CHAPTER IX.

CONDITIONAL AND HYPOTHETICAL PROPOSITIONS.

173. The distinction between Conditional Propositions and Hypothetical Propositions[268]—Propositions commonly written in the form If A is B, C is D belong to two very different types. For they may be the expression either of simple judgments or of compound judgments (as distinguished in section [55]).

[268] For the distinction indicated in the present section I was in the first instance indebted to an essay, written in 1884, by Mr W. E. Johnson. This essay has not been published in its original form; but the substance of it has been included in some papers on The Logical Calculus by Mr Johnson which appeared in Mind in 1892.

In the first place, A being B and C being D may be two events or two combinations of properties, concerning which it is affirmed that whenever or wherever the first occurs the second will occur also. For example, If an import duty is a source of revenue, it does not afford protection ; If a child is spoilt, his parents suffer ; If a straight line falling upon two other straight lines makes the alternate angles equal to one another, the two straight lines are parallel to one another ; If a lighted match is applied to gunpowder, there will be an explosion ; Where the carcase is, there shall the eagles be gathered together. What is affirmed in all such cases as these is a connexion between phenomena; it may be either a co-inherence of attributes in a common subject, or a relation in time or space between certain occurrences. Propositions belonging to this type may be called distinctively conditional.

But again, A is B and C is D may be two propositions of independent import, the relation between which cannot be 250 directly resolved into any time or space relation or into an affirmation of the co-inherence of attributes in a common subject. In other words, a relation may be affirmed between the truth of two judgments as holding good once and for all without distinction of place or time or circumstance. For example, If it be a sin to covet honour, I am the most offending soul alive ; If patience is a virtue, there are painful virtues ; If there is a righteous God, the wicked will not escape their just punishment ; If virtue is involuntary, so is vice ; If the earth is immoveable, the sun moves round the earth. Propositions belonging to this type may be called hypothetical as distinguished from conditional, or they may be spoken of still more distinctively as true hypotheticals or pure hypotheticals.[269]

[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.

175. Conditional Propositions and Categorical Propositions.—We may go on to consider what is the essential nature of the distinction between conditional propositions and 254 categorical propositions, and in particular whether the distinction is one of verbal form only or one that corresponds to a real distinction between judgments.

If a vital distinction is to be drawn between the two forms, it must be on one or other of the two following grounds, namely, either (i) that the categorical is to be interpreted assertorically while the conditional is to be interpreted modally, or (ii) that the categorical is to be interpreted as implying the existence of its subject while the conditional is not to be interpreted as implying the occurrence of its antecedent.

(i) There is much to be said for adopting a convention by which the categorical form would be interpreted assertorically and the conditional form modally. The adoption of this convention would, however, necessitate some modification of the forms of ordinary speech, for, as we have already seen, the proposition All S is P is in current use sometimes apodeictic, while the proposition If any S is P then it is Q may (though perhaps rarely) be merely assertoric. Whether the one form or the other is used really depends a good deal on linguistic considerations. Consider, for instance, the propositions, All isosceles triangles have the angles at their base equal to one another, If the angles at the base of a triangle are equal to one another, that triangle is isosceles. These propositions fall naturally into the categorical and conditional forms respectively, simply because there happens to be no single adjective (like “isosceles”) which connotes “having two equal angles.” It is clear, however, that the use of the one form rather than the other is not intended to imply any fundamental difference in the character of the relation asserted. If either of the propositions in its ordinary use is apodeictic, so is the other; if either is merely assertoric, so is the other.

It is to be added that if we adopt the convention under consideration then the universal categorical is inferable from the universal conditional, but not vice versâ ; while, on the other hand, the problematic conditional (which corresponds to the particular) is inferable from the particular categorical, but not vice versâ. Thus, All PQ is R is subaltern to If any P is Q it 255 is R, while If any P is Q it may be R is subaltern to Some PQ is R.

(ii) We may pass on to consider whether categoricals and conditionals are to be differentiated in respect of their existential import.

We have seen in section [163] that if categoricals are interpreted modally they are not to be regarded as necessarily implying the existence of their subjects; and certainly conditionals, interpreted modally, are not to be regarded as necessarily implying the occurrence of their antecedents. Hence if both propositional forms are interpreted modally, we have no differentiation as regards their existential import.

It further seems clear that, so far as universal are concerned, a conditional proposition—even though interpreted as merely assertoric—is not to be regarded as necessarily implying the actual occurrence of its antecedent. Hence, whether, on the assertoric interpretation of both, the two forms are to be existentially differentiated depends upon our existential interpretation of the categorical.

(a) If a universal categorical is interpreted as necessarily implying the actual existence of its subject, then we have a marked distinction between the two forms.[273] If any P is Q then it is also R cannot be resolved into All PQ is R, since the latter implies the existence of PQ, while the former does not.

[273] This is Ueberweg’s view, “The categorical judgment, in distinction from the hypothetical, always includes the pre-supposition of the existence of the subject” (Logic, § 122).

(b) If, on the other hand, universal categoricals are not interpreted as necessarily implying the existence of their subjects, then universal conditionals and universal categoricals (both being interpreted assertorically) may be resolved into one another. We may say indifferently All S is P or If anything is S it is P ; If ever A is B then on all such occasions C is B or All occasions of A being B are occasions of C being D.

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.

[277] The four propositions are precisely equivalent to the four categoricals,—All PQ is R, No PQ is R, Some PQ is R, Some PQ is not R.

(b) On the modal interpretation, the distinction between 258 apodeictic and problematic takes the place of that between universal and particular; and if we maintain the distinction between affirmative and negative, we have the four following propositions corresponding to the ordinary square of opposition:

If any P is Q, that P must be R ;  Am
If any P is Q, that P cannot be R ;  Em
If any P is Q, that P may be R ;  Im
If any P is Q, that P need not be R.  Om

It will be convenient to have distinctive symbols to denote modal propositions, and those that we have here introduced will serve to bring out the analogies between modals and the ordinary assertoric forms.

In the above schedule, subject to a certain condition mentioned below, Am and Om, and also Em and Im, are contradictories according to the definition given in section [84]; Am and Em are contraries; Am and Im, and also Em and Om, are subalterns; and Im and Om are subcontraries.

The condition referred to relates to the interpretation of the propositions as regards the implication of the possibility of their antecedents. Thus, in order that Am and Om (or Em and Im) may be true contradictories it is necessary that apodeictic and problematic propositions shall be interpreted differently in this respect. If, for example, Am is interpreted as not implying the possibility of its antecedent then its full import is to deny the possibility of the combination P and Q without R. Its contradictory must affirm this possibility. Om will not, however, do this unless it is interpreted as implying the possibility of the combination P, Q.

It is necessary to call attention to this complication, but hardly necessary to work out in detail the results which follow from the various principles of interpretation that might be adopted. If the student will do this for himself, he will find that the results correspond broadly with those obtained in section [159].[278]

[278] In connexion with the problem of opposition we may touch briefly on the relation between the apodeictic proposition If any P is Q that P must be R and the assertoric proposition Some PQ is not R. These propositions are not contradictories, for they may both be false. They cannot, however, both be true; and the latter, if it can be established, affords a valid ground for the denial of the former. Mr Bosanquet appears not to admit this, but to maintain, in opposition to it, that the enumerative particular is of no value as overthrowing the abstract universal. “When we have said that If (i.e., in so far as) a man is good, he is wise, it is idle to reply that Some good men are not wise. This is to attach an abstract principle with unanalysed examples. What we must say in order to deny the above-mentioned abstract judgment is something of this kind: If or Though a man is good, yet it does not follow that he is wise, that is, Though a man is good, yet he need not be wise” (Logic, i. p. 316). But surely if we find that some good men are not wise, we are justified in saying that though a man is good yet he need not be wise. Of course the converse does not hold. We might be able to shew that wisdom does not necessarily accompany goodness by some other method than that of producing instances. But if we can produce undoubted instances, that amply suffices to confute the apodeictic conditional.

259 177. Immediate Inferences from Conditional Propositions.—In a conditional proposition the antecedent and the consequent correspond respectively to the subject and the predicate of a categorical proposition. In conversion, therefore, the old consequent must be the new antecedent, and in contraposition the negation of the old consequent must be the new antecedent.

(a) On the assertoric interpretation, the analogy with categoricals is so close that it is unnecessary to treat immediate inferences from conditionals in any detail. One or two examples may suffice. Taking the A proposition, If any P is Q then it is R, we have for its converse Sometimes if a P is R it is also Q, and for its contrapositive If any P is not R then it is not Q. Taking the E proposition If any P is Q then it is not R, we have for its converse If any P is R then it is not Q, and for its contrapositive Sometimes if a P is not R it is Q. The validity of these inferences is of course affected by the existential interpretation of the propositions just as in the case of the categoricals. It will be noticed that in some immediate inferences (for example, the contraposition of A) the conditional form has an advantage over the ordinary categorical form inasmuch as it avoids the use of negative terms, the employment of which is so strongly objected to by Sigwart and some other logicians.[279]

[279] Compare page [136].

(b) If conditionals are interpreted modally, then the apodeictic form takes the place of the universal, and the 260 problematic takes the place of the particular. On this basis, the converse of If any P is Q that P must be R would be If any P is R that P may be Q, and the contrapositive would be If any P is not R that P cannot be Q.

Are these inferences legitimate? On the interpretation that a modal proposition implies nothing as to the possibility of its antecedent, then our answer must be in the affirmative, as regards the contraposition of Am. The full import both of the original proposition and of the contrapositive is to deny the possibility of the combination P and Q without R. On the same interpretation, however, the conversion of Am is not valid. For the converse implies that if PR is possible then PQ is possible, while the possibility of PR combined with the impossibility of PQ is compatible with the truth of the original proposition. It can be shewn similarly that, while the conversion of Em is valid, its contraposition is invalid.

If we were to vary the interpretation, the results would be different.

The correspondence between the results shewn above and our results respecting the conversion and contraposition of the assertoric A and E propositions, on the interpretation that no proposition implies the existence of its subject (see page [225]), is obvious. The truth is that the interpretation of modals in respect to the possibility of their antecedents gives rise to problems precisely analogous to those arising out of the interpretation of assertoric propositions in respect to the actuality of their subjects. It is unnecessary that we should work out the different cases in detail.

Amongst immediate inferences from a conditional proposition, its reduction to categorical form, so far as this is valid, is generally included. This is a case of what has been called change of relation, meaning thereby an immediate inference in which we pass from a given proposition to another which belongs to a different category in the division of propositions according to relation (see section [54]). The more convenient term transversion is used by Miss Jones for this process.

How far conditionals can be inferred from categoricals and vice versâ depends on their interpretation. If both types of 261 propositions are interpreted assertorically or both modally, and if they are interpreted similarly as regards the implication of the existence (or possibility) of their subjects (or antecedents), then the validity of passing from either type to the other cannot be called in question. Some doubt may, however, be raised as to whether in this case we have an inference at all or merely a verbal change. This is a distinction to which attention will be called later on.

If conditionals are interpreted modally and categoricals assertorically then (apart from any complications that may arise from existential implications) A can be inferred from Am or E from Em, but not vice versâ. On the other hand, Im can be inferred from I, or Om from O, but not vice versâ.

We have another case of transversion when we pass from conditional to disjunctive, or from disjunctive to conditional. The consideration of this case must be deferred until we have discussed disjunctives.

178. The Import of Hypothetical Propositions.—The pure hypothetical may be written symbolically in the form If A is true then C is true, or more briefly, If A then C, where A and C stand for propositions of independent import. It is clear that this proposition affirms nothing as regards the truth or falsity of either A or C taken separately. We may indeed frame the proposition, knowing that C is false, with the express object of showing that A is false also. What we have is of course a judgment not about either A or C taken separately, but about A and C in relation to one another.

The main question at issue in regard to the import of the hypothetical proposition is whether it is merely assertoric or is modal. The contrast may be simply put by asking whether, when we say If A then C, our intention is merely to deny the actuality of the conjunction of A true with C false or is to declare this conjunction to be an impossibility.

The contrast between these two interpretations can be brought out most clearly by asking how the proposition If A then C is to be contradicted. If our intention is merely to deny the actuality of the conjunction of A true with C false, then the contradictory must assert the actuality of this conjunction; if 262 our intention is to deny the possibility of the conjunction, then the contradictory will merely assert its possibility. In other words, on the assertoric interpretation the contradictory will be A is true but C is false ;[280] on the modal interpretation it will be If A is true C may be false.[281]

[280] We may look at it in this way. Let AC denote the truth of both A and C, ACʹ the truth of A and the falsity of C, and so on. Then there are four à priori possibilities, namely, AC, ACʹ, AʹC, AʹCʹ, one or other of which must hold good, but any pair of which are mutually inconsistent. The proposition If A then C merely excludes ACʹ, and still leaves AC, AʹC, AʹCʹ, as possible alternatives. In denying it, therefore, we must definitely affirm ACʹ, and exclude the three other alternatives. Hence the contradictory as above stated.

[281] A certain assumption is necessary, in order that this result may be correct. The opposition of hypotheticals on the modal interpretation will be discussed in more detail in section [179].

Hypotheticals intended to be interpreted assertorically are to be met with in ordinary discourse, but they are unusual. There appear to be two cases: (a) When we know that one or other of two propositions is true but do not know (or do not remember) which, we may express our knowledge in the form of a hypothetical, If X is not true then Y is true, and such hypothetical will be merely assertoric. For example, If the flowers I planted in this bed were not pansies they were violets. Here the intention is merely to deny the actuality of the flowers being neither pansies nor violets. (b) We may deny a proposition emphatically by a hypothetical in which the proposition in question is combined as antecedent with a manifestly false consequent; and such hypothetical will again be merely assertoric. For example, If what you say is true, I’m a Dutchman ; If that boy comes back, I’ll eat my head (vide Oliver Twist); I’m hanged if I know what you mean. In these examples the intention is to deny the actuality (not the possibility) of the conjunctions,—What you say is true and I am not a Dutchman ; That boy will come back and I shall not eat my head ; I am not hanged and I know what you mean; and since the elements of the conjunctions printed in italics are admittedly true, the force of the propositions is to deny the truth of the other elements, that is to say, to affirm,—What you say is not true, That boy will not come back, I do not know what you mean. Similarly 263 we may sometimes employ the hypothetical form of expression as an emphatic way of declaring the truth of the consequent (an antecedent being chosen which is admittedly true); for example, If he cannot act, at any rate he can sing. Here once more the hypothetical is merely assertoric.

It cannot, however, be maintained that any of the above are typical hypotheticals; and the claim that our natural interpretation of hypotheticals is ordinarily modal may be justified on the ground that we do not usually consider it to be necessary to affirm the antecedent in order to be able to deny a hypothetical. We have seen that, in order to deny the assertoric hypothetical If A then C, we must affirm A and deny C ; but we should usually regard it as sufficient for denial if we can shew that there is no necessary connexion between the truth of A and that of C, whether A is actually true or not.

We shall then in the main be in agreement with ordinary usage if we interpret hypotheticals modally, and the adoption of such an interpretation will also give hypotheticals a more distinctive character. In what follows the hypothetical form will accordingly be regarded as modal, except in so far as an explicit statement is made to the contrary.[282]

[282] Either C is true or A is not true is usually regarded as the disjunctive equivalent of the hypothetical If A is true then C is true. The relation between these two propositions will be discussed further later on. It is, however, desirable to point out at once that, if the equivalence is to hold good, both the propositions must be interpreted assertorically or both modally. There is a good deal to be said for differentiating the two forms by regarding the hypothetical as modal and the disjunctive as merely assertoric. This method of treatment is explicitly adopted by Mr McColl. He writes (using the symbolism, a : b for If a then b, a + b for a or b, aʹ for the denial of a)—“The expression a : b may be read a implies b or If a is true, b must be true. The statement a : b implies aʹ + b. But it may be asked are not the two statements really equivalent; ought we not therefore to write a : b = aʹ + b? Now if the two statements are really equivalent their denials will also be equivalent. Let us see if this will be the case, taking as concrete examples: ‘If he persists in his extravagance he will be ruined’; ‘He will either discontinue his extravagance or he will be ruined.’ The denial of a : b is (a : b)ʹ and this denial may be read—‘He may persist in his extravagance without necessarily being ruined.’ The denial of aʹ + b is abʹ which may be read—‘He will persist in his extravagance and he will not be ruined.’ Now it is quite evident that the second denial is a much stronger and more positive statement than the first. The first only asserts the possibility of the combination abʹ ; the second asserts the certainty of the same combination. The denials of the statements a : b and aʹ + b having thus been proved to be not equivalent, it follows that the statements a : b and aʹ + b are themselves not equivalent, and that, though aʹ + b is a necessary consequence of a : b, yet a : b is not a necessary consequence of aʹ + b” (see Mind, 1880, pp. 50 to 54; one or two slight verbal changes have been made in this quotation).

264 Some writers who adopt the modal interpretation of hypotheticals speak of the consequent as being an inference from the antecedent. There are no doubt some hypotheticals to which this description accurately applies. Thus, we may have hypotheticals which are formal in the sense in which that term has been used in section [31], the consequent being, for instance, an immediate inference from the antecedent, or being the conclusion of a syllogism of which the premisses constitute the antecedent. The following are examples,—If all isosceles triangles have the angles at the base equal to one another, then no triangle the angles at whose base are unequal can be isosceles ; If all men are mortal and the Pope is a man, then the Pope must be mortal.

But more usually the consequent of a hypothetical proposition cannot be inferred from the antecedent alone. The aid is required of suppressed premisses which are taken for granted, the premiss which alone is expressed being perhaps the only one as to the truth of which any doubt is regarded as admissible. It would, therefore, be better to speak of the consequent as being the necessary consequence of the antecedent, than as being an inference from it. When we speak of C as being an inference from A, there is a suggestion that A affords the complete justification of C, whereas when we speak of it as a necessary consequence, this suggestion is at any rate less prominent.[283]

[283] Miss Jones (General Logic, p. 45) divides hypotheticals into formal or self-contained hypotheticals and referential hypotheticals. In the former, “the consequent is an inference from the antecedent alone”; in the latter, “the consequent is inferred not from the antecedent alone, but from the antecedent taken in conjunction with some other unexpressed proposition or propositions.”

179. The Opposition of Hypothetical Propositions.—Regarding hypotheticals as always affirming a necessary consequence, it may reasonably be held that they do not admit of distinctions of quality. Sigwart accordingly lays it down that all hypotheticals are affirmative. “Passing to hypothetical judgments 265 containing negations, we find that the form ‘If A is, B is not’ represents the negation of a proposition as the necessary consequence of an affirmation, thus affirming that the hypotheses A and B are incompatible.”[284] The force of this argument must be admitted. There is, however, some convenience in distinguishing between hypotheticals according as they lead up, in the consequent, to an affirmation or a denial; and in the formal treatment of hypotheticals, we shall be better able to preserve an analogy with categoricals and conditionals if we denote the proposition If X is true then Y is true by the symbol Am, and the proposition If X is true then Y is not true by the symbol Em.

[284] Logic, i. p. 226.

Whether or not we decide thus to recognise distinctions of quality in the case of hypotheticals, we certainly cannot recognise distinctions of quantity. The antecedent of a hypothetical is not an event which may recur an indefinite number of times, but a proposition which is simply true or false. We have already seen that the same proposition cannot be sometimes true and sometimes false, since propositions referring to different times are different propositions.[285]

[285] This, as Mr Johnson has pointed out, must be taken in connexion with the recognition of propositions as involving multiple quantification. “Thus we may indicate a series of propositions involving single, double, triple … quantification, which may reach any order of multiplicity: (1) ‘All luxuries are taxed’; (2) ‘In some countries all luxuries are taxed’; (3) ‘At some periods it is true that in all countries all luxuries are taxed’.… with respect to each of the types of proposition (1), (2), (3).… I contend that, when made explicit with respect to time or place, etc., it is absurd to speak of them as sometimes true and sometimes false” (Mind, 1892, p. 30 n.).

Do not distinctions of modality, however, take the place of distinctions of quantity? Up to this point, we have practically confined our attention to the apodeictic hypothetical, If A then C. This proposition is denied by the proposition If A is true still C need not be true (that is to say, The truth of C is not a necessary consequence of the truth of A). Can this latter proposition be described as a problematic hypothetical? Clearly it is not a hypothetical at all if we begin by defining a hypothetical as the affirmation of a necessary consequence. There seems, however, no need for this limitation. We may define a 266 hypothetical as a proposition which starting from the hypothesis of the truth (or falsity) of a given proposition affirms (or denies) that the truth (or falsity) of another proposition is a necessary consequence thereof. But, whether or not we adopt this definition, there can be no doubt that the proposition If A then possibly C appropriately finds a place in the same schedule of propositions as If A then necessarily C. In such a schedule we have the four forms,—

If A is true then C is true ;  Am

If A is true then C is not true ;  Em

If A is true still C may be true ;  Im

If A is true still C need not be true.  Om

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.”

[289] This holds good whether we adopt the assertoric or the modal interpretation. On the former interpretation, the import of both the propositions If A then C and If Cʹ then Aʹ is to negative ACʹ ; on the latter interpretation, the import of both is to deny the possibility of the conjunction ACʹ.

From the negative hypothetical If A is true then C is not true we can infer If C is true then A is not true. This is analogous to conversion in the case of categoricals.

269 From the affirmative If A then C, we may obtain by conversion If C then possibly A ; but this is only on the interpretation that both propositions imply the possibility of the truth of their antecedents.[290] The reader will notice that to pass from If A then C to If C then A would be to commit a fallacy analogous to simply converting a categorical A proposition; and this is perhaps the most dangerous fallacy to be guarded against in the use of hypotheticals.[291]

[290] Compare section [158]. The various results obtained in section 158 may be applied mutatis mutandis to modal hypotheticals. The reader may consider for himself the contraposition of Em.

[291] On the assertoric interpretation If A then C merely negatives ACʹ, while If C then A merely negatives AʹC, and hence it is clear that neither of these propositions involves the other; on the modal interpretation the result is the same, for the truth of C may be a necessary consequence of the truth of A, while the converse does not hold good.

A consideration of immediate inferences enables us to shew from another point of view that If A then C and If A then Cʹ are not true contradictories. For the contrapositives If A then Cʹ, If C then Aʹ, are equivalent to one another; and whenever two propositions are equivalent, their contradictories must also be equivalent. But If A then C is not equivalent to If C then A.

If distinctions of quality are admitted, then the process of obversion is applicable to hypotheticals. For example, If A is true then C is not true = If A is true then Cʹ is true. It is nearly always more natural and more convenient to take hypotheticals in their affirmative rather than in their negative form; and hence in the case of hypotheticals more importance attaches to the process of contraposition than to that of conversion.

If the falsity of C is assumed to be possible, then we may pass by inversion from If A then C to It is possible for both A and C not to be true ; or, putting the same thing in a different way, we may by inversion pass from If A then C to If the falsity of C is possible then the falsity of both A and C is possible.[292] It is of course a fallacy to argue from If A then C to If Aʹ then Cʹ.

[292] The inversion of Em may be worked out similarly. Here, as elsewhere, the process of inversion, although of little or no practical importance, raises problems that are of considerable theoretical interest.

Turning to problematic hypotheticals, we find that from the proposition If A is true C may be true, we obtain by conversion If C is true A may be true ; and from the proposition If A is 270 true C need not be true we obtain by contraposition If C is true A need not be true. Here the analogy with categoricals is again very close.

181. Hypothetical Propositions and Categorical Propositions.—A true hypothetical proposition has been defined as a proposition expressing a relation between two other propositions of independent import, not between two terms; and it follows that a true hypothetical is not, like a conditional, easily reducible to categorical form. So far as we can obtain an equivalent categorical, its subject and predicate will not correspond with the antecedent and consequent of the hypothetical. Thus, the proposition If A then C may, according to our interpretation of it, be expressed in one or other of the following forms; A is a proposition the truth of which is incompatible with the falsity of C ; A is a proposition from the truth of which the truth of C necessarily follows. It will be observed that, apart from the fact that these propositions are not of the ordinary categorical type,[293] the predicate is not in either of them equivalent to the consequent of the hypothetical.[294] No doubt a hypothetical proposition may be based on a categorical proposition of the ordinary type. But that is quite a different thing from saying that the two propositions are equivalent to one another.

[293] Since they are compound, not simple, propositions. The expression of compound propositions in categorical form is not convenient, and it is better to reserve the hypothetical and disjunctive forms for such propositions, the categorical and conditional forms being used for simple propositions.

[294] Amongst other differences the contrapositives of both these propositions differ from the contrapositive of the hypothetical. For, on either interpretation of the hypothetical, its contrapositive is If C is not true then A is not true, whilst the contrapositives of the above propositions are respectively,—A proposition whose truth is compatible with the falsity of the proposition C is not the proposition A, A proposition from which the proposition C is not a necessary consequence is not the proposition A.

The relation between hypothetical and disjunctive propositions will be discussed in the following [chapter].

182. Alleged Reciprocal Character of Conditional and Hypothetical Judgments.—Mr Bosanquet argues that the hypothetical judgment (and under this designation he would include the conditional as well as what we have called the true 271 hypothetical) “when ideally complete must be a reciprocal judgment. If A is B, it is C must justify the inference If A is C, it is B. We are of course in the habit of dealing with hypothetical judgments which will not admit of any such conversion, and the rules of logic accept this limitation … If in actual fact … AB is found to involve AC while AC does not involve AB, it is plain that what was relevant to AC was not really AB but some element αβ within it … Apart from time on the one hand and irrelevant elements on the other, I cannot see how the relation of conditioning differs from that of being conditioned … In other words, if there is nothing in A beyond what is necessary to B, then B involves A just as much as A involves B. But if A contains irrelevant elements, then of course the relation becomes one-sided … The relation of Ground is thus essentially reciprocal, and it is only because the ‘grounds’ alleged in every-day life are burdened with irrelevant matter or confused with causation in time, that we consider the Hypothetical Judgment to be in its nature not reversible” (Logic, I. pp. 261–3).

The question here raised is analogous to that of the possibility of plurality of causes which is discussed in inductive logic. It may perhaps be described as a wider aspect of the same question. So long as a given consequence has a plurality of grounds, it is clear that the hypothetical proposition affirming it to be a consequence of a particular one of these grounds cannot admit of simple conversion, for the converted proposition would hold good only if the ground in question were the sole ground.

Mr Bosanquet urges that the relation between ground and consequence will become reciprocal by the elimination from the antecedent of all irrelevant elements. It should be added that we can also secure reciprocity by the expansion of the consequent so that what follows from the antecedent is fully expressed. Thus, if we have the hypothetical If A then γ, which is not reciprocal, it is possible that A may be capable of analysis into αβ, and γ of expansion into γδ, so that either of the hypotheticals If α then γ, If αβ then γδ, is reciprocal. In the former case we have a more exact statement of the ground, all extraneous 272 elements being eliminated; in the latter case we have a more complete statement of the consequence. Sometimes, moreover, the latter of these alternatives may be practicable while the former is not.

This may be tested by reference to a formal hypothetical. The proposition If all S is M and all M is P, then all S is P is not reciprocal. We may make it so by expanding the consequent so that the proposition becomes If all S is M and all M is P, then whatever is either S or M is P and is also M or not S. But how in this case it would be possible to eliminate the irrelevant from the antecedent it is difficult to see. Our object is to eliminate M from the consequent, and if in advance we were to eliminate it from the antecedent the whole force of the proposition would be lost. And the same is true of non-formal hypotheticals, at any rate in many cases. Instances of reciprocal conditionals may be given without difficulty, for example, If any triangle is equilateral, it is equiangular. Such propositions are practically U propositions. We may also find instances of pure hypotheticals that are reciprocal; but, on the whole, while agreeing with a good deal that Mr Bosanquet says on the subject, I am disposed to demur to his view that the reciprocal hypothetical represents an ideal at which we should always aim. We have seen that there are two possible ways of securing reciprocity, whether or not they are always practicable; but the expansion of the consequent would generally speaking be extremely cumbrous and worse than useless, while the elimination from the antecedent of everything not absolutely essential for the realisation of the consequent would sometimes empty the judgment of all practical content for a given purpose. With reference to the case where AB involves AC, while AC does not involve AB, Mr Bosanquet himself notes the objection,—“But may not the irrelevant element be just the element which made AB into AB as distinct from AC, so that by abstracting from it AB is reduced to AC, and the judgment is made a tautology, that is, destroyed?” (p. 261). This argument, although somewhat overstated, deserves consideration. The point upon which I should be inclined to lay stress is that in criticising a judgment we ought to have regard 273 to the special object with which it has been framed. Our object may be to connect AC with AB, including whatever may be irrelevant in AB. Consider the argument,—If anything is P it is Q, If anything is Q it is R, therefore, If anything is P it is R. It is clear that if we compare the conclusion with the second premiss, the antecedent of the conclusion contains irrelevancies from which the antecedent of the premiss is free. Yet the conclusion may be of the greatest value to us while the premiss is by itself of no value. If our aim were always to get down to first principles, there would be a good deal to be said for Mr Bosanquet’s view, though it might still present some difficulties; but there is no reason why we should identify the conditional or the hypothetical proposition with the expression of first principles.

It is to be added that, if Mr Bosanquet’s view is sound, we ought to say equally that the A categorical proposition is imperfect, and that in categoricals the U proposition is the ideal at which we should aim. In categoricals, however, we clearly distinguish between A and U; and so far as we give prominence to the reciprocal modal, whether conditional or hypothetical, we ought to recognise its distinctive character. We may at the same time assign to it the distinctive symbol Um.

EXERCISES.

183. Give the contrapositive of the following proposition: If either no P is R or no Q is R, then nothing that is both P and Q is R. [K.]

184. There are three men in a house, Allen, Brown, and Carr, who may go in and out, provided that (1) they never go out all at once, and that (2) Allen never goes out without Brown.
Can Carr ever go out? [LEWIS CARROLL.]

185. There are two propositions, A and B.
Let it be granted that
If A is true, B is true. (i)
Let there be another proposition C, such that
If C is true, then if A is true B is not true. (ii)
274 (ii) amounts to this,—
If C is true, then (i) is not true.
But, ex hypothesi, (i) is true.
Therefore, C cannot be true; for the assumption of C involves an absurdity.
Examine this argument. [LEWIS CARROLL.]

[If the problem in section 184 is regarded as a problem in conditionals, this is the corresponding problem in hypotheticals.]

186. Assuming that rain never falls in Upper Egypt, are the following genuine pairs of contradictories?
(a) The occurrence of rain in Upper Egypt is always succeeded by an earthquake; the occurrence of rain in Upper Egypt is sometimes not succeeded by an earthquake.
(b) If it is true that it rained in Upper Egypt on the 1st of July, it is also true that an earthquake followed on the same day; if it is true that it rained in Upper Egypt on the 1st of July, it is not also true that an earthquake followed on the same day.
If the above are not true contradictories, suggest what should be substituted. [B.]

187. Give the contrapositive and the contradictory of each of the following propositions:
(1) If any nation prospers under a Protective System, its citizens reject all arguments in favour of free-trade;
(2) If any nation prospers under a Protective System, we ought to reject all arguments in favour of free-trade. [J.]

188. Examine the logical relation between the two following propositions; and enquire whether it is logically possible to hold (a) that both are true, (b) that both are false: (i) If volitions are undetermined, then punishments cannot rightly be inflicted; (ii) If punishments can rightly be inflicted, then volitions are undetermined. [J.]