Studies and Exercises in Formal Logic

by John Neville Keynes

[²⁷⁷] The four propositions are precisely equivalent to the four categoricals,—All PQ is R, No PQ is R, Some PQ is R, Some PQ is not R.

(b) On the modal interpretation, the distinction between 258 apodeictic and problematic takes the place of that between universal and particular; and if we maintain the distinction between affirmative and negative, we have the four following propositions corresponding to the ordinary square of opposition:

If any P is Q, that P must be R ;  A_m
If any P is Q, that P cannot be R ;  E_m
If any P is Q, that P may be R ;  I_m
If any P is Q, that P need not be R.  O_m

It will be convenient to have distinctive symbols to denote modal propositions, and those that we have here introduced will serve to bring out the analogies between modals and the ordinary assertoric forms.

In the above schedule, subject to a certain condition mentioned below, A_m and O_m, and also E_m and I_m, are contradictories according to the definition given in section [84]; A_m and E_m are contraries; A_m and I_m, and also E_m and O_m, are subalterns; and I_m and O_m are subcontraries.

The condition referred to relates to the interpretation of the propositions as regards the implication of the possibility of their antecedents. Thus, in order that A_m and O_m (or E_m and I_m) may be true contradictories it is necessary that apodeictic and problematic propositions shall be interpreted differently in this respect. If, for example, A_m is interpreted as not implying the possibility of its antecedent then its full import is to deny the possibility of the combination P and Q without R. Its contradictory must affirm this possibility. O_m will not, however, do this unless it is interpreted as implying the possibility of the combination P, Q.

It is necessary to call attention to this complication, but hardly necessary to work out in detail the results which follow from the various principles of interpretation that might be adopted. If the student will do this for himself, he will find that the results correspond broadly with those obtained in section [159].[278]

[²⁷⁸] In connexion with the problem of opposition we may touch briefly on the relation between the apodeictic proposition If any P is Q that P must be R and the assertoric proposition Some PQ is not R. These propositions are not contradictories, for they may both be false. They cannot, however, both be true; and the latter, if it can be established, affords a valid ground for the denial of the former. Mr Bosanquet appears not to admit this, but to maintain, in opposition to it, that the enumerative particular is of no value as overthrowing the abstract universal. “When we have said that If (i.e., in so far as) a man is good, he is wise, it is idle to reply that Some good men are not wise. This is to attach an abstract principle with unanalysed examples. What we must say in order to deny the above-mentioned abstract judgment is something of this kind: If or Though a man is good, yet it does not follow that he is wise, that is, Though a man is good, yet he need not be wise” (Logic, i. p. 316). But surely if we find that some good men are not wise, we are justified in saying that though a man is good yet he need not be wise. Of course the converse does not hold. We might be able to shew that wisdom does not necessarily accompany goodness by some other method than that of producing instances. But if we can produce undoubted instances, that amply suffices to confute the apodeictic conditional.

259 177. Immediate Inferences from Conditional Propositions.—In a conditional proposition the antecedent and the consequent correspond respectively to the subject and the predicate of a categorical proposition. In conversion, therefore, the old consequent must be the new antecedent, and in contraposition the negation of the old consequent must be the new antecedent.

(a) On the assertoric interpretation, the analogy with categoricals is so close that it is unnecessary to treat immediate inferences from conditionals in any detail. One or two examples may suffice. Taking the A proposition, If any P is Q then it is R, we have for its converse Sometimes if a P is R it is also Q, and for its contrapositive If any P is not R then it is not Q. Taking the E proposition If any P is Q then it is not R, we have for its converse If any P is R then it is not Q, and for its contrapositive Sometimes if a P is not R it is Q. The validity of these inferences is of course affected by the existential interpretation of the propositions just as in the case of the categoricals. It will be noticed that in some immediate inferences (for example, the contraposition of A) the conditional form has an advantage over the ordinary categorical form inasmuch as it avoids the use of negative terms, the employment of which is so strongly objected to by Sigwart and some other logicians.[279]