Particular conditionals, so far as they are merely assertoric, are almost without exception based upon specific experience. Hence they may not unreasonably be interpreted as implying the occurrence of their antecedents, as, for example, in the 256 proposition, “Sometimes when Parliament meets, it is opened by the Sovereign in person.” The existential interpretation of categoricals for which a preference was expressed in the preceding chapter may therefore be adopted for conditionals also, so far as they are merely assertoric; and the two forms become mutually interchangeable.
On the whole, except in so far as we adopt the convention indicated under (i) above, there seems no reason for drawing a vital distinction between judgments according as they are expressed in the conditional or the categorical form.[274] Many of the conditionals of ordinary discourse are indeed so obviously equivalent to categoricals that they hardly seem to require a separate consideration.[275] At the same time, as we have seen, some statements fall more naturally into the one form and some into the other. The more complex the subject-term, the greater is the probability that the natural form of the proposition will be conditional.
[274] It has been argued that, starting from the categorical form, we cannot pass to the conditional, if the subject of the proposition is a simple term. The basis of this argument is that the antecedent of a conditional requires two terms, and that in the case supposed these are not provided by the categorical. Thus, Miss Jones (Elements of Logic, p. 112) takes the example, “All lions are quadrupeds.” It will not do, she says, to reduce this to the form, “If any creatures are lions, they are quadrupeds,” since this involves the introduction of a new term, and passing back again to the categorical form, we should have “All creatures which are lions are quadrupeds,” a proposition not equivalent to our original proposition. If, however, “creature” is regarded as part of the connotation of “lion,” there is no reason for refusing to allow that the two propositions are equivalent to one another. Similarly, in any concrete instance, by taking some part of the connotation of the subject of our categorical proposition, we can obtain the additional term required for its reduction to the conditional form. Where we are dealing with purely symbolic expressions, and this particular solution of the difficulty is not open to us, we may have recourse to the all-embracing term “anything,” such a proposition as All S is P being reduced to the form If anything is S it is P.
[275] The examples given at the commencement of section [173] are reducible to the following categoricals: Import duties which are sources of revenue do not afford protection ; All spoilt children have suffering parents ; All pairs of straight lines which are such that another straight line falling upon them makes the alternate angles equal to one another are parallel ; All occasions of the application of a lighted match to gunpowder are occasions of an explosion ; Any place where there is a carcase is a place where the eagles will gather together.
176. The Opposition of Conditional Propositions.—This question needs a separate discussion according as conditionals are interpreted (a) assertorically, or (b) modally.
257 (a) If conditionals are interpreted assertorically, then the ordinary distinctions both of quality and of quantity can be applied to them in just the same way as to categoricals. We may regard the quality of a conditional as determined by the quality of its consequent; thus, the proposition If any P is Q then that P is not R may be treated as negative.[276] As regards quantity, conditionals are to be regarded as universal or particular, according as the consequent is affirmed to accompany the antecedent in all or merely in some cases.
[276] The negative force of this proposition would be more clearly brought out if it were written in the form If any P is Q then it is not the case that it is also R. The categorical equivalent is No PQ is R.
We have then the four types included in the ordinary four-fold schedule:—
If any P is Q, it is also R ; A
If any P is Q, it is not also R ; E
Sometimes if a P is Q, it is also R ; I
Sometimes if a P is Q, it is not also R. O
These propositions constitute the ordinary square of opposition, and if conditionals are assimilated to categoricals so far as their existential import is concerned, then the opposition of conditionals on the assertoric interpretation seems to require no separate discussion.[277] It may, however, be pointed out that there is more danger of contradictories being confused with contraries in the case of conditionals than in the case of categoricals. If A is B then C is not D is very liable to be given as the contradictory of If A is B then C is D. But it is clear on consideration that both these propositions may be false. For example, the two statements—If the Times says one thing, the Westminster Gazette says another; If the Times says one thing, the Westminster Gazette says the same, i.e., does not say another—might be, and as a matter of fact are, both false; the two papers are sometimes in agreement and sometimes not.