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.]
CHAPTER X.
DISJUNCTIVE (OR ALTERNATIVE) PROPOSITIONS.
189. The terms Disjunctive and Alternative as applied to Propositions.—Propositions of the form Either X or Y is true are ordinarily called disjunctive. It has been pointed out, however, that two propositions are really disjoined when it is denied that they are both true rather than when it is asserted that one or other of them is true; and the term alternative, as suggested by Miss Jones (Elements of Logic, p. 115), is obviously appropriate to express the latter assertion. We should then use the terms conjunctive, disjunctive, alternative, remotive, for the four following combinations respectively: X and Y are both true, X and Y are not both true, Either X or Y is true,295 Neither X nor Y is true.
295 Some writers indeed regard the proposition Either X or Y is true as expressing a relation between X and Y which is disjunctive in the above sense as well as alternative; but the disjunctive character of this proposition as regards X and Y is at any rate open to dispute, whilst its alternative character is unquestionable (see section 191).
Whilst, however, the name alternative is preferable to disjunctive for the proposition Either X or Y is true, the latter name has such an established position in logical nomenclature that it seems inadvisable altogether to discontinue its use in the old sense. It may be pointed out further that an alternative contains a veiled disjunction (namely, between not-X and not-Y) even in the stricter sense; for the statement that Either X or Y is true is equivalent to the statement that Not-X and not-Y are not both true. Hence, although generally using the term alternative, I shall not entirely discard the term disjunctive as synonymous with it.
276 190. Two types of Alternative Propositions.—In the case of propositions which are ordinarily described as simply disjunctive a distinction must be drawn similar to that drawn in the preceding chapter between conditionals and true hypotheticals. For the alternatives may be events or combinations of properties one or other of which it is affirmed will (always or sometimes) occur, e.g., Every blood vessel is either a vein or an artery, Every prosperous nation has either abundant natural resources or a good government ; or they may be propositions of independent import whose truth or falsity cannot be affected by varying conditions of time, space, or circumstance, and which must therefore be simply true or false, e.g., Either there is a future life or many cruelties go unpunished, Either it is no sin to covet honour or I am the most offending soul alive.
Any proposition belonging to the first of the above types may be brought under the symbolic form All (or some) S is either P or Q, and may, therefore, be regarded as an ordinary categorical proposition with an alternative term as predicate. It is usual and for some reasons convenient to defer the discussion of the import of alternative terms until propositions of this type are being dealt with. Such propositions might otherwise be dismissed after a very brief consideration.296
296 It should be particularly observed that although the proposition Every S is P or Q may be said to state an alternative, it cannot be resolved into a true alternative combination of propositions. Such a resolution is, however, possible if the proposition (while remaining affirmative and still having an alternative predicate) is singular or particular: for example, This S is P or Q = This S is P or this S is Q ; Some S is P or Q = Some S is P or some S is Q.
Corresponding to this, we may note that an affirmative categorical proposition with a conjunctive predicate is equivalent to a conjunction of propositions if it is singular or universal, but not if it is particular. Thus, This S is P and Q = This S is P and this S is Q ; All S is P and Q = All S is P and all S is Q. From the proposition Some S is P and Q we may indeed infer Some S is P and some S is Q ; but we cannot pass back from this conclusion to the premiss, and hence the two are not equivalent to one another.
It may be added that a negative categorical proposition with an alternative predicate cannot be said to state an alternative at all, since to deny an alternation is the same thing as to affirm a conjunction. Thus the proposition No S is either P or Q can only be resolved into a conjunctive synthesis of propositions, namely, No S is P and no S is Q.
277 Alternative propositions of the second type are compound (as defined in section 55). They contain an alternative combination of propositions of independent import: and they have for their typical symbolic form Either X is true or Y is true, or more briefly, Either X or Y, where X and Y are symbols representing propositions (not terms). So far as it is necessary to give them a distinctive name, they have a claim to be called true alternative propositions, since they involve a true alternative synthesis of propositions, and not merely an alternative synthesis of terms.
It will be convenient to speak of P and Q as the alternants of the alternative term P or Q, and of X and Y as the alternants of the alternative proposition Either X or Y.
191. The Import of Disjunctive (Alternative) Propositions.—The two main questions that arise in regard to the import of alternative propositions are (1) whether the alternants of such propositions are necessarily to be regarded as mutually exclusive, (2) whether the propositions are to be interpreted as assertoric or modal.
(1) We ask then, in the first place, whether in an alternative proposition the alternants are to be interpreted as formally exclusive of one another; in other words, whether in the proposition All S is either A or B it is necessarily (or formally) implied that no S is both A and B,297 and whether in the proposition X is true or Y is true it is necessarily (or formally) implied that X and Y are not both true. It is desirable to notice at the outset that the question is one of the interpretation of a propositional form, and one that does not arise except in connexion with the expression of judgments in language. Hence the solution will be, at any rate partly, a matter of convention.
297 This is an alternative proposition of the first type, and the same question is raised by asking whether the term A or B includes AB under its denotation or excludes it; in other words, whether the denotation of A or B is represented by the shaded portion of the first or of the second of the following diagrams:
278 The following
considerations may help to make this point clearer. Let X and
Y represent two judgments. Then the following are two possible
states of mind in which we may be with regard to X and Y:
(a) we may know that one or other of them is true, and that
they are not both true;
(b) we may know that one or other of them is true, but may
be ignorant as to whether they are or are not both true.
Now whichever interpretation (exclusive or non-exclusive) of the propositional form X or Y is adopted, there will be no difficulty in expressing alternatively either state of mind. On the exclusive interpretation, (a) will be expressed in the form X or Y, (b) in the form XY or XYʹ or XʹY (Xʹ representing the falsity of X, and Yʹ the falsity of Y). On the non-exclusive interpretation, (a) will be expressed in the form XʹY or XYʹ, (b) in the form X or Y. There can, therefore, be no intrinsic ground based on the nature of judgment itself why X or Y must be interpreted in one of the two ways to the exclusion of the other.
As then we are dealing with a question of the interpretation of a certain form of expression, we must look for our solution partly in the usages of ordinary language. We ask, therefore, whether in ordinary speech we intend that the alternants in an alternative proposition should necessarily be understood as excluding one another?298 A very few instances will enable us to decide in the negative. Take, for example, the proposition, “He has either used bad text-books or he has been badly taught.” No one would naturally understand this to exclude the possibility of a combination of bad teaching and the use of bad text-books. Or suppose it laid down as a 279 condition of eligibility for some appointment that every candidate must be a member either of the University of Oxford, or of the University of Cambridge, or of the University of London. Would anyone regard this as implying the ineligibility of persons who happened to be members of more than one of these Universities? Jevons (Pure Logic, p. 68) instances the following proposition: “A peer is either a duke, or a marquis, or an earl, or a viscount, or a baron.” We do not consider this statement incorrect because many peers as a matter of fact possess two or more titles. Take, again, the proposition, “Either the witness is perjured or the prisoner is guilty.” The import of this proposition, as it would naturally be interpreted, is that the evidence given by the witness is sufficient, supposing it is true, to establish the guilt of the prisoner; but clearly there is no implication that the falsity of this particular piece of evidence would suffice to establish the prisoner’s innocence.
298 There are no doubt many cases in which as a matter of fact we understand alternants to be mutually exclusive. But this is not conclusive as shewing that even in these cases the mutual exclusiveness is intended to be expressed by the alternative proposition. For it will generally speaking be found that in such cases the fact that the alternants exclude one another is a matter of common knowledge quite independently of the alternative proposition; as, for example, in the proposition, He was first or second in the race. This point is further touched upon in Part III, Chapter 6.
But it may be urged that this does not definitely settle the question of the best way of interpreting alternative propositions. Granted that in common speech the alternants may or may not be mutually exclusive, it may nevertheless be argued that in the use of language for logical purposes we should be more precise, and that an alternative statement should accordingly not be admitted as a recognised logical proposition except on the condition that the alternants mutually exclude one another.
We may admit that the argument from the ordinary use of speech is not final. But at any rate the burden of proof lies with those who advocate a divergence from the usage of everyday language; for it will not be denied that, other things being equal, the less logical forms diverge from those of ordinary speech the better. Moreover, condensed forms of expression do not conduce to clearness, or even ultimately to conciseness.299 280 For where our information is meagre, a condensed form is likely to express more than we intend, and in order to keep within the mark we must indicate additional alternatives. On this ground, quite apart from considerations of the ordinary use of language, I should support the non-exclusive interpretation of alternatives. The adoption of the exclusive interpretation would certainly render the manipulation of complex propositions much more complicated.
299 Obviously a disjunctive proposition is a more condensed form of expression on the exclusive than on the non-exclusive interpretation. Compare Mansel’s Aldrich, p. 242, and Prolegomena Logica, p. 288. “Let us grant for a moment the opposite view, and allow that the proposition All C is either A or B implies as a condition of its truth No C can be both. Thus viewed, it is in reality a complex proposition, containing two distinct assertions, each of which may be the ground of two distinct processes of reasoning, governed by two opposite laws. Surely it is essential to all clear thinking that the two should be separated from each other, and not confounded under one form by assuming the Law of Excluded Middle to be, what it is not, a complex of those of Identity and Contradiction” (Aldrich, p. 242). It may be added that one paradoxical result of the exclusive interpretation of alternatives is that not either P or Q is not equivalent to neither P nor Q.
A further paradoxical result is pointed out by Mr G. R. T. Ross in an article on the Disjunctive Judgment in Mind (1903, p. 492), namely, that on the exclusive interpretation the disjunctives A is either B or C and A is either not B or not C are identical in their import; for in each case the real alternants are B but not C and C but not B. Thus, to take an illustration borrowed from Mr Ross, the two following propositions are (on the interpretation in question) identical in their import,—“Anyone who affirms that he has seen his own ghost is either not sane or not telling what he believes to be the truth,” “Anyone who affirms that he has seen his own ghost is either sane or truthful.”
Mr Bosanquet and other writers who advocate the exclusive interpretation of disjunctives appear to have chiefly in view the expression in disjunctive form of a logical division or scientific classification. I should of course agree that such a division or classification is imperfect if the members of which it consists are not mutually exclusive as well as collectively exhaustive. This condition must also be satisfied when we make use of the disjunctive judgment in connexion with the doctrine of probability.300 It will, however, hardly be proposed to confine the disjunctive judgment to these uses. We frequently have occasion to state alternatives independently of any scientific classification or any calculation of probability; and we must not regard the bare form of the disjunctive judgment as expressing anything that we are not prepared to recognise as universally involved in its use.
300 In this connexion the further condition of the “equality” in a certain sense of the alternants has in addition to be satisfied.
It is of course always possible to express an alternative 281 statement in such a way that the alternants are formally incompatible or exclusive. Thus, not wishing to exclude the case of A being both B and C we may write A is B or bC ;301 or, wishing to exclude that case, A is Bc or bC. But in neither of these instances can we say that the incompatibility of the alternants is really given by the alternative proposition. It is a merely formal proposition that No A is both B and bC or that No A is both Bc and bC. The proposition Every A is Bc or bC does, however, tell us that no A is both B and C ; and when from our knowledge of the subject-matter it is obvious that we are dealing with alternants that are mutually exclusive (and no doubt this is a very frequent case), we have in the above form a means of correctly and unambiguously expressing the fact. Where it is inconvenient to use this form, it is open to us to make a separate statement to the effect that No A is both B and C. All that is here contended for is that the bare symbolic form A is either B or C should not be interpreted as being equivalent to A is either Bc or bC.
301 Where b = not-B, and c = not-C. What is contained in this paragraph is to some extent a repetition of what is given on page 278.
(2) We may pass on to consider the second main question that arises in connexion with the import of disjunctive (alternative) propositions, namely, whether such propositions are to be interpreted as modal or as merely assertoric.
In chapter 9 it was urged that the modal interpretation of the typical hypothetical proposition If A then C must be regarded as the more natural one, on the ground that we should not ordinarily think it necessary to affirm the truth of A in order to contradict the proposition, as would be necessary if it were interpreted assertorically.302 Similarly the enquiry as to how we should naturally contradict the typical alternative propositions Every S is either P or Q, Either X or Y is true, may help us in deciding upon the interpretation of these propositions.
On the assertoric interpretation, the contradictories of the propositions in question are Some S is neither P nor Q, Neither X nor Y is true ; on the modal interpretation, they are An S need not be either P or Q, Possibly neither X nor Y is true. 282 There can be no doubt that this last pair of propositions would not as a rule be regarded as adequate to contradict the pair of alternatives; and on this ground we may regard the assertoric interpretation of alternatives as most in accordance with ordinary usage. There is also some advantage in differentiating between hypotheticals and alternatives by interpreting the former modally and the latter assertorically, except in so far as a clear indication is given to the contrary. It is not of course meant that modal alternatives are never as a matter of fact to be met with or that they cannot receive formal recognition; they can always be expressed in the distinctive forms Every S must be either P or Q, Either X or Y is necessarily true.
192. Scheme of Assertoric and Modal Propositions.—By differentiating between forms of propositions in the manner indicated in preceding sections we have a scheme by which distinctive expression can be given to assertoric and modal propositions respectively, whether they are simple or compound.
Thus the categorical form of proposition might be restricted to the expression of simple assertoric judgments; the conditional form to that of simple modal judgments; the disjunctive (alternative)303 form to that of compound assertoric judgments; and the hypothetical form to that of compound modal judgments.
303 We are of course referring here to disjunctive (alternative) propositions of the second type only, alternative propositions of the first type being treated as categoricals with alternative predicates. See section 190.
I have not in the present treatise attempted to adopt this scheme to the exclusion of other interpretations of the different propositional forms; but I have had it in view throughout, and I put it forward as a scheme the adoption of which might afford an escape from some ambiguities and misunderstandings.
193. The Relation of Disjunctive (Alternative) Propositions to Conditionals and Hypotheticals.—It may be convenient if we briefly consider this question independently of the distinctions indicated in the preceding section, the assumption being made that these different types of propositions are interpreted either all assertorically or all modally. On this assumption, alternative propositions are reducible to the conditional or the true hypothetical form according to the type to which they belong. Thus, 283 the proposition, “Every blood vessel is either a vein or an artery,” yields the conditional, “If any blood vessel is not a vein then it is an artery”; the true compound alternative proposition, “Either there is a future life or many cruelties go unpunished,” yields the true hypothetical, “If there is no future life then many cruelties go unpunished.”
It may be asked whether an alternative proposition does not require a conjunction of two conditionals or hypotheticals in order fully to express its import. This is not the case, however, on the view that the alternants are not to be interpreted as necessarily exclusive. It is true that even on this view an alternative proposition, such as Either X or Y, is primarily reducible to two hypotheticals, namely, If not X then Y and If not Y then X. But these are contrapositives the one of the other, and therefore mutually inferable. Hence the full meaning of the alternative proposition is expressed by means of either of them.
On the exclusive interpretation, the alternative proposition Either X or Y yields primarily four hypotheticals, namely, If X then not Y and If Y then not X in addition to the two given above. But these again are contrapositives the one of the other. Hence the full import of the alternative proposition will now be expressed by a conjunction of the two hypotheticals, If X then not Y and If not X then Y.
This is denied by Mr Bosanquet, who holds that the disjunctive proposition yields a positive assertion not contained in either of the hypotheticals. “‘This signal light shews either red or green.’ Here we have the categorical element, ‘This signal light shews some colour,’ and on the top of this the two hypothetical judgments, ‘If it shews red it does not shew green,’ ‘If it does not shew red it does shew green.’ You cannot make it up out of the two hypothetical judgments alone; they do not give you the assertion that ‘it shews some colour.’”304 But surely the second of the two hypotheticals contains this implication quite as clearly and definitely as the disjunctive does.305