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NECESSITY. 299 numbers makes no assertion about a number of instances. It has indeed been suggested that propositions such as the Law of Contradiction might be more properly expressed in a form analogous to arithmetical propositions ; that we should say, not : Every proposition is either true or false ; but : Proposition is either true or false, just as we say : Man is mortal. But there seems reason to suspect that these propositions are really universal in a sense in which arithmetical propositions are not so, and that ' proposition ' is not a property of propositions in the same sense in which any number is a property of the collection of which it is predi- cated. For even granted that ' Man is mortal ' has a mean- ing, how can we get from this to the proposition ' All men are mortal,' except by adding that the property of mortality is always connected with the other properties of humanity, wherever these latter occur ? Whereas from the proposition that 7 + 5 = 12, you can arrive at the conclusion that all collections of five and seven are equal to collections of twelve, without the premiss that 7 -I- 5 = 12, wherever they occur ; for the reason, which seems to be true though it will hardly be thought convincing, that 5 and 7 never do occur. For myself, I cannot perceive that ' Man is mortal ' has any meaning at all except that ' Man is always mortal ' ; and similarly with the Law of Contradiction, since propositions do not occur in time and therefore cannot be said to be always either true or false, the ultimate expression of it would seem to be that all propositions are either true or false. We must, therefore, say that some necessary propositions are not universal in the sense that they make an assertion about a sum of instances, whereas other necessary proposi- tions are universal in this sense. This universality too, then, will not furnish the meaning of that necessity which belongs to necessary truths. But is there, perhaps, some third kind of universality which is common both to the propositions of Arithmetic and to the Law of Contradiction, and indeed to all propositions which have a prima facie claim to be considered necessary truths ? There is, I think, a sense in which, not indeed strict universality, but a certain generality may be claimed for all of them. They may all be said to be propositions of a wide application ; and a discussion of what exactly this wide application is will furnish my answer to the question what is meant by that necessity which may be truly ascribed to necessary truths. It will then only remain to inquire what, if anything, there is in common between this so-called ' ideal ' necessity and causal or ' real ' necessity.