_Of Opposition._
§ 449. Opposition is an immediate inference grounded on the relation
between propositions which have the same terms, but differ in quantity
or in quality or in both.
§ 450. In order that there should be any formal opposition between two
propositions, it is necessary that their terms should be the
same. There can be no opposition between two such propositions as
these--
(1) All angels have wings.
(2) No cows are carnivorous.
§ 451. If we are given a pair of terms, say A for subject and B for
predicate, and allowed to affix such quantity and quality as we
please, we can of course make up the four kinds of proposition
recognised by logic, namely,
A. All A is B.
E. No A is B.
I. Some A is B.
O. Some A is not B.
§ 452. Now the problem of opposition is this: Given the truth or
falsity of any one of the four propositions A, E, I, O, what can be
ascertained with regard to the truth or falsity of the rest, the
matter of them being supposed to be the same?
§ 453. The relations to one another of these four propositions
are usually exhibited in the following scheme--
A . . . . Contrary . . . . E
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Subaltern Contradictory Subaltern
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
I . . . Sub-contrary . . . O
§ 454. Contrary Opposition is between two universals which differ in
quality.
§ 455. Sub-contrary Opposition is between two particulars which differ
in quality.
§ 456. Subaltern Opposition is between two propositions which differ
only in quantity.
§ 457. Contradictory Opposition is between two propositions which
differ both in quantity and in quality.
§ 458. Subaltern Opposition is also known as Subalternation, and of
the two propositions involved the universal is called the Subalternant
and the particular the Subalternate. Both together are called
Subalterns, and similarly in the other forms of opposition the two
propositions involved are known respectively as Contraries,
Sub-contraries and Contradictories.
§ 459. For the sake of convenience some relations are classed under
the head of opposition in which there is, strictly speaking, no
opposition at all between the two propositions involved.
§ 460. Between sub-contraries there is an apparent, but not a real
opposition, since what is affirmed of one part of a term may often
with truth be denied of another. Thus there is no incompatibility
between the two statements.
(1) Some islands are inhabited.
(2) Some islands are not inhabited.
§ 461. In the case of subaltern opposition the truth of the universal
not only may, but must, be compatible with that of the particular.
§ 462. Immediate Inference by Relation would be a more appropriate
name than Opposition; and Relation might then be subdivided into
Compatible and Incompatible Relation. By 'compatible' is here meant
that there is no conflict between the _truth_ of the two
propositions. Subaltern and sub-contrary opposition would thus fall
under the head of compatible relation; contrary and contradictory
relation under that of incompatible relation.
Relation
______________|_____________
| |
Compatible Incompatible
______|_____ _____|_______
| | | |
Subaltern Sub-contrary Contrary Contradictory.
§ 463. It should be noticed that the inference in the case of
opposition is from the truth or falsity of one of the opposed
propositions to the truth or falsity of the other.
§ 464. We will now lay down the accepted laws of inference with regard
to the various kinds of opposition.
§ 465. Contrary propositions may both be false, but cannot both be
true. Hence if one be true, the other is false, but not vice versâ.
§ 466. Sub-contrary propositions may both be true, but cannot both be
false. Hence if one be false, the other is true, but not vice versâ.
§ 467. In the case of subaltern propositions, if the universal be
true, the particular is true; and if the particular be false, the
universal is false; but from the truth of the particular or the
falsity of the universal no conclusion can be drawn.
§ 468. Contradictory propositions cannot be either true or false
together. Hence if one be true, the other is false, and vice versâ.
§ 469. By applying these laws of inference we obtain the following
results--
If A be true, E is false, O false, I true.
If A be false, E is unknown, O true, I unknown.
If E be true, O is true, I false, A false.
If E be false, O is unknown, I true, A unknown.
If O be true, I is unknown, A false, E unknown.
If O be false, I is true, A true, E false.
If I be true, A is unknown, E false, O unknown.
If I be false, A is false, E true, O true.
§ 470. It will be seen from the above that we derive more information
from deriving a particular than from denying a universal. Should this
seem surprising, the paradox will immediately disappear, if we reflect
that to deny a universal is merely to assert the contradictory
particular, whereas to deny a particular is to assert the
contradictory universal. It is no wonder that we should obtain more
information from asserting a universal than from asserting a
particular.
§ 471. We have laid down above the received doctrine with regard to
opposition: but it is necessary to point out a flaw in it.
When we say that of two sub-contrary propositions, if one be false,
the other is true, we are not taking the propositions I and O in their
now accepted logical meaning as indefinite (§ 254), but rather in
their popular sense as 'strict particular' propositions. For if I and
O were taken as indefinite propositions, meaning 'some, if not all,'
the truth of I would not exclude the possibility of the truth of A,
and, similarly, the truth of O would not exclude the possibility of
the truth of E. Now A and E may both be false. Therefore I and O,
being possibly equivalent to them, may both be false also. In that
case the doctrine of contradiction breaks down as well. For I and O
may, on this showing, be false, without their contradictories E and A
being thereby rendered true. This illustrates the awkwardness, which
we have previously had occasion to allude to, which ensures from
dividing propositions primarily into universal and particular, instead
of first dividing them into definite and indefinite, and particular (§
256).
§ 472. To be suddenly thrown back upon the strictly particular view of
I and O in the special case of opposition, after having been
accustomed to regard them as indefinite propositions, is a manifest
inconvenience. But the received doctrine of opposition does not even
adhere consistently to this view. For if I and O be taken as strictly
particular propositions, which exclude the possibility of the
universal of the same quality being true along with them, we ought not
merely to say that I and O may both be true, but that if one be true
the other must also be true. For I being true, A is false, and
therefore O is true; and we may argue similarly from the truth of O to
the truth of I, through the falsity of E. Or--to put the Same thing in
a less abstract form--since the strictly particular proposition means
'some, but not all,' it follows that the truth of one sub-contrary
necessarily carries with it the truth of the other, If we lay down
that some islands only are inhabited, it evidently follows, or rather
is stated simultaneously, that there are some islands also which are
not inhabited. For the strictly particular form of proposition 'Some A
only is B' is of the nature of an exclusive proposition, and is really
equivalent to two propositions, one affirmative and one negative.
§ 473. It is evident from the above considerations that the doctrine
of opposition requires to be amended in one or other of two
ways. Either we must face the consequences which follow from regarding
I and O as indefinite, and lay down that sub-contraries may both be
false, accepting the awkward corollary of the collapse of the doctrine
of contradiction; or we must be consistent with ourselves in regarding
I and O, for the particular purposes of opposition, as being strictly
particular, and lay down that it is always possible to argue from the
truth of one sub-contrary to the truth of the other. The latter is
undoubtedly the better course, as the admission of I and O as
indefinite in this connection confuses the theory of opposition
altogether.
§ 474. Of the several forms of opposition contradictory opposition is
logically the strongest. For this three reasons may be given--
(1) Contradictory opposites differ both in quantity and in quality,
whereas others differ only in one or the other.
(2) Contradictory opposites are incompatible both as to truth and
falsity, whereas in other cases it is only the truth _or_
falsity of the two that is incompatible.
(3) Contradictory opposition is the safest form to adopt in
argument. For the contradictory opposite refutes the adversary's
proposition as effectually as the contrary, and is not so hable to a
counter-refutation.
§ 475. At first sight indeed contrary opposition appears stronger,
because it gives a more sweeping denial to the adversary's
assertion. If, for instance, some person with whom we were arguing
were to lay down that 'All poets are bad logicians,' we might be
tempted in the heat of controversy to maintain against him the
contrary proposition 'No poets are bad logicians.' This would
certainly be a more emphatic contradiction, but, logically considered,
it would not be as sound a one as the less obtrusive contradictory,
'Some poets are not bad logicians,' which it would be very difficult
to refute.
§ 476. The phrase 'diametrically opposed to one another' seems to be
one of the many expressions which have crept into common language from
the technical usage of logic. The propositions A and O and E and I
respectively are diametrically opposed to one another in the sense
that the straight lines connecting them constitute the diagonals of
the parallelogram in the scheme of opposition.
§ 477. It must be noticed that in the case of a singular proposition
there is only one mode of contradiction possible. Since the quantity
of such a proposition is at the minimum, the contrary and
contradictory are necessarily merged into one. There is no way of
denying the proposition 'This house is haunted,' save by maintaining
the proposition which differs from it only in quality, namely, 'This
house is not haunted.'
478. A kind of generality might indeed he imparted even to a singular
proposition by expressing it in the form 'A is always B.' Thus we may
say, 'This man is always idle'--a proposition which admits of being
contradicted under the form 'This man is sometimes not idle.'