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universal, are called contraries, as, e.g., " Y is X" and " Y is not X”; and if particular, subcontraries, as, e. 8.,
Some Y is X” and Some Y is not X.” Where propositions differ in quantity only, as A and I, or E and 0, the particular propositions are called subalterns, as, e. g., “Y is X" and "Some Y is X”; and “Y is not X” and Some Y is not X."
There are, therefore, four kinds of opposition recognized by logicians, viz. : (1) the opposition of contradictories; (2) that of contraries , (3) that of subcontraries, and (4) that of subalterns to their corresponding universals; which, with their relations to each other, are admi. rably expressed in the following table, which has come to us from ancient times:
(1) CONTRADICTORIES. The most complete kind of opposition is that of contradictories. These cannot both be either true or false: i. e., if one is true, the other is false; or, if one is false, the other is true. For if it be true that “ All men are sinners,” it cannot be true that “Some men are not sinners"; and, conversely, if it be true that “ Some are not righteous,” it cannot be true that “ All men are righteous." In other words, between contradictories there is no intermediate proposition conceivable; one must be true and the other false. This is called the law of Excluded Middle.
(2) CONTRARIES. Contraries cannot both be true; for if it be true that “ Every man is an animal,” it must be false that “ No man is an animal." But both may
But both may be false, as, for example, the propositions that “ All men are learned,” and that “ No men are learned ”; which are both false, for some are learned and some are not. In other words, contrary propositions do not exclude the truth of either of the particular propositions between the same terms.
(3) SUBCONTRARIES. Subcontraries are contrasted with contraries by the principle that they may be both true, but cannot both be false. Thus it may be true that “ Some men are just,” and also that Some men are not just”; but if it be false that “ Some men are just," it must be true that “No man is just, ” -- which is the contradictory,—and, a fortiori, that “Some men are not just,” — which is the subcontrary.
(4) SUBALTERNATE OPPOSITION. With regard to subaltern propositions, their truth follows from the corresponding universal propositions; for if “ all men are animals, some men are animals,” and if no man is an ape,
some men are not apes.” But from the truth of a subaltern proposition we cannot infer the truth of the corresponding universal, as, e.g., from the proposition “ Some men are false," the proposition “ All men are false ”; or from the proposition “ Some men are not false," the proposition that “No man is false."
$ 90. OBSERVATIONS UPON CONTRARY AND CONTRADICTORY OPPOSITIONS.- Accurately speaking, these constitute the only kinds of opposition. Subcontraries are, in fact, not opposites; and the same is true of subalterns and their corresponding universals.
It will be observed it does not follow from the principle of contrary opposition that of two terms regarded as subject and predicateas, e. g., Y and X-either the latter or its negative may always be predicated of the former, or, in other words, that Y must be either X, or not X; for, in fact, some Y may be X, and some Y not X, as will obviously appear from the following diagrams:
Hence there arises, seemingly, a puzzling contradiction between this principle and the law of Excluded Middle-as it is often stated. Thus, it is said, “ Rock must be either hard or not hard” (Jevons, Lessons in Logic, p. 119), or, generally," Y is either X or not X." But obviously this, unless accidentally, is not true; for some rock may be hard and some soft; or some Y may be X, and some not X. And so we cannot say of “men” either that they are learned or that they are not learned; for some are the one and some the other. parent contradiction arises from the misstate. ment of the law of Excluded Middle; which is itself nothing more or less than the principle governing contradictories, as expressed above. We may, indeed, where a subject term (as, e. g., Y) denotes an individual or single thing (real or fictitious), affirm of it that it is either X or not X; but if Y denotes a class of more than one we cannot so affirm.'
But the ap
1 Even Hobbes falls into the error of Jevons on this point. “ Positive and negative terms,” he says, are contradictory to one another, so that they cannot both be the name of the same thing. Besides, of contradictory names, one is the name of anything whatsoever (i. l., of any conceivable thing), for whatsoever is, is either a man, or not a man, white, or not white, and so of the rest.” But, it may be asked, “Does the name 'biped' denote (universally) either man, or not man?" or “the name 'man', either white man, or man not white?"
The confusion results from the technical view that regards $91. CONVERSION OF PROPOSITIONS.—A proposition is said to be converted when its terms are transposed, i.e., when the subject is made the predicate and the predicate the subject (8 54). Such conversion is admissible only when illative, i. e., where the truth of the converse is implied in that of the original prop. osition. When such conversion can be made without otherwise changing the proposition it is called a simple conversion; otherwise, it is called a conversion per accidens. Thus A (“Y is X") cannot be converted simply, because the subject only is distributed; we therefore cannot say that “ All X is Y,” but only that “Some X is Y,” which is called conversion per accidens. But E (" Y is not X")-as both subject and predicate are distributed-may be converted simply; or, in other words, we may say
the Particular Proposition as a form distinct from the Universal, and its source would be removed if, as elsewhere suggested, this form of the proposition should be rejected (8 52, n.). We might then adopt, as equally accurate and profound, the remaining observation of Hobbes, that “the certainty of this axiom, namely, that of two contradictory names one is the name of anything whatsoever, the other not, is the original and foundation of all ratiocination, that is, of all philosophy” (Logic, Sec. 8), which is in accord with the view of Aristotle : “For the same thing to be present and not to be present, at the same time, in the same subject, and in the same sense, is impossible.
For by nature this is the first principle of all the other axioms" (Metaphysics, R. iii., chap. iii.).