| Every A is B | and | No A is B |
and the following are contradictories,
| Every A is B | to | Some A is not B |
| No A is B | to | Some A is B |
A contrary, therefore, is a complete and total contradictory; and a little consideration will make it appear that the decisive distinction between contraries and contradictories lies in this, that contraries may both be false, but of contradictories, one must be true and the other false. We may say, ‘Either P is true, or something in contradiction of it is true;’ but we cannot say, ‘Either P is true, or every thing in contradiction of it is true.’ It is a very common mistake to imagine that the denial of a proposition gives a right to affirm the contrary; whereas it should be, that the affirmation of a proposition gives a right to deny the contrary. Thus, if we deny that Every A is B, we do not affirm that No A is B, but only that Some A is not B; while, if we affirm that Every A is B, we deny No A is B, and also Some A is not B.
But, as to contradictories, affirmation of one is a denial of the other, and denial of one is affirmation of the other. Thus, either Every A is B, or Some A is not B: affirmation of either is denial of the other, and vice versá.
Let the student now endeavour to satisfy himself of the following. Taking the four preceding propositions, A, E, I, O, let the simple letter signify the affirmation, the same letter in parentheses the denial, and the absence of the letter, that there is neither affirmation nor denial.
| From A | follow | (E), I, (O) | From (A) | follow | O |
| From E | (A), (I), O | From (E) | I | ||
| From I | (E) | From (I) | (A), E, O | ||
| From O | (A) | From (O) | A, (E), I |
These may be thus summed up: The affirmation of a universal proposition, and the denial of a particular one, enable us to affirm or deny all the other three; but the denial of a universal proposition, and the affirmation of a particular one, leave us unable to affirm or deny two of the others.
In such propositions as ‘Every A is B,’ ‘Some A is not B,’ &c., A is called the subject, and B the predicate, while the verb ‘is’ or ‘is not,’ is called the copula. It is obvious that the words of the proposition point out whether the subject is spoken of universally or partially, but not so of the predicate, which it is therefore important to examine. Logical writers generally give the name of distributed subjects or predicates to those which are spoken of universally; but as this word is rather technical, I shall say that a subject or predicate enters wholly or partially, according as it is universally or particularly spoken of.
1. In A, or ‘Every A is B,’ the subject enters wholly, but the predicate only partially. For it obviously says, ‘Among the Bs are all the As,’ ‘Every A is part of the collection of Bs, so that all the As make a part of the Bs, the whole it may be.’ Thus, ‘Every horse is an animal,’ does not speak of all animals, but states that all the horses make up a portion of the animals.