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]
[304] Essentials of Logic, p. 124.
[305] Mr Bosanquet’s opinion that “the disjunction seems to complete the system of judgments,” and that in some way it rises superior to other forms of judgment, is apparently based on the view that it is by the aid of the disjunctive judgment that we set forth the exposition of a system with its various subdivisions. Apart, however, from the fact that a disjunctive judgment does not necessarily contain such an exposition, Mr Bosanquet’s doctrine appears to regard a classification of some kind as representing the ideal of knowledge; and this can hardly be allowed. We cannot, for example, regard the classifications of such a science as botany as of equal importance with the expressions of laws of nature, such as the law of universal gravitation. And the ultimate laws on which all the sciences are based are not expressed in the form of disjunctive propositions.
284 Returning to the distinctions indicated in the preceding section, it is hardly necessary to add that if the hypothetical If not X then Y is interpreted modally, while the alternative Either X or Y is interpreted assertorically, then the alternative can be inferred from the hypothetical, but not vice versâ.
EXERCISES.
194. Shew how an alternative proposition in which the alternants are not known to be mutually exclusive (e,g., Either X or Y or Z is true) may be reduced to a form in which they necessarily are so. Write the new proposition in as simple a form as possible. [K.]
195. Shew why the following propositions are not contradictories: Wherever A is present, B is present and either C or B is also present ; In some cases where A is present, either B or C or B is absent. How must each of these propositions in turn be amended in order that it may become the true contradictory of the other? [K.]
196. No P is both Q and R. Reduce this proposition (a) to the form of a conditional proposition, (b) to the form of an alternative proposition. Give the contradictory of the original proposition, of its conditional equivalent, and of its alternative equivalent; and test your results by enquiring whether the three contradictories thus obtained are equivalent to one another. [K.]