one must be true; and then the Axiom of Non-Contradiction adds, that the other must be false. Now, A and O are two such Judgments, and likewise E and I; so also the two Singular Judgments, Socrates is wise, and Socrates is not wise. Between either of these pairs, no "third" or intermediate Judgment is conceivable. Hence the universal rule for this sort of Opposition, that Contradictories cannot both be true, and cannot both be false. Therefore, as both cannot be true, if I posit (affirm) one, I immediately infer that the other is sublated (denied); and as both cannot be false, if I sublate one, the other is posited. For example; —if E is not true, that No quadruped is rational, I must be true, that Some quadrupeds are rational. Observe that two Judgments properly contradict each other only when that which is affirmed by the one is denied by the other, 1. in the same manner; 2. in the same respect; 3. in the same degree; and 4. at the same time. Thus, to borrow some examples from Aldrich, -1. A dead body is, and is not, a man; that is, it is a dead man, but not a living one. 2. Zoilus is, and is not, black; that is, black-haired, but red-faced. 3. Socrates is, and is not, long-haired; that is, he is so, if you compare him with Scipio, but is not so, if you compare him with Xenophon. 4. Nestor is, and is not, an old man, according as you speak of him when in childhood, or when he was at the siege of Troy. The second sort of Opposition is that of Contrariety, which exists between two Universal Judgments, that differ in Quality, but are alike in Quantity; that is, between A and E. Here the Axiom of Excluded Middle does not apply; for between A and E, there is a "Middle " or intermediate Judgment, namely, I. Though it is not true, either that all men are wise, or that no man is wise, it is true that some men are wise. Hence both Contraries may be false, so that I cannot infer the truth of one from the falsity of the other. On the other hand, as one of these Contraries affirms what the other denies, the Axiom of Non-Contradiction applies; both Contraries cannot be true; and, therefore, from the truth of one I can immediately infer the falsity of the other. Accordingly, the rule is, Contraries may be false together, but both cannot be true. Therefore, from positing either A or E, I can immediately infer that the other is sublated; but from sublating either, I cannot infer that the other is posited. men some men as The third sort of Opposition is that of Sub-Contrariety, which exists between two Particular Judgments, that differ in Quality, but are alike in Quantity; that is, between I and O. To these, the Axiom of Excluded Middle is applicable; for there is no third, or intermediate, Judgment conceivable between Some are, and Some are not. Accordingly, both cannot be false, but one must be true. On the other hand, if I and O are considered as Propositions, that is, if the Judgments are expressed in words, the Axiom of Non-Contradiction does not apply to them; for both may be true. Though some men are learned, it is also true that some men are not learned. But observe, that the " some 66 in the latter case are not the same in the former; though expressed by the same words, they are thought as different. To make the former Proposition true, "some men" may be thought to be "graduates of Oxford"; to make the latter true, 66 some men may mean "American Indians." As Propositions, then, and possibly as Judgments, the two assertions do not contradict each other, but may both be true. Hence the rule, that Sub-Contraries may be true together, but cannot both be false. Therefore, by sublating either I or O, we immediately infer that the other is posited; but by positing either, we cannot infer that the other is sublated. Of course, SubContraries can be called "opposites" only in a qualified and technical sense; they are actually congruent, or, to adopt one of Hamilton's newly-coined words, they are "compossible." The fourth sort of Opposition is that of Subalternation, which exists between Judgments alike in Quality, but different in Quantity; that is, between A and I, and between E and O. Here, again, it is evident that the "Opposition" is merely technical, the two Judgments being not merely consistent, but so nearly allied that the Particular can be inferred from its Universal by the Axiom of Identity. Since all includes some, if we affirm A, All A are B, we thereby also affirm I, Some A are B; and in like manner, to posit E is also to posit O. The same Axiom compels us to think, that sublating I sublates A also, and sublating O sublates E also. In this sort of inference, the Universal may be called the Subalternans, and the Particular, the Subalternate. Hence we have this rule for inference by Subalternation, that if the Subalternans is true, the Subalternate is true also; and if the Subalternate is false, the Subalternans is false also. Summing up, we have the following list of Immediate Inferences by Opposition. If A is true, O is false, E false, and I true. If E is false, I is true, the others unknown. the others unknown. If O is true, A is false, Hence it appears, that from the truth of a Universal or the falsehood of a Particular, we may infer the character of all the opposed Judgments; but from the falsehood of a Universal or truth of a Particular, we can know the character only of the Contradictory. of the Terms of a Judgment, or by dropping their Infinitation, the Judgments thus produced being, in certain cases, æquipollent, or equivalent to those from which they were derived. E. Universal Negative. I. Particular Affirmative. | O. Particular Negative. No X is Y. Some X are Y. No quadruped is rational. Some swans are black. Some X are not Y. Some men are not fa mous. Rule. TWO KINDS OF INFINITATION. To change the Infinitation of | Rule.-To change the Infinitation of the Predicate, either by infinitating it or by dropping its Infinitation, change the Quality of the Judgment; the Quantity of the Judgment remains unaltered. the Subject, convert the Judgment, and then either change the Quality, or change the Infinitation of the (old) Predicate also. Here, also, the Quantity is unaltered. |