« AnteriorContinuar »
mative has the same meaning as the exceptive negative, and vice versa.
In the first set of examples above given there can be no doubt about the meaning: in each case we are told that the Germans did one thing and that the others did the other. In the second set, however, the meaning is not so clear. When we say that all but the brave deserve the fair, or, to make the example less unnatural, all but the brave deserve to die, or that the brave are the only ones who do not deserve to die, do we mean that every brave man deserves to live, or merely that so far as courage is concerned, brave men do not deserve death ? In the latter case a brave man might deserve it on other grounds. So when we say that
denied of the subject. Both definitions thus assume that the propositions are affirmative. As examples of exclusive propositions Jevons gives “ Elements alone are metals” and “ None but elements are metals". He states that they are equivalent and assumes that they are both affirmative. As an example of exceptive propositions he gives All the planets except Venus and Mercury are beyond the earth’s orbit”. But suppose that instead of affirming this we should deny all the facts asserted, the proposition would then read : None of the planets except Venus and Mercury are beyond the earth's orbit. This form is precisely identical with “ None but elements are metals”, which Jevons regards as an affirmative exclusive proposition. The form is clearly exceptive and clearly negative, and there is no reason why the proposition should be regarded as either exclusive or affirmative, unless the distinction between the two kinds of proposition is abolished altogether. This is actually done by Minto, as follows : “ The formula for ExclusIVE PROPOSITIONS. • None but the brave deserve the fair'; • No admittance except on business'; “Only Protestants can sit on the throne of England'. These propositions exemplify different ways in common speech of naming a subject exclusively, the predication being made of all outside a certain term.' (P. 76.) The trouble with this description is that where the subject is 'named exclusively ', as in the example about Protestants, the predication as it stands is not made about all outside the term', but about those inside it. On the other hand, when the predication is made about all outside the term 'as in the two other examples, the subject is not 'named exclusively'; for that which is named is not the subject.
the brave alone deserve the fair or that none but the brave deserve the fair, do we mean that every brave man deserves a fair wife no matter what he may be in other respects, or merely that to deserve one he must at least be brave?
The first set of propositions are unambiguous because they are purely historical statements about certain individuals as such. The second are ambiguous because they express conditions about kinds of objects and they do not make it plain whether the condition mentioned is or is not the only one upon which the case depends. *
* Exclusive and exceptive propositions can be varied a good deal in quantity. When we say The Germans alone remained we (1) specify clearly the smaller group (of Germans as distinguished from the rest of the persons involved, and (2a) say something about each member of the specified smaller group (they remained) and (26) about each of the rest (they did not remain). The proposition is thus in every possible respect universal.
When we say The brave alone deserve the fair, we (1) distinguish clearly enough between our groups, (26) we say something about all who are not brave, and (2a) if we are interpreted as saying anything at all about those who are brave,-namely, that they have complied with one condition—we say it about all of them. This proposition is thus also universal in every respect.
When we say Some of the Germans were the only persons who remained, we still (1) specify the smaller group clearly and (26) still say something about all the persons outside of it ; but (2a) the individuals within it of whom we speak are no longer definitely designated. The proposition is thus in one respect particular. When we say The Germans and some others alone remained, 1 is still definite, za is universal, and 26 particular. When we say Some of the Germans and some of the others alone remained, the groups are still clearly distinguished, but the original proposition is broken up into two exclusives, each particular in so far as it fails to specify the distinction between those who did and those who did not stay. Each of these exclusives is equivalent to both I and O: some stayed and some did not.
When we say The soldiers of one nation alone remained, a statement is made (2) about each member of each group. To this extent the proposition is universal. But as (1) the smaller group is no longer definitely specified, the proposition is in this respect particular.
The precise meaning as to quantity of an exclusive or exceptive proposition, like that of any other, may be indefinite.
So far we have been dealing with what are called categorical propositions; those in which something is, or at least
seems from the form of the proposition to be, Disjunctives and Hypo- stated without alternative and without conditheticals.
tion. Propositions in which it is affirmed that one or other of several alternative states of affairs exists are called disjunctive or alternative, e.g., Every man is either married or single; He is either a fool or a knave; Either he is a knave or I have been grossly deceived; Either A or B did it; He is either not here or not there.* Propositions in which it is affirmed that if some specified state of affairs exists another specified state of affairs also exists are called hypothetical, e.g., If he is not a fool he is a knave; If he is a knave I have been grossly deceived; If he is not in the room he is not in the house.
The part of a hypothetical proposition which specifies the condition, either of something being so, or of our knowing it, is called the antecedent, the part which specifies what follows from that condition is called the consequent.
Disjunctive propositions state that one of two things must be true; but do they imply that both cannot be true? This question has been discussed at much length. If a man is married he cannot possibly be single. We know this from the nature of things, but there is no reason in the nature of
* A negative proposition asserts the existence of a state of affairs just as much as an affirmative.
† “ This is the familiar form of the disjunctive judgment. . . It is usual to mention along with it the copulative judgment (“S is both p and q and r'), and the remotive judgment (“S is neither p nor q nor r'); but in spite of the external analogy of form, neither of these has the same logical value as the disjunctive ; the first is only a collection of positive, the second of negative, judgments with the same subject and different predicates, which latter are not placed in any logically important relation to each other. The disjunctive judgment alone expresses a special relation between its members : it gives its subject no predicate at all, but prescribes to it the alternative between a definite number of different predicates.” Lotze, • Logic”, § 69. (Clarendon Press.)
things why a person cannot be both fool and knave. When it is asserted that he is either one or the other, is it necessarily implied by the form of the statement that he is not both ? Fowler says: It seems to me that in the expression either
-' we distinctly exclude the possibility of both alternatives being true, as well as of both being false. In fact, when we do not wish to exclude the possibility of both being true, we add the words or both ’, thus: 'He is either a fool or a knave, or both ʼ; I shall come either to-day or to-morrrow, or perhaps both days'.' With this view Thomas Aquinas, Kant, Hamilton, Boole, Bradley, and others agree. Whately, Mansel, Mill, Jevons, Keynes, † and others maintain on the other hand that such propositions merely mean that both alternatives cannot be false, though both may be true.
Says Keynes: “
Suppose it laid down as a 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 any one regard this as implying the ineligibility of persons who happened to be members of more than one of these universities ?'
The question is, of course, one of the interpretation of language, not of logical processes. So far as logic is concerned any one is at liberty to use language in any sense he pleases, provided that he explains beforehand the sense in which he means to use it; but since there is a real difference in usage it seems to me better in this case, as in the case of the word ' some', to assume that the words are used with the greatest caution and imply nothing but what is stated. Let us, therefore, agree, at least for the purposes of this book, that when we say that one or other of several alternatives is true we do not necessarily imply that both cannot be true, though of course we do imply that both cannot be false.
* “Deductive Logic”, p. 118, Ninth Ed. (Clarendon Press).
† See Jevons, Principles of Science ”, p. 68, and Keynes, -ó Formal Logic”, § 140.
Like exclusive and exceptive propositions, hypothetical and disjunctive propositions are different in form and must be distinguished from each other, though they can be made to express the same meaning. The following table shows the relations between them:
Hypothetical A is either B or C
= If A is not B, it is C.
= If A is not C, it is B. A is either not B or not C If A is B, it is not C.
If A is C, it is not B. A is either B or not C = If A is not B, it is not C.
= If A is C, it is B.
In each case there are two hypothetical propositions, either of which is equivalent to the disjunctive, and each of which is exactly equivalent to the other. To say If A is not B it is C means precisely the same thing as to say If A is not C it is B; and so with the rest.
If any one were asked the use of disjunctive and hypothetical propositions, the first answer that occurred to him would probably be: To express knowledge combined with doubt. To use Venn's illustration, if I say that A. B. is either a barrister or a solicitor, I express my knowledge that he is a lawyer and my doubt as to his precise standing at the bar. The same thought would be expressed in the hypothetical proposition, 'If he is not a barrister he is a solicitor'.
But disjunctive and hypothetical propositions are not always used to express doubt. When, for example, we say that in the United States every person is either married or single, the statement does not express the slightest doubt as to the condition of any given individual in this respect. Its real force is to explain the laws or social customs of the country, under which a person is regarded as single until some prescribed condition has been fulfilled, and then as married. The statement would hardly hold of an oriental