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.