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.

t "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 neces

sarily implied by the both? Fowler says:

form of the statement that he is not 'It seems to me that in the expression either or' 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'; '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, "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 relations between them:

Disjunctive.

A is either B or C

A is either not B or not C

A is either B or not C

The following table shows

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'.

Its

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. 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

society in which concubinage was recognized. Such propositions, therefore, express knowledge, not ignorance; but it is a knowledge of the laws or general conditions of existence prevailing in any sphere or universe', not of the precise state of some particular individual in that universe.

In the example just discussed the proposition affirming the existence of a general law happened to be disjunctive. It is more common to affirm such laws in hypothetical, or even in universal categorical propositions, e.g., If a man is insulted he becomes angry, or Insulted men become angry; When it rains hard the streets are wet, or Hard rains wet the streets; The nearer bodies get together the more they attract each other, or Contiguous bodies attract each other more than those that are farther apart. *

* On p. 87 there are examples taken from Keynes of several other universal propositions of this kind. Such propositions, as we there saw, do not necessarily imply the existence of things as they are described in the grammatical subjects of the propositions; but they do imply the existence of a universe whose laws they more or less accurately express. There doubtless are universal propositions founded upon direct observation of the things named in them and intended to imply the existence of those things as well as to describe them; e.g., None of the Stuarts were good sovereigns; Each of the United States contains colored citizens. Such propositions cannot be put into hypothetical form. But universals arrived at by deductive reasoning, or reasoning from general considerations, are probably always capable of being put into hypothetical form and seldom or never necessarily imply the existence of the things described by their subjects, though they probably do imply the existence of the things named in the equivalent hypothetical propositions, e.g., Seniors are wiser than Sophomores; Every husband has a wife. Turned into hypothetical form these propositions would run: If a person belongs to the Senior class he is wiser than if he belonged only to the Sophomore class. If a man is married, he has a wife. The general consideration in the first of these examples lies in the supposed law of the college universe, that two more years of college life must add something to one's wisdom. It is a statement which will be as valid as it is now as long as colleges and human nature remain what they are. It is not concerned specially with the present, the past, or the future existence of Seniors and Sophomores and colleges, the present term of the verb to be, like the phrase must be, being used in a perfectly

timeless sense. The proposition merely states the effects supposed to result from certain causal agencies whenever and wherever they may be supposed to exist. The statement that every husband has a wife is based upon a similar consideration of the nature of things. We mean by a husband a man who is married, and we know perfectly well that as the world is constituted men can marry only women, that is, wives. It is this general fact which the proposition expresses; it does not necessarily imply that any one is married.

As a

Particular propositions, unlike universals, are not usually deduced from general considerations; though sometimes they may be e.g., 'Some Sophomores must be wiser than the average Senior '. rule, however, particular propositions are based upon the direct observation of individuals to which we are forced to resort when general considerations are inapplicable, and they naturally imply the existence of the individuals observed. The propositions Some Sophomores are wiser than some Seniors, Some husbands are not happy, do not lay down general laws of the universe or state the effects that certain causes necessarily produce. For this reason they would hardly ever be put into disjunctive form, for though such a form is possible in this case it is not very clear, and has no special value when the implication of general law is omitted; e.g., A student either is not a Sophomore or is a member of a group of persons some of whom are wiser than some Seniors. Put into hypothetical form particular propositions have considerable significance, for they serve to deny the existence of the kind of law that universals of opposite quality assert; e.g., If a person is a Sophomore he may be wiser than a Senior. This is the hypothetical form of the particular proposition Some Sophomores are wiser than some Seniors'. They both serve to deny the universal law expressed in the universal categorical proposition, No Sophomore is wiser than a Senior', or in the hypothetical proposition, 'If a student is a Sophomore he is not as wise as a Senior'.

Universal laws expressible in the above forms can also be expressed by the phrases must be, are necessarily, etc., and denied by the phrases need not be, are not necessarily, etc.