wise,' and therefore as the equivalent of 'Not (Socrates is wise).' But when quantitative relations are involved, or when the proposition is indesignate, * nothing but confusion can result from equating 'Not (S is P)' to 'S is-not P.'

Let us consider the case where 'S is P' is indesignate, and take, by way of illustration, the proposition 'Frenchmen are vivacious.' It is surely misleading to assert forthwith that the denial of 'Frenchmen are vivacious' is 'Frenchmen are-not vivacious.' For if the indesignate proposition is here taken as particular, then, in denying 'Some Frenchmen are vivacious' by 'Some Frenchmen are-not vivacious,' we should be mistaking what is a mere subcontrary opposition for logical denial.

We conclude, then, that 'Not (S is P)' or 'S is P,' and not the declaration 'S is-not P,' is the only general form which the denial of 'S is P' can logically take; and that when this denial is specified in definite propositional form, it is a statement having the same subject and predicate as the proposition which it denies.

As regards the more special application of the laws of Contradictory Opposition, since A and O are contradictories, and also E and I, we see that, in the case of the first of the two laws, it consists in arguing that if a certain proposition is accepted, then another proposition, having the same subject and predicate, but differing from it in quality and quantity, cannot be accepted; and so, mutatis mutandis, for the application of the second law of Contradictory Opposition.

Subcontrariety.

The Rules of Subcontrariety :

Rule 1. Of two Subcontraries, if one is accepted, there is no logical ground either for the acceptance or for the rejection of the other.

Rule 2. Of two Subcontraries, if one is rejected, the other must be accepted.

Proof of Rule 1. -To accept the I proposition is to state that one S at least is a P, while making no statement about all the S's. In accepting I we do not state or imply either that the proposition 'All the S's are P's' must be rejected, or that it must be accepted. The acceptance of I does not involve either the rejection or the acceptance of A. Consequently the acceptance of I does not involve either the acceptance or the rejection of the contradictory of A. But the contradictory of A is O. Therefore the acceptance of I does not involve either the acceptance or the rejection of O. Similarly it may be shown that the acceptance of O does not involve either the acceptance or the rejection of I.

Proof of Rule 2:

If I is rejected, E must be accepted (Contradiction).

If E is accepted, O must be accepted (Identity, vide infra on
Subalternation).

... If I is rejected, O must be accepted.

So, again,

If O is rejected, A must be accepted (Contradiction).
If A is accepted, I must be accepted (Identity).

... If O is rejected, I must be accepted.

The rules of Subcontrariety, as above enumerated, cease to hold when, in place of the undistributed reading for 'Some,' we substitute either the exclusive or the indefinite reading, or the reading which expresses a limitation of knowledge.

If 'Some'='One at least, but not all' (exclusive), then not only are the I and O propositions not inconsistent-i.e., not only can they be accepted together-but if one of them is accepted, then the other must also be accepted. For to say that Some S's are P's is to state that not all the S's are P's; and to say that Some S's are-not P's is to state that not all the S's are-not P's-i.e., that Some S's are P's.

If 'Some'='One at least, possibly all' (indefinite), the I and O propositions cannot be accepted together. If the proposition 'One S at least is a P, possibly all the S's are P's' is accepted, then the proposition 'One S at least is not a P, possibly all the S's are not P's' must be rejected, and vice versa. For the statement 'One S at least is a P' is inconsistent with the statement 'possibly all the S's are-not P's'; and the statement 'One S at least is-not a P' is inconsistent with the statement 'Possibly all the S's are P's.' If I state that at least one member of a class is certain to pass (or not to pass) a specified examination, I cannot consistently go on to say that all may possibly fail (or pass).

In the case also of the reading 'One at least, but not assuredly (or "knowedly ") all, there is an inconsistency involved in accepting the I and O propositions together as true, though the inconsistency is here less easy to detect. Thus, as a despairing member of a class of examinees, I might assert : 'One at least of us will fail, though I do not know that we all shall,' and then, meeting a fellow-candidate in whose ability I have full confidence, I might say to him: 'One at least of us will pass, * though I don't know that we all shall,' and my two statements would be verbally compatible, for, knowing that I shall fail, I certainly do not know that we all shall pass. But my second statement, if not actually disingenuous, at least cannot be said to express a limitation of knowledge. My knowledge as to the truth of the statement that we shall all pass is * Here 'pass' is used as an abbreviation for 'not fail.'

not limited if I know that one at least will fail. But it is precisely this limitation of knowledge which the reading we are considering professes to express. There is, therefore, a contradiction involved, on this reading, in accepting the I and O propositions as true together. To harmonize the two I must exchange statement of limitation (of knowledge) for limitation of statement. I must adopt the undistributed reading, and say: 'One at least of us will pass, but more than that I don't say; about the question of our all passing I make no statement.'

Contrary Opposition.

The Rules of Contrary Opposition are the following:

Rule 1. Of two contraries, if one is accepted the other must be

rejected.

Rule 2. Of two contraries, if one is rejected, there is no logical ground either for the acceptance or for the rejection of the other.

These rules may now be justified as follows:

1. Contraries cannot be accepted together; for, supposing

this possible, then

As A is accepted, I must be accepted (Principle of
Identity: vide infra on Subalternation).

And, as E is accepted, O must be accepted (ibid.).
... A and O are accepted together, as also are E and I.
But this is impossible, by the Principle of Non-Contra-
diction.

2. It may be quite consistent to reject both contraries; for their contradictories (which are subcontraries) may

both be accepted. See Subcontrariety, Rule 1.

The question may be asked why, in controversy, it is preferable to attempt the refutation of a statement by proving its contradictory rather than by proving its contrary. The simple answer is that the contrary of a proposition is harder to prove than its contradictory. But this is not all. The additional element of assertion which the strengthened opposition superinduces upon the pure contradiction not only plays no part in the refutation of the statement, but may give the adversary an opening for counterattack which he would not have possessed had the refutation taken place through contradiction simply.

Subalternation.

The following are the Rules of Subalternation :

Rule 1. If the universal proposition is accepted, the particular proposition must also be accepted.

1

If A is accepted, I must be accepted (Principle of
Identity).

If E is accepted, O must be accepted (ibid.).

Rule 2. If the universal proposition is rejected, there is no logical ground either for accepting or for rejecting the particular proposition.

Proof of Rule 2:

The acceptance of O affords no logical ground either for the acceptance or for the rejection of I (Subcontrariety, 1). But the acceptance of O is logically equivalent to the rejection of A (Contradiction).

... the rejection of A affords no logical ground either for the acceptance or for the rejection of I.

Rule 3. If the particular proposition is accepted, there is no logical ground either for accepting or for rejecting the universal proposition.

For a justification of this Rule, see above, p. 174 (Proof of the First Rule of Subcontrariety).

Rule 4. If the particular is rejected, then the universal must also be rejected.

Proof: If I is rejected, E must be accepted (Contradiction). ... A must be rejected (Contrariety).

If O is rejected, A must be accepted (Contradiction).

... E must be rejected (Contrariety).

The following Table of Opposition will serve to summarize the results of the previous discussion :

N.B.-'Some' = 'One at least, but not statedly all.' By 'neither' we mean that there is no logical ground either for the acceptance or for the rejection of the proposition in question.

The Opposition of Disjunctive Propositions.

The contradictory of the disjunctive proposition 'Either Por Q' in its non-exclusive form is 'Neither P nor Q.' When the 'Either or' is read exclusively its contradictory is 'Either both P and Q, or else neither P nor Q.' In this second case 'Either one or the other' is contradicted by 'Either both or else neither.' According to the exclusive view, 'Neither P nor Q' could not be accepted as the contradictory of 'Either P or Q,' since both propositions could be rejected, the accepted statement being 'Both P and Q.'

Again, according to the exclusive reading, 'Neither P nor Q' and 'Both P and Q' may both be considered as contraries of the disjunctive proposition 'Either Por Q.' For 'Either Por Q' and 'Neither P nor Q' cannot both be accepted, and yet, as we have already seen, it may be quite consistent to reject both. Similarly, 'Either Por Q' and 'Both P and Q'cannot both be accepted, but it may be quite consistent to reject them both-namely, when the accepted statement is 'Neither P nor Q.'

The Opposition of Hypotheticals.

The Scheme of Opposition here takes different forms corresponding to the different types of Hypothetical Proposition.

1. (a) The Apodeictic Scheme (Formal).

H. If Pis accepted, the acceptance of P is necessary.
H. If Pis accepted, the rejection of P is necessary.
H1. If Pis accepted, the rejection of Q is not necessary.
Ho. If P is accepted, the acceptance of Q is not necessary.

Example.-H.

H.

H1.

If 'All S's are P's' is accepted, the acceptance of 'All S's are P's' is necessary.

If 'All S's are P's' is accepted, the rejection of 'All S's are P's' is necessary.

If 'All S's are P's' is accepted, the rejection of 'All S's are P's' is not necessary.

(I.e., Either 'All S's are P's' must be accepted, or else there is no ground either for accepting or for rejecting it.)

H. If 'All S's are P's' is accepted, the acceptance of 'All S's are P's' is not necessary.

(I.e., Either 'All S's are P's' must be rejected, or else there is no ground either for accepting or for rejecting it.)