I. Contradictories. There is no condition of solubility of a pair of Contradictories, inasmuch as the propositions so opposed cannot be simultaneously true. II. and III. Contraries and Subcontraries. If the propositions in any pair of Contraries be false, those in its corresponding pair of Subcontraries are true. We may therefore confine our attention to the pairs of subcontraries : 1. Some-X is not Some-Y, Certain-X is not Certain-Y is the resulting illation. I append a couple of examples. Some-civilian is not Some-military. Therefore, if certain members of parliament be the civilians who constitute a portion of the military, the class civilian is divisible into military M.P.'s and not-military, and the class military is divisible into military M.P.'s and not-civilian. Again, All-human is Some-biped. Some-biped is not Any-human. Therefore, if birds be the bipeds who are not human, the class biped is divisible into birds and men, and the other bipeds, if any. IV. Subalterns. Let All-X is Some-Y and Some-Y is not Any-X be simultaneous ly true. All-X is Certain-Y) are the resulting These solutions are of more importance in applied logic than their form would lead us to suppose. In concluding our remarks on these instances of immediate inference, we may call attention to the fact, that in every case in which an inference is drawn immediately from two propositions, we have only to lay down the illative result as the original proposition, and the two opposed propositions will form a double immediate inference therefrom. We now proceed to consider Illation by Privation. The following Table will exhibit all possible assertorial inferences from the eight general propositions, by writing for one or both terms its contradictory (or infinite complement). 1. All-X is All-Y. (Any-X is not Any-not-Y, Therefore, Any-not-X is not Any-Y, (All-not-X is All-not-Y. Example to 1. The just are all the holy.' 'No just man is unholy.' 'No holy man is unjust.' 'All unjust men are all the unholy.' 2. All-X is Some-Y. (Any-X is not Any-not-Y, Therefore, Any-not-X is not Some-Y, Example to 2. All the righteous are happy.' 'None of the righteous are unhappy.' 'Some happy men are not unrighteous.' 'All who are not happy are not righteous.' 3. Some-X is All-Y. Some-X is not Any-not-Y, Therefore, Any-not-X is not Any-Y, All-not-X is Some-not-Y. Example to 3. Some happy persons are all the righteous.''Some-happy persons are not unrighteous.' 'No unhappy person is righteous.' 'All who are unrighteous are unhappy.' 4. Therefore, Some-X is Some-Y. (Some-X is not Any-not-Y, Any-not-X is not Some-Y. Example to 4. 'Some desirable events are probable.' 'Some desirable events are not im probable.' 'Some probable events are not undesirable.' These are all the immediate (assertorial) inferences by privation that can be drawn from affirmative propositions. It will be observed that each proposition except the fourth affords three inferences and no more. When we proceed however to negative propositions, the circumstances are very different. No negative proposition can yield more than two assertorial illations, and one negative proposition affords none at all. 5. Some-X is not Some-Y. On this proposition no categorical immediate inference by privation can be drawn. Therefore, (Some-X is Some-not-Y, Any-not-X is not Some-not-Y. Examples of 6. 'Some possible cases are not probable.' 'Some possible cases are improbable.' 'Some improbable cases are not impossible.' 7. Any-X is not Some-Y. (Some-not-X is Some-Y, Therefore, Some-not-X is not Any-not-Y. Examples of 7. 'No probable case is somepossible.' 'Some impossible cases are some possible.' Some improbable cases are not impossible.' |