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noticed that the contradictory of a universal proposition is always either a particular or a singular, and that of a particular a universal. Universal propositions of opposite quality (* All Englishmen like roast beef', 'No Englishmen like roast beef') cannot be contradictories; for while they cannot possibly both be true, they may both be false. Such propositions are always 'Contraries'.
The lesson to be learned from these facts is that guarded statements are often quite as useful as sweeping statements and much safer. If two opponents make guarded statements both may be right; if they make sweeping statements they cannot both be right, but both may be wrong; and if either of them makes a sweeping statement, the other need not make a statement equally sweeping in order to prove him wrong, for a universal proposition can be disproved by a single exception. Cautious statements may not always be very interesting, but they are not likely to be ridiculous.
The terms contrary and contradictory are the only ones in this connection which for ordinary purposes are worth remembering. There are others, however, whose meaning is made clear enough in the following traditional ' square of opposition':
A and E are each called a subalternans (active).
* The table which I have given on p. 112 (not the square) shows the relations of the various propositions when it is taken for granted that objects of the kind described in their subjects exist. We never talk about things that we do not assume for the moment at least to exist in some universe or other. But when a universal proposition is used as the equivalent of a hypothetical to express a general law, then the thing as it is described in the subject of the universal categorical proposition is not what we are really talking about and the proposition does not neces. sarily imply its existence. This has been already explained. (See p. 87.) It has also been pointed out that particular propositions usually do imply th xistence of things as they are described in the subject. Particular propositions therefore imply something that one kind of universal propositions do not imply. In this case therefore the truth of the universal does not necessarily imply the truth of the particular. By way of example let us suppose that the universal proposition Candidates arriving late are fined' is equivalent to the hypothetical • If any candidate arrives late he will be fined'. This statement may be true as stating a rule of the board whether any candidate happens to be late or not. But if the particular proposition · Some candidates, or some of the candidates, arriving late are fined' means that there are candidates arriving late and some of them are fined; then this statement is not true unless there are candidates arriving late. It may thus be possible that the universal proposition is true when the particular is false.
What the logical opposites of exclusive and exceptive propositions are depends upon their interpretation. We have seen that when propositions in this form
With exclurelate concrete facts about individual objects sives and
exceptives. they usually imply something about the objects specifically mentioned as well as about the other members of the class in question, but that when they express some
Similarly if particular propositions imply the existence of candidates arriving late and there are not any, the propositions I and O will both be false at the same time, as they cannot be when the existence of the objects named is taken for granted throughout.
Keynes thinks that it can be shown in a similar way that “the ordinary doctrine of contrariety does not hold good”. If universal propo. sitions do not imply the existence of the kind of things described in their subjects, “ All S is P and no S is Pare not inconsistent with one another, but the force of asserting both of them is to deny that there are any S’s”. (p. 195.) To take his example, if there is no such thing as an honest miller it is true that an honest miller has a golden thumb, and it is true that he has not.
I think that here Dr. Keynes is mistaken. The statement that an honest miller has a golden thumb is equivalent to the hypothetical: “If 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.
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:
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 sloowing 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.
(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
Symbols. of symbols is suggested.
Let a small circle represent any object S of the kind discussed, and a plain stroke through it o 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. It is evident that the stroke cannot be both plain and crossed.
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:
A: All S is P φφφ
I and O: Some Sis P and some is not 0 0 0 0 0
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: φφοο 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
In other words, if we know that Proposition O is true