Plato’s view of the negative is erroneous. Logical maxim of contradiction.
The view which Plato here takes of the negative deserves the greater notice, because, if it were adopted, what is called the maxim of contradiction would be divested of its universality. Given a significant proposition with the same subject and the same predicate, each taken in one and the same signification — its affirmative and its negative cannot both be true. But if by the negative, you mean to make a new affirmation, different from that contained in the affirmative — the maxim just stated cannot be broadly maintained as of universal application: it may or may not be valid, as the case happens to stand. The second affirmation may be, as a matter of fact, incompatible with the first: but this is not to be presumed, from the mere fact that it is different from the first: proof must be given of such incompatibility.
Examination of the illustrative propositions chosen by Plato — How do we know that one is true, the other false?
We may illustrate this remark by looking at the two propositions which Plato gives as examples of true and false. Theætêtus is sitting down — Theætêtus is flying. Both the examples are of affirmative propositions: and it seems clear that Plato, in all this reasoning, took no account of negative propositions: those which simply deny, affirming nothing. The second of these propositions (says Plato) affirms what is not, as if it were, respecting the subject But how do we know this to be so? In the form of the second proposition there is nothing to show it: there is no negation of any thing, but simply affirmation of a different positive attribute. Although it happens, in this particular case, that the two attributes are incompatible, and that the affirmation of the one includes the negation of the other — yet there is nothing in the form of either proposition to deny the other:— no formal incompatibility between them. Both are alike affirmative, with the same subject, but different predicates. These two propositions therefore do not serve to illustrate the real nature of the negative, which consists precisely in this formal incompatibility. The proper negative belonging to the proposition — Theætêtus is sitting down — would be, Theætêtus is not sitting down. Plato ought to maintain, if he followed out his previous argument, that Not-Sitting down is as good a Form as Sitting-down, and that it meant merely — Different from Sitting down. But instead of doing this Plato gives us a new affirmative proposition, which, besides what it affirms, conceals an implied negation of the first proposition. This does not serve to illustrate the purpose of his reasoning — which was to set up the formal negative as a new substantive attribute, different from its corresponding affirmative. As between the two, the maxim of contradiction applies: both cannot be true. But as between the two propositions given in Plato, that maxim has no application: they are two propositions with the same subject, but different predicates; which happen in this case to be, the one true, the other false — but which are not formally incompatible. The second is not false because it differs from the first; it has no essential connection with the first, and would be equally false, even if the first were false also.
The function of the negative is to deny. Now denial is not a species of affirmation, but the reversal or antithesis of affirmation: it nullifies a belief previously entertained, or excludes one which might otherwise be entertained, — but it affirms nothing. In particular cases, indeed, the denial of one thing may be tantamount to the affirmation of another: for a man may know that there are only two suppositions possible, and that to shut out the one is to admit the other. But this is an inference drawn in virtue of previous knowledge possessed and contributed by himself: another man without such knowledge would not draw the same inference, nor could he learn it from the negative proposition per se. Such then is the genuine meaning of the negative; from which Plato departs, when he tells us that the negative is a kind of affirmation, only affirming something different — and when he illustrates it by producing two affirmative propositions respecting the same subject, affirming different attributes, the one as matter of fact incompatible with the other.
Necessity of accepting the evidence of sense.
But how do we know that the first proposition — Theætêtus is sitting down — affirms what is:— and that the second proposition — Theætêtus is flying — affirms what is not? If present, our senses testify to us the truth of the first, and the falsehood of the second: if absent, we have the testimony of a witness, combined with our own past experience attesting the frequency of facts analogous to the one, and the non-occurrence of facts analogous to the other. When we make the distinction, then, — we assume that what is attested by sense or by comparisons and inductions from the facts of sense, is real, or is: and that what is merely conceived or imagined, without the attestation of sense (either directly or by way of induction), is not real, or is not. Upon this assumption Plato himself must proceed, when he takes it for granted, as a matter of course, that the first proposition is true, and the second false. But he forgets that this assumption contradicts the definition which, in this same dialogue,[132] he had himself given of Ens — of the real or the thing that is. His definition was so comprehensive, as to include not only all that could be seen or felt, but also all that had capacity to be known or conceived by the mind: and he speaks very harshly of those who admit the reality of things perceived, but refuse to admit equal reality to things only conceived. Proceeding then upon this definition, we can allow no distinction as to truth or falsehood between the two propositions — Theætêtus is sitting down — Theætêtus is flying: the predicate of the second affirms what is, just as much as the predicate of the first: for it affirms something which, though neither perceived nor perceivable by sense, is distinctly conceivable and conceived by the mind. When Plato takes for granted the distinction between the two, that the first affirms what is, and the second what is not — he unconsciously slides into that very recognition of the testimony of sense (in other words, of fact and experience), as the certificate of reality, which he had so severely denounced in the opposing materialist philosophers: and upon the ground of which he thought himself entitled, not merely to correct them as mistaken, but to reprove them as wicked and impudent.[133]
[132] Plato, Sophist. pp. 247 D-E, 248 D-E.
[133] Plato, Sophist. p. 246 D.
Errors of Antisthenes — depended partly on the imperfect formal logic of that day.