[269] The above distinction has been adopted in some recent treatises on Logic, but it must be borne in mind that most logicians use the terms conditional and hypothetical as synonymous or else draw a distinction between them different from the above.
The parts of the conditional and also of the true hypothetical are called the antecedent and the consequent. Thus, in the proposition If A is B, C is D, the antecedent is A is B, the consequent is C is D.
It is impossible formally to distinguish between conditionals and hypotheticals so long as we keep to the expression If A is B, C is D, since this may be either the one or the other. The following forms, however, are unmistakeably conditional: Whenever A is B, C is D ; In all cases in which A is B, C is D ; If any P is Q then that P is R.[270] The form If A is true then C is true is, on the other hand, distinctively hypothetical. A and C here stand for propositions or judgments, not terms, and the words “is true” are introduced in order to make this explicit. It is quite sufficient, however, to write the true hypothetical in the form If A then C.
[270] Conditionals can generally be reduced to the last of these three forms without much difficulty, and such reduction is sometimes useful. A consideration of the concrete examples already given will, however, shew that a certain amount of manipulation may be required in order to effect the reduction. The following are examples: If any child is spoilt, then that child will have suffering parents ; If any two straight lines are such that another straight line falling upon them makes the alternate angles equal to one another, then those two straight lines are parallel to one another.
251 Since a conditional proposition usually contains a reference to some concurrence in time or space, the if of the antecedent may as a rule be replaced either by when or by where, as the case may be, without any change in the significance of the proposition; but the same cannot be said in the case of the true hypothetical. This consideration will often suffice to resolve any doubt that may arise in concrete cases as to the particular type to which any given proposition belongs. Another and more fundamental criterion may be found in the answer to the question whether or not the antecedent and consequent are propositions of independent import, whose meaning will not be impaired if they are considered apart from one another. If the answer is in the affirmative, then the proposition is hypothetical. Thus, taking examples of hypotheticals already given, we find that the antecedents, It is a sin to covet honour, Patience is a virtue, Virtue is involuntary, and the consequents, I am the most offending soul alive, There are painful virtues, Vice is involuntary, all retain their full meaning though separated from one another. If, on the other hand, the consequent necessarily refers us back to the antecedent in order that it may be fully intelligible, then the proposition is conditional. Thus, taking by itself the consequent in the first conditional given on [page 249], namely, it does not afford protection, we are at once led to ask what is here meant by it. The answer is—that import duty. But what import duty? An adequate answer can be given only by introducing into the consequent the whole of the antecedent,—an import duty which is a source of revenue does not afford protection. We now have the full force of our original conditional proposition in the form of a single categorical. It will be found that if other conditionals are treated in the same way, they resolve themselves similarly into categoricals of the form All PQ is R.[271] 252 The problem of the reduction of conditionals and hypotheticals to categorical form will be considered in more detail [later] on in this chapter, and it will be shewn that whilst such reduction is always possible, and generally simple and natural, in the case of conditionals, it is not possible at all (with terms corresponding to the original antecedent and consequent) in the case of hypotheticals.[272]
[271] As another example, we may take the conditional proposition, If the weather is dry, the British root-crops are light. Here it may at first sight appear that the consequent is a proposition of independent import. The proposition, The British root-crops are light, is, however, a judgment incompletely stated. For it contains a time-reference that needs to be made explicit. The conditional really means, If in any year the weather is dry, the British root-crops in that year are light ; and this is equivalent to the categorical, Any year in which the weather is dry is a year in which the British root-crops are light. By looking at the conditional in this way, we see the necessity of referring back to the antecedent in order that the consequent may be fully expressed.
[272] The question may be raised whether a proposition of the form, If this P is Q, it is R, is properly to be described as a singular conditional or as a hypothetical. The answer is that a proposition of this form affords a kind of junction between the conditional and the hypothetical; it is derivable from the conditional, If any P is Q, it is R ; but it is itself hypothetical. The antecedent and the consequent are propositions of independent import; and the proposition as a whole is not directly reducible (as is the conditional, If any P is Q, it is P) to categorical form. Thus, the proposition, If any P is Q, it is R, may prima facie be reduced to the form Any P that is Q is R ; but the proposition, If this P is Q, it is R, certainly cannot be identified with the singular categorical, This P which is Q is R.
174. The Import of Conditional Propositions.—It is sometimes held that the real differentia of all propositions of the form If A is B, C is D is “to express human doubt.” Clearly, however, there is no intention to express doubt as regards the relation between the antecedent and the consequent; and the doubt must, therefore, be supposed to relate to the actual occurrence of the antecedent. But so far at any rate as conditionals are concerned, the doubt which they may thus imply must be considered incidental rather than the fundamental or differentiating characteristic belonging to them. The if of the conditional may, as we have seen, usually be replaced by when without altering the significance of the proposition, and in this case the element of doubt is no more prominent than in the categorical proposition. From the material standpoint, conditionals may or may not involve the actual occurrence of their antecedents. Whenever the connexion between the antecedent and the consequent can be inferred from the nature of the antecedent independently of specific experience (and this may be the more usual case), then the actual happening of the 253 antecedent is not involved; but if our knowledge of the connexion does depend on specific experience (as it sometimes may), then such actual happening is materially involved. For example, the statement, “If we descend into the earth, the temperature increases at a nearly uniform rate of 1° Fahr. for every fifty feet of descent down to almost a mile,” is based upon knowledge gained by actual descents into the earth having been made, and apart from such experience the truth of the statement would not have been known.
The question of main importance in regard to the import of conditional propositions is whether such propositions are to be interpreted as modal or as merely assertoric. Confining ourselves for the present to the universal affirmative, that is, to the form If any P is Q then it is R, are we affirming a necessary relation between P being Q and its being R, or are we merely affirming that it so happens that every P that is Q is also R? This is really in another form the distinction already drawn between unconditionally universal propositions and empirically universal propositions, and our answer must again be that the same form of words may express the one judgment or the other. There can be no doubt that the proposition, If the angles at the base of a triangle are equal to one another, that triangle is isosceles, is intended to be interpreted modally as expressing a necessary connexion, while the proposition, If any book is taken down from that shelf, it will be found to be a novel, would be intended to be interpreted merely assertorically.
In ordinary discourse conditionals are as a rule modal; but this is not universally the case. Unless, therefore, we are prepared to depart from ordinary usage (and there is a good deal to be said for such departure), we must recognise both assertoric conditionals and modal conditionals, and this distinction must be borne in mind in all that follows. We shall find that practically the same problem arises in regard to true hypotheticals, and we shall have to consider it further in that connexion.