The happening of his arrival │ │an event from which it may be
to-morrow, │is│ inferred to be _im_probable
│ │ that he will stay till Monday,
whereas the following,
The happening of his arrival │ │an event from which it may be
to-morrow, │is _not_│ 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│_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.