the thing must have existed before as both contraries equally — just as Anaxagoras says that everything is mixed in everything; and also Democritus, for he too says that Void and Plenum are present equally in any part, and yet the latter is, and the former is not.
The method of convincing them:
To those, then, who base their judgement on these considerations, we shall say that although in one sense their theory is correct, in another they are mistaken. For “being” has two meanings, so that there is a sense in which something can be generated from “not-being”, and a sense in which it cannot; and a sense in which the same thing can at once be and not be; but not in the same respect. For the same thing can “be” contraries at the same time potentially, but not actually. And further, we shall request them to conceive another kind also of substance of existing things, in which there is absolutely no motion or destruction or generation.
“But the reason why these men hold this view is that although they studied the truth about reality, they supposed that reality is confined to sensible things, in which the nature of the Indeterminate, i.e. of Being in the sense which we have explained, is abundantly present. (Thus their statements though plausible, are not true; this form of the criticism is more suitable than that which Epicharmus applied to Xenophanes.) And further, observing that all this indeterminate substance is in motion, and that no true predication can be made of that which changes, they supposed that it is impossible to make any true statement about that which is in all ways and entirely changeable. For it was from this supposition that there blossomed forth the most extreme view of those which we have mentioned, that of the professed followers of Heraclitus, and such as Cratylus held, who ended by thinking that one need not say anything, and only moved his finger; and who criticized Heraclitus for saying that one cannot enter the same river twice, for he himself held that it cannot be done even once.”
The method of refuting this view:
“But we shall reply to this theory also that although that which is changeable supplies them, when it changes, with some real ground for supposing that it “is not,” yet there is something debatable in this; for that which is shedding any quality retains something of that which is being shed, and something of that which is coming to be must already exist. And in general if a thing is ceasing to be, there will be something there which is; and if a thing is coming to be, that from which it comes and by which it is generated must be; and this cannot go on to infinity.”
“Let it be granted that there is nothing permanent in respect of quantity; but it is by the form that we recognize everything.”
Book IV. Section VI.
“That the most certain of all beliefs is that opposite statements are not both true at the same time, and what follows for those who maintain that they are true, and why these thinkers maintain this, may be regarded as adequately stated. And since the contradiction of a statement cannot be true at the same time of the same thing, it is obvious that contraries cannot apply at the same time to the same thing. For in each pair of contraries one is a privation no less than it is a contrary — a privation of substance. And privation is the negation of a predicate to some defined genus. Therefore if it is impossible at the same time to affirm and deny a thing truly, it is also impossible for contraries to apply to a thing at the same time; either both must apply in a modified sense, or one in a modified sense and the other absolutely.”
Book IV. Section VII.
Arguments to prove the law of the excluded middle:
Nor indeed can there be any intermediate between contrary statement, but of one thing we must either assert or deny one thing, whatever it may be. This will be plain if we first define truth and falsehood. To say what is is not, or that what is not is not, is true; and therefore also he who says that a thing is or is not will say either what is true or what is false. But neither what is nor what is not is said not to be or to be.
Further, an intermediate between contraries will be intermediate either as grey is between black and white, or as “neither man nor horse” is between man and horse. If in the latter sense, it cannot change (for change is from not-good to good, or from good to not-good); but in fact it is clearly always changing; for change an only be into the opposite and the intermediate. And if it is a true intermediate, in this case too there would be a kind of change into white not from not white; but in fact this is not seen. Further, the understanding either affirms or denies every object of understanding or thought (as is clear from the definition) whenever it is right or wrong. When, in asserting or denying, it combines the predicates in one way, it is right; when in the other it is wrong.
Again, unless it is maintained merely for argument’s sake, the intermediate must exist beside all contrary terms; so that one will say what is neither true nor false. And it will exist beside what is and what is not; so that there will be a form of change beside generation and destruction.
Again, there will also be an intermediate in all classes in which the negation of a term implies the contrary assertion; e.g., among numbers there will be a number which is neither odd nor not-odd. But this is impossible, as is clear from the definition.
Again, there will be an infinite progression, and existing things will be not only half as many again, but even more. For again it will be possible to deny the intermediate in reference both to its assertion and to its negation, and the result will be something; for its essence is something distinct.
Again, when a man is asked wither a thing is white and says “no,” he has denied nothing except that it is <white>, and its not-being <white> is a negation.
Not this view has occurred to certain people in just the same way as other paradoxes have also occurred; for when they cannot find a way out from eristic arguments, they submit to the argument and admit that the conclusion is true. Some, then, hold the theory for this kind of reason, and others because they require an explanation for everything. In dealing with all such persons the starting-point is from definition; and definition results from the necessity of their meaning something; because the formula, which their term implies, will be a definition. The doctrine of Heraclitus, which says that everything is and is not, seems to make all things true; and that of Anaxagoras seems to imply an intermediate in contraction, so that all things are false; for when things are mixed, the mixture is neither good nor not-good; and so no statement is true.
Aristotle with a Bust of Homer,