Part 2

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│No A is B denies │Some A is not B

No A is B affirms and contains Some A is not B │Every A is B and denies │Some A is B

Some A is B does not │Every A is B │but denies No A is B contradict │Some A is not B│

Some A is not B does not │No A is B │but denies Every A is B contradict │Some 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.

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.

2. In E, or ‘No A is B,’ both subject and predicate enter wholly. ‘No A whatsoever is any one out of all the Bs;’ ‘search the whole collection of Bs, and every B shall be found to be something which is not A.’

3. In I, or ‘Some A is B,’ both subject and predicate enter partially. ‘Some of the As are found among the Bs, or make up a part (the whole possibly, but not known from the preceding) of the Bs.’

4. In O, or ‘Some A is not B,’ the subject enters partially, and the predicate wholly. ‘Some As are none of them any whatsoever of the Bs; every B will be found to be no one out of a certain portion of the As.’

It appears then that,

In affirmatives, the predicate enters partially.

In negatives, the predicate enters wholly.

In contradictory propositions, both subject and predicate enter differently in the two.