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 298 G. E. MOORE : can not be that which makes a necessary truth what it is. If certainty be used in any other sense, it may be discussed more conveniently, after we have considered universality. The universal certainly would seem a more likely candi- date, than either of the others, for the honour of identifica- tion with the necessary. They have been ranked together by Kant as joint marks of the a priori. But here again it is necessary to make a distinction of meaning. For, in the first place, a truth may be said to be universal, in the sense already considered as meant by eternal, namely that it is always true. This, we found, would not serve to distinguish any one truth from any other. We must, then, find some other meaning for universality if it is to be identified with necessity. And we have obviously got a universality of some sort, which is not this, in the Law of Contradiction. For it asserts that every proposition is either true or false ; and inasmuch as it thus applies to every instance of the class ' proposition ' it may be said to be universal. But this suggests a distinction which is not without importance. For what is true of every proposition is that it is true or false ; it is not true of any proposition that every proposition is true or false ; but it is this latter which is said to be neces- sary. The necessary, therefore, is not universal in the sense of being a property common to all the instances of a certain kind. If, then, we are to say that necessity is connected with universality, we must say it in the sense that every necessary proposition is one which asserts that some property is to be found in every instance in which some other property is found. But is this true of all necessary propositions ? It would seem it is not true of arithmetical propositions, for instance, of the proposition that 5 + 7 = 12. For here we assert nothing about a number of instances. There are not several instances of 5 and of 7 ; there is but one 5, one 7 and one 12. And yet we assert a connexion between them which is commonly held to be necessary. It is indeed true of every collection of things which number five that, if you add to them a collection which numbers seven, the whole collection will number twelve. But different collections of five things, are not different fives ; and though a proposition about collections of five things may be universal in the sense in which the Law of Contradiction is universal, that is no evidence that a proposition about five itself is so. It is not, then, true that every proposition about a universal is a universal proposition. For every number is a universal in the sense that it is a property of many different collections ; and yet a proposition asserting the connexions between