These four propositions correspond to those included in the ordinary square of opposition; and, if we start with the assumption that A is possibly true,[286] the ordinary relations of opposition hold good between them. Am and Om, Em and Im are pairs of contradictories; Am and Em are contraries; Am and Im, Em and Om, are pairs of subalterns; Imand Om are subcontraries.
[286] By this is meant that we start with the assumption that A is possibly true independently of the affirmation of any one of the propositions in question. The reader must particularly notice that this assumption is quite different from the assumption that each of the propositional forms implies as part of its import that A is possibly true; otherwise the results reached in this paragraph may appear to be inconsistent with those reached in the following paragraph.
If, however, it is not assumed that A is possibly true, then the problem is more complicated, since the character of the relations is affected by the manner in which the propositions are interpreted in respect to the possibility of their antecedents. The results are substantially the same as in the case of modal conditionals (section [176]), and correspond with those obtained in section [159], where the analogous problem in regard to categoricals (assertorically interpreted) is discussed. Thus, in order that Am, and Om, Em and Im, may be contradictories, apodeictic and problematic propositions must be interpreted differently as regards the implication or non-implication of the possible truth of their antecedents; while, on the other hand, in order that Am and Im, Em and Om, may be subalterns, 267 problematic propositions must not be interpreted as implying the possible truth of their antecedents unless apodeictic propositions are interpreted similarly in this respect. If we interpret neither apodeictic nor problematic hypotheticals as implying the possible truth of their antecedents, then the contradictory of If A, then C may be expressed in the form Possibly A, but not C (or, as it may also be formulated, A is possibly true, and if it is true, still C need not be true).
It would occupy too much space to discuss in detail all the problems that might be raised in this connexion. The principles involved have been sufficiently indicated; and the reader will find no difficulty in working out other cases for himself. We may, however, touch briefly on the relation between the propositions If A then C and If A then not C, shewing in particular that on no supposition are they true contradictories.
If these two propositions are interpreted assertorically, then so far from being contradictories, they are subcontraries. For, supposing A happens not to be true, then it cannot be said that either of them is false: the statement If A then C merely excludes ACʹ, and If A then Cʹ merely excludes AC ; hence two possibilities are left, AʹC or AʹCʹ, neither of which is inconsistent with either of the propositions.[287] On the other hand, the propositions cannot both be false, since this would mean the truth of both ACʹ and AC.
[287] The validity of the above result will perhaps be more clearly seen by substituting for the hypotheticals their (assertoric) disjunctive equivalents, namely, Either A is not true or C is true, Either A is not true or C is not true. As a concrete example, we may take the propositions, “If this pen is not cross-nibbed, it is corroded by the ink,” “If this pen is not cross-nibbed, it is not corroded by the ink.” Supposing that the pen happens to be cross-nibbed, we cannot regard either of these propositions as false. It will be observed that their disjunctive equivalents are, “This pen is either cross-nibbed or corroded by the ink,” “This pen is either cross-nibbed or not corroded by the ink.” Take again the propositions, “If the sun moves round the earth, some astronomers are fallible.” “If the sun moves round the earth, all astronomers are infallible.” The truth of the first of these propositions will not be denied, and on the interpretation of hypotheticals with which we are here concerned the second cannot be said to be false. It may be taken as an emphatic way of denying that the sun does move round the earth.
Returning to the modal interpretation of the propositions, then if interpreted as implying the possible truth of their 268 antecedents, they are contraries. They cannot both be true, but may both be false. It may be that neither the truth nor the falsity of C is a necessary consequence of the truth of A.[288]
[288] It has been argued that If A then C must have for its contradictory If A then not C, since the consequent must either follow or not follow from the antecedent. But to say that C does not follow from A is obviously not the same thing as to say that not-C follows from A.
Once more, if interpreted modally but not as implying the possible truth of their antecedents, the propositions may both be true as well as both false. This case is realised when we establish the impossibility of the truth of a proposition by shewing that, if it were true, inconsistent results would follow.
180. Immediate Inferences from Hypothetical Propositions.—The most important immediate inference from the proposition If A then C is If Cʹ then Aʹ. This inference is analogous to contraposition in the case of categoricals, and may without any risk of confusion be called by the same name. We may accordingly define the term contraposition as applied to hypotheticals as a process of immediate inference by which we obtain a new hypothetical having for its antecedent the contradictory of the old consequent, and for its consequent the contradictory of the old antecedent. If we recognise distinctions of quality in hypotheticals, then (as regards apodeictic hypotheticals) this process is valid in the case of affirmatives only. It will be observed that from the contrapositive we can pass back to the original proposition; and from this it follows that the original proposition and its contrapositive are equivalents.[289] The following are examples: “If patience is a virtue, there are painful virtues” = “If there are no painful virtues, patience is not a virtue”; “If there is a righteous God, the wicked will not escape their just punishment” = “If the wicked escape their just punishment, there is no righteous God.”