A somewhat different argument is implied by Sigwart when he says, “If A = mortal, where will justice, virtue, law, order, distance find a place? They are neither mortal beings, nor yet not-mortal beings, for they are not beings at all.” The answer seems clear. They are not-(mortal beings), and therefore not-A. As a rule, it is needless to exclude explicitly from a species what does not even belong to some higher genus. But the fact of the exclusion remains.
Granting then that in practice we rarely, if ever, employ a negative name except with reference to some proximate genus, we nevertheless hold that not-A is perfectly intelligible whatever the universe of discourse may be and however wide it may be. For it denotes in that universe whatever is not denoted by the corresponding positive name. Moreover in formal processes we should be unnecessarily hampered if not allowed to pass unreservedly from X is not A to X is not-A.[68]
[68] Writers who take the view which we are here criticising must in consistency deny the universal validity of the process of immediate inference called obversion. Thus Lotze, rightly on his own view, will not allow us to pass from spirit is not matter to spirit is not-matter ; in fact he rejects altogether the form of judgment S is not-P (Logic, § 40). Some writers, who follow Lotze on the general question here raised, appear to go a good deal further than he does, not merely disallowing such a proposition as virtue is not-blue but also such a proposition as virtue is not blue, on the ground that if we say “virtue is not blue,” there is no real predication, since the notion of colour is absolutely foreign to an unextended and abstract concept such as “virtue.” Lotze, however, expressly draws a distinction between the two forms S is non-Q and S is not Q, and tells us that “everything which it is wished to secure by the affirmative predicate non-Q is secured by the intelligible negation of Q” (Logic, § 72; cf. § 40). On the more extreme view it is wrong to say that Virtue is either blue or it is not blue ; but Lotze himself does not thus deny the universality of the law of excluded middle.
61 From this point of view attention may be called to the difference in ordinary use between such forms as unholy, immoral, discourteous and such forms as non-holy, non-moral, non-courteous. The latter may be used with reference to any universe of discourse, however extensive. But not so the former; in their case there is undoubtedly a restriction to some universe of discourse that is more or less limited in its range. We can, for example, speak of a table as non-moral, although we cannot speak of it as immoral. A want of recognition of this distinction may be partly responsible for the denial that any terms can properly be described as infinite or indefinite.[69]
[69] It should be added that in the ordinary use of language the negative prefix does not always make a term negative as here defined. Thus, as Mill points out, “the word unpleasant, notwithstanding its negative form, does not connote the mere absence of pleasantness, but a less degree of what is signified by the word painful, which, it is hardly necessary to say, is positive.” On the other hand, some names positive in form may, with reference to a limited universe of discourse, be negative in force; e.g., alien, foreign. Another example is the term Turanian, as employed in the science of language. This term has been used to denote groups lying outside the Aryan and Semitic groups, but not distinguished by any positive characteristics which they possess in common.
40. Contradictory Terms.—A positive name and the corresponding negative are spoken of as contradictory. We may define contradictory terms as a pair of terms so related that between them they exhaust the entire universe to which reference is made, whilst in that universe there is no individual of which both can be affirmed at the same time. It is desirable to repeat here that contradiction can exist primarily between 62 judgments or propositions only, so that as applied to terms or ideas the notion of contradiction must be interpreted with reference to predication. A and not-A are spoken of as contradictory because they cannot without contradiction be predicated together of the same subject. Thus it is in their exclusive character that they are termed contradictory; as between them exhausting the universe of discourse they might rather be called complementary.[70]
[70] Dr Venn (Empirical Logic, p. 191) distinguishes between formal contradictories and material contradictories, according as the relation in which the pair of terms stand to one another is or is not apparent from their mere form. Thus A and not-A are formal contradictories; so are human and non-human. Material contradictories, on the other hand, are not constructed “for the express purpose of indicating their mutual relation.” No formal contradiction, for example, is apparent between British and Foreign, or between British and Alien ; and yet “within their range of appropriate application—which in the latter case includes persons only, and in the former case is extended to produce of most kinds—these two pairs of terms fulfil tolerably well the conditions of mutual exclusion and collective exhaustion.”
41. Contrary Terms.—Two terms are usually spoken of as contrary[71] to one another when they denote things which can be regarded as standing at opposite ends of some definite scale in the universe to which reference is made; e.g., first and last, black and white, wise and foolish, pleasant and painful.[72] Contraries differ from contradictories in that they admit of a mean, and therefore do not between them exhaust the entire universe of discourse. It follows that, although two contraries cannot both be true of the same thing at the same time, they may both be false. Thus, a colour may be neither black nor 63 white, but blue; a feeling may be neither pleasant nor painful, but indifferent.
[71] De Morgan uses the terms contrary and contradictory as equivalent, his definition of them corresponding to that given in the preceding section.
[72] It has been already pointed out that the negative prefix does not always make a term really negative in force. Thus pleasant and unpleasant are not contradictories, for they admit of a mean; when we say that anything is unpleasant, we intend something more than the mere denial that it is pleasant. It should be added that a pair of terms of this kind may also fail to be contraries as above defined, since while admitting of a mean they may at the same time not denote extremes. Unpleasant, for example, denotes only that which is mildly painful: unless intended ironically, it would be a misuse of terms to speak of the tortures of the Inquisition as merely unpleasant. Compare Carveth Read, Logic, p. 49.