[107] This chapter will be mainly concerned with the opposition of categorical propositions; and, as regards categoricals, complications arising in connexion with their existential interpretation will for the present be postponed.
79. The Square of Opposition.—In dealing with the subject of this chapter it will be convenient to begin with the ancient square of opposition which relates exclusively to the traditional schedule of propositions. It will, however, ultimately be found desirable to give more general accounts of what is to be understood by the terms contradictory, contrary, &c., so that they may be adapted to other schedules of propositions.
Two propositions are technically said to be opposed to each other when they have the same subject and predicate respectively, but differ in quantity or quality or both.[108]
[108] This definition, according to which opposed propositions are not necessarily incompatible with one another, is given by Aldrich (p. 53 in Mansel’s edition). Ueberweg (Logic, § 97) defines opposition in such a way as to include only contradiction and contrariety; and Mansel remarks that “subalterns are improperly classed as opposed propositions” (Aldrich, p. 59). Modern logicians, however, usually adopt Aldrich’s definition, and this seems on the whole the best course. Some term is wanted to signify the above general relation between propositions; and though it might be possible to find a more convenient term, no confusion is likely to result from the use of the term opposition if the student is careful to notice that it is here employed in a technical sense.
Taking the propositions SaP, SiP, SeP, SoP, in pairs, we find that there are four possible kinds of relation between them.
(1) The pair of propositions may be such that they can neither both be true nor both false. This is called contradictory opposition, and subsists between SaP and SoP, and between SeP and SiP. 110
(2) They may be such that whilst both cannot be true, both may be false. This is called contrary opposition. SaP and SeP.
(3) They may be such that they cannot both be false, but may both be true. Subcontrary opposition. SiP and SoP.
(4) From a given universal proposition, the truth of the particular having the same quality follows, but not vice versâ.[109] This is subaltern opposition, the universal being called the subalternant, and the particular the subalternate or subaltern. SaP and SiP. SeP and SoP.
[109] This result and some of our other results may need to be modified when, later on, account is taken of the existential interpretation of propositions. But, as stated in the note at the beginning of the chapter, all complications resulting from considerations of this kind are for the present put on one side.