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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 citi

Such propositions cannot be put into hypothetical form. 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

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


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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.

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.



In the last two chapters we dealt with the deeper interpretation of propositions. We must now discuss another question which is practically one of interpretation, but which is not at all deep. Such a discussion is important enough to be found in all the text-books of logic; and yet the only end which it serves is to force the reader to think about the obvious meaning of his words and to show him how easy it is to make foolish blunders when we rattle off words without thinking.

The Opposition of Propositions', as the phrase is used in logic, means merely the mutual implications of propositions which differ in quantity or quality or both. To be 'opposed’ in this sense it is therefore not necessary that two propositions should be inconsistent. This use of the term opposition is not happy, but since it is common we must understand it.

Anybody who will exercise a little patience ought to be able to work out an answer to this question : Assuming the truth or the falsity of one of the propositions A, With comE, I, and 0, what can we know about the truth sitions. or falsity of the others ? If it is true that all the members of the present senior class are in good health (proposition A), is it true or false that some of them are in good health (proposition I), that none of them are in good health (proposition E), that some of them are not in good health (proposition O)?

For the sake of helping the reader to verify his reasoning I shall give a table showing the relations of the various cate

mon propo

I true,

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gorical propositions. This table, like any other that may be found in a text-book of logic, ought to be understood, but not learned by heart. The only thing in logic that ever ought to be learned by heart is a definition, for we must depend upon memory for a precise meaning of words, but even a definition ought not to be learned in this way, if there is


way in which a person can remember and restate its precise meaning. With a logical table the case is entirely different. It is valuable only because clear thinking is required to construct it. It is not worth remembering; and to commit it to heart like a multiplication table is a pure waste of time. If A is true, E is false,

O false.
A “ false,

I false, O true. " I

A“ doubtful, E false, O doubtful.

A “ false, E doubtful, I doubtful.
A is false, E is doubtful, I doubtful, O true.
E" A “ doubtful, I true, O doubtful.

E true,

O true.
A“ true,

E false, I true. This table is concerned only with the relations of universal propositions and particulars. It tells us nothing about the relation of either universals or particulars to propositions which tell something about some designated individual or class of individuals within the larger group. If we designate all propositions dealing with a designated individual or class within the larger group by the letter S we get the followng: If A is true, S affirmative is true and S negative is false.

S “ doubtful" S

doubtful. O S doubtful S

doubtful. " A is false, S


doubtful. E S doubtful S

doubtful. I S false 'S

true. S

true S





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If S aff. is true, A is doubtful, E false, I true, O doubtful.
If S neg. is true, A is false, E doubtful, I doubtful, O true. *

It should be noticed that the truth of a universal proposition involves the truth of the corresponding singulars and particulars, and the falsity of a singular or particular involves the falsity of the corresponding universal, but not vice versa.

Of propositions which differ only in quality, if both are particular one must be true and both may be; if both are universal one must be false and both may be. Of the propositions A, E, I, O, having the same subject and predicate, it is only when they differ in both quantity and quality (i.e., A and O, E and I), that one must necessarily be true and the other false. Such propositions are called contradictories. Universal propositions of different quality (i.e., A and E), are called contraries.

To‘contradict' a statement is to deny its truth. make any statement whatever (e.g., that the moon is made of green cheese, that every Englishman likes roast beef), and if I say “That is not true', I contradict you, and the important thing to notice is that if either of us is right the other is wrong, and, vice versa, if either of us is wrong the other is right. We cannot both be right, and we cannot both be wrong.

To contradict a statement it is not necessary to say “That is not true’ in these exact words. Any statement contradicts another if the two are so related that when either of them is true the other must be false and vice versa.

In formal logic, however, contradictory propositions are supposed also to have the same terms in the subject and in the predicate, thus: “The moon is made of green cheese' and

The moon is not made of green cheese', ` All Englishmen like roast beef' and ' Some Englishmen do not like roast beef'. In comnection with this last example it must be


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* It is often said that for logical purposes singular propositions can be treated as universals. In the present case they must be treated rather as particulars, though not precisely. In actual experience the matter pre. sents no difficulties,

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