general law or condition it is only the unspecified members of the class about which something is necessarily implied. 'Only the red pills are to be taken' means that the red pills are to be taken, and the others left; ' Only the good are happy' means that goodness is essential to happiness, so that the bad are never happy; but it does not mean that every good man is happy.

To take the last case first; apart from the implication of a general law which may be contradicted by saying that virtue is not at all essential to happiness, the proposition is merely equivalent to the statement that no bad people are happy, or that all happy people are good; and the logical opposites of these propositions are obvious.

In the other case (Only the red pills are to be taken), where an exceptive or exclusive proposition is equivalent to two ordinary propositions (The red are to be taken and the others are not to be taken), it would be false if any of the following were true:

a miller is honest his thumb turns into gold'. Of course this is intended to imply that honest millers are not to be found, but it also implies the existence of some occult causal relation between honesty in a miller and a golden thumb, and a person cannot deny the statement without implying that this conception of the universe is fictitious.

On the whole subject see Keynes, pp. 186–210.

Since hypothetical propositions and universals equivalent to them are usually intended to imply the existence of some general law, they are sufficiently contradicted by any proposition denying the existence of such a law. The equivalent statements 'If a man is rich he is stingy' and 'All rich men are stingy' can be always contradicted by the statement 'It is not always so'; but if the supposed causal relation between riches and stinginess were the real subject of interest it would be sufficient to say 'It is not necessarily so'; and this might be proved even though no concrete exception to the universal categorical proposition could be found. To put the matter otherwise: Hypothetical propositions tell what under certain circumstances must be. To contradict them it is sufficient to say that it need not be. This can be proved by showing that it sometimes is not, but if a concrete exception could not be found it might be proved in some other way.

(1) Neither the red are to be taken nor the others not to be taken The others alone are to be taken.

(2) The red or some of the red are not to be taken. (3) The others or some of them are to be taken.

(4) Either the red or some of them are not to be taken or the others or some of them are to be taken.

The last of these four propositions-the disjunctive-is the only one that must necessarily be true when the original proposition is false. It therefore is its contradictory. The first of the four is the most extreme statement in the other direction. It therefore is the contrary of the original. The other two are contraries or contradictories of the parts into which the original is resolved.

From all this it can be seen that a logical opposite of an exclusive or exceptive proposition is itself rarely exclusive or exceptive.

To give concreteness to what has been said about the opposition of propositions the following system

of symbols is suggested.

Symbols.

Let a small circle represent any object S of the kind discussed, and a plain stroke through it indicate the presence of a given attribute P, while a stroke with a small bar or tick across it indicates the absence of that attribute. is evident that the stroke cannot be both plain and crossed.

It

Suppose all the objects of the kind discussed to be represented by a number of small circles. When anything is said about all the objects of the kind under discussion draw a plain or ticked stroke as the case may be through each of the little circles. When something is said about only some of the objects leave some of the circles unmarked. The result is as follows:

To show how these symbols help. Let us represent the supposed fact that All S is P-that every circle has a plain stroke through it-thus: $ $ $ $ It does not matter how many of these circles we draw so long as the plain stroke is drawn through each one of them, to show that there are no exceptions.

Now suppose the question to arise: How many of these S's have a crossed stroke? We need only glance at the circles to see that there are none marked that way and no unmarked circles that might be marked that way. Hence we say: None of the circles can be marked with a crossed stroke; none of the S's can be non-P; no S is non-P. Thus the symbols enable us to see that this follows from the supposed fact that each of the S's is P. In the same way if each of the S's is P-if each circle has a plain stroke-we need only look at the above figures to see that it is also true that at least some of the circles have plain strokes-that some S's are P (Proposition I); false that some of the circles have not plain strokes-that some S's are not P (Proposition O); and still more false that none of the circles have plain strokes -that no S's are P (Proposition E).

Again, let us suppose that some S's are not P (Proposition O) and represent it by drawing a crossed stroke through some but not all of the circles-it does not matter how many: oo We leave some of the circles unmarked because there are some that the proposition does not say anything about. We know that in reality each of these must have one character or the other, but we do not attempt to represent it until we know which character it is.

What now can we say about the truth or falsity of the statement that no S's are P (Proposition E)-that none of the circles should really be marked with a plain stroke? All we can say is that the marks already there will not tell us. In other words, if we know that Proposition O is true

and if that is all we know, we must remain in doubt about the truth or falsity of E. So likewise with I; so long as we do not know whether those unmarked circles should really be marked with a plain stroke or with a crossed stroke we cannot say whether it is true or false that some S's are not P. We can tell, however, about the truth or falsity of A; for if A were true and all S's were P, all the circles would have to be marked with a plain stroke, and that is not possible so long as at least some of them are marked with a crossed stroke. Hence we can see from the symbols that represent the truth of O that A must be false. And so of the rest.

So far no particular S has been definitely and individually designated. To indicate some particular individual or subgroup of individuals use a small black dot or blacken the circle. All the remaining categorical propositions can then be symbolized.

Singular A: Socrates is P

Singular E: Plato is not P

Exceptive A: All the S's but B are P Φ Φ

or фф

The first of these figures indicates All the S's but B are P (and B is not)', the second indicates All the S's but B are P (and it is not said whether B is P or not)'.

Exceptive E: No S but B is P

ΦΦ

Exclusive A: B is the only S which is P

or

Exclusive E: B is the only S which is not P

Φ

It is to be noticed that a person interpreting these diagrams could not distinguish between a proposition and its ‘obverse'; for example, between the affirmative proposition All S is P and the negative No S is non-P, or between the negative No S is P and the affirmative All S is non-P. This is an advantage rather than a defect; indeed the whole value of the symbols rests upon such facts, for the difference between a proposition and its obverse expresses a difference of shading or accent in the thought, but not a difference in the objects thought about. The relations of the objects remain the

same whether they are told about in one way or in another, and the diagrams symbolize these relations as they are supposed to exist in the objects. They point to the reality with which thought is concerned and to which it must always conform whatever its shading, rather than to the particular shading which the thought may happen to take or the words in which it happens to be expressed, and they can be used to test the thought no matter what its shading or form of expression.

The fact that these diagrams express no difference between a proposition and its obverse suggests the question that is sometimes discussed whether proposition A is not after all negative rather than affirmative. When we say that every nation prefers its own interests to the good of humanity (All S is P), do we have in mind all the nations that do this, or the fact that none can be found which does not ? Certainly we cannot be sure that the statement is true until we find that there is no nation which does not (No S is non-P). Perhaps we can say that when proposition A expresses a hasty and unverified generalization it is affirmative, when it is derived deductively from general considerations it may also be affirmative, but when it is reached cautiously in the absence of general considerations it is usually negative. When we seek to verify a general statement, we do not count the cases in which it holds, but we look for exceptions.*

*This system of diagrams seems to me to indicate the opposition of propositions better than Euler's (explained elsewhere), partly because it provides a diagram for every proposition, while his only provides for the first four, partly because the same diagram represents a proposition and its obverse, partly because the diagrams for all the propositions that express different facts are distinctly different, but mainly because it preserves the distinction between things and attributes, and represents the presence or absence of the latter in the former rather than the partial or complete inclusion or exclusion of one class by another.