Aristotle, Metaphysics

[1012a] [1] Further, the understanding either affirms or denies every object of understanding or thought (as is clear from the definition¹）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.²

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 something³; for its essence is something distinct.

Again, when a man is asked whether a thing is white and says "no," he has denied nothing except that it is <white>, and its not-being <white> is a negation.

Now 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. [20] 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.⁴ The doctrine of Heraclitus, which says that everything is and is not,⁵ seems to make all things true; and that of Anaxagoras⁶ seems to imply an intermediate in contradiction, 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.

It is obvious from this analysis that the one-sided and sweeping statements which some people make cannot be substantially true—some maintaining that nothing is true (for they say that there is no reason why the same rule should not apply to everything as applies to the commensurability of the diagonal of a square⁷), and some that everything is true.These theories are almost the same as that of Heraclitus. For the theory which says that all things are true and all false also makes each of these statements separately;