| The happening of his arrival to-morrow, | is not | an event from which it may be inferred as probable that he will stay till Monday, |
would be expressed thus: ‘If he should come to-morrow, that is no reason why he should stay till Monday.’
Moreover, the negative words not, no, &c., have two kinds of meaning which must be carefully distinguished. Sometimes they deny, and nothing more: sometimes they are used to affirm the direct contrary. In cases which offer but two alternatives, one of which is necessary, these amount to the same thing, since the denial of one, and the affirmation of the other, are obviously equivalent propositions. In many idioms of conversation, the negative implies affirmation of the contrary in cases which offer not only alternatives, but degrees of alternatives. Thus, to the question, ‘Is he tall?’ the simple answer, ‘No,’ most frequently means that he is the contrary of tall, or considerably under the average. But it must be remembered, that, in all logical reasoning, the negation is simply negation, and nothing more, never implying affirmation of the contrary.
The common proposition that two negatives make an affirmative, is true only upon the supposition that there are but two possible things, one of which is denied. Grant that a man must be either able or unable to do a particular thing, and then not unable and able are the same things. But if we suppose various degrees of performance, and therefore degrees of ability, it is false, in the common sense of the words, that two negatives make an affirmative. Thus, it would be erroneous to say, ‘John is able to translate Virgil, and Thomas is not unable; therefore, what John can do Thomas can do,’ for it is evident that the premises mean that John is so near to the best sort of translation that an affirmation of his ability may be made, while Thomas is considerably lower than John, but not so near to absolute deficiency that his ability may be altogether denied. It will generally be found that two negatives imply an affirmative of a weaker degree than the positive affirmation.
Each of the propositions, ‘A is B,’ and ‘A is not B,’ may be subdivided into two species: the universal, in which every possible case is included; and the particular, in which it is not meant to be asserted that the affirmation or negation is universal. The four species of propositions are then as follows, each being marked with the letter by which writers on logic have always distinguished it.
| A Universal Affirmative | Every A is | B |
| E Universal Negative | No A is | B |
| I Particular Affirmative | Some A is | B |
| O Particular Negative | Some A is not | B |
In common conversation the affirmation of a part is meant to imply the denial of the remainder. Thus, by ‘some of the apples are ripe,’ it is always intended to signify that some are not ripe. This is not the case in logical language, but every proposition is intended to make its amount of affirmation or denial, and no more. When we say, ‘Some A is B,’ or, more grammatically, ‘Some As are Bs,’ we do not mean to imply that some are not: this may or may not be. Again, the word some means, ‘one or more, possibly all.’ The following table will shew the bearing of each proposition on the rest.
| Every A is B affirms and contains Some A is B and denies | No A is B Some A is not B |
| No A is B affirms and contains Some A is not B and denies | Every A is B Some A is B |
| Some A is B does not contradict | Every A is B Some A is not B | but denies No A is B |
| Some A is not B does not contradict | No A is B Some A is B | but denies Every A is B |
Contradictory propositions are those in which one denies any thing that the other affirms; contrary propositions are those in which one denies every thing which the other affirms, or affirms every thing which the other denies. The following pair are contraries.