The possibility of implicit secondary quantification, although no such quantification is explicitly indicated, is a not unfruitful source of fallacy in the employment of propositions having singular subjects. If we take such propositions as Browning is obscure, Epimenides is a liar, This flower is blue, and give as their contradictories Browning is not obscure, Epimenides is not a liar, This flower is not blue, shall we say that the original proposition or its contradictory is true in case Browning is sometimes (but not always) obscure, or in case Epimenides sometimes (but not often) speaks the truth, or in case the flower is partly (but not wholly) blue? There is certainly a considerable risk in such instances as these of confusing contradictory and contrary opposition, and this will be avoided if we make the secondary quantification of the propositions explicit at the outset by writing them in the form Browning is always (or sometimes) obscure, &c.[114] The contradictory will then be particular or universal accordingly.
[114] Or we might reduce them to the forms,—All (or some) of the poems of Browning are obscure, All (or some) of the statements of Epimenides are false, All (or some) of the surface of this flower is blue.
83. The Opposition of Modal Propositions.—So far in this chapter our attention has been confined to assertoric propositions. For the present, a very brief reference to the opposition 117 of modals will suffice. The main points involved will come up for further consideration [later on].
We have seen that the unconditionally universal proposition, whether expressed in the ordinary categorical form All S is P, or as a conditional If anything is S it is P, affirms a necessary connexion, by which is meant not merely that all the S’s are as a matter of fact P’s, but that it is inherent in their nature that they should be so. The statement that some S’s are not P’s is inconsistent with this proposition, but is not its contradictory, since both the propositions might be false: the S’s might all happen to be P’s, and yet there might be no law of connexion between S and P. The proposition in question being apodeictic will have for its contradictory a modal of another description, namely, a problematic proposition; and this may be written in the form S need not be P, or If anything is S still it need not be P, according as our original proposition is expressed as a categorical or as a conditional
Similarly, the contradictory of the hypothetical If P is true then Q is true, this proposition being interpreted modally, is If P is true still Q need not be true.
84. Extension of the Doctrine of Opposition.[115]—If we do not confine ourselves to the ordinary square of opposition, but consider any pair of propositions (whatever may be the schedule to which they belong), it becomes necessary to amplify the list of formal relations recognised in the square of opposition, and also to extend the meaning of certain terms. We may give the following classification:
[115] The illustrations given in this section presuppose a knowledge of immediate inferences. The section may accordingly on a first reading be postponed until part of the following chapter has been read.
(1) Two propositions may be equivalent or equipollent, each proposition being formally inferable from the other. Hence if either one of the propositions is true, the other is also true; and if either is false, the other is also false. For example, as will presently be shewn, All S is P and All not-P is not-S stand to each other in this relation.
(2) and (3) One of the two propositions may be formally inferable from the other, but not vice versâ. If we are 118 considering two given propositions Q and R, this yields two cases: for Q may carry with it the truth of R, but not conversely; or R may carry with it the truth of Q, but not conversely. Ordinary subaltern propositions with their subalternants fall into this class; and it will be convenient to extend the meaning of the term subaltern, so as to apply it to any pair of propositions thus related, whether they belong to the ordinary square of opposition or not. It will indeed be found that any pair of simple propositions of the forms A, E, I, O, that are subaltern in the extended sense, are equivalent to some pair that are subaltern in the more limited sense.[116] Thus All S is P and Some P is S, which are subaltern in the extended sense, are equivalent to All S is P and Some S is P. All S is P and Some not-S is not P are another pair of subalterns. Here it is not so immediately obvious in what direction we are to look for a pair of equivalent propositions belonging to the ordinary square of opposition. No not-P is S and Some not-P is not S will, however, be found to satisfy the required conditions.
[116] This will of course not hold good when we apply the term subaltern to compound propositions, e.g., to the pair Some S is not P and some P is not S, Some S is not P or some P is not S.