But although all logical (or mathematical) propositions can be expressed wholly in terms of logical constants together with variables, it is not the case that, conversely, all propositions that can be expressed in this way are logical. We have found so far a necessary but not a sufficient criterion of mathematical propositions. We have sufficiently defined the character of the primitive ideas in terms of which all the ideas of mathematics can be defined, but not of the primitive propositions from which all the propositions of mathematics can be deduced. This is a more difficult matter, as to which it is not yet known what the full answer is. We may take the axiom of infinity as an example of a proposition which, though it can be enunciated in logical terms, [Pg 202] cannot be asserted by logic to be true. All the propositions of logic have a characteristic which used to be expressed by saying that they were analytic, or that their contradictories were self-contradictory. This mode of statement, however, is not satisfactory. The law of contradiction is merely one among logical propositions; it has no special pre-eminence; and the proof that the contradictory of some proposition is self-contradictory is likely to require other principles of deduction besides the law of contradiction. Nevertheless, the characteristic of logical propositions that we are in search of is the one which was felt, and intended to be defined, by those who said that it consisted in deducibility from the law of contradiction. This characteristic, which, for the moment, we may call tautology, obviously does not belong to the assertion that the number of individuals in the universe is , whatever number may be. But for the diversity of types, it would be possible to prove logically that there are classes of terms, where is any finite integer; or even that there are classes of terms. But, owing to types, such proofs, as we saw in Chapter XIII., are fallacious. We are left to empirical observation to determine whether there are as many as individuals in the world. Among "possible" worlds, in the Leibnizian sense, there will be worlds having one, two, three, ... individuals. There does not even seem any logical necessity why there should be even one individual[43]—why, in fact, there should be any world at all. The ontological proof of the existence of God, if it were valid, would establish the logical necessity of at least one individual. But it is generally recognised as invalid, and in fact rests upon a mistaken view of existence—i.e. it fails to realise that existence can only be asserted of something described, not of something named, so that it is meaningless to argue from "this is the so-and-so" and "the so-and-so exists" to "this exists." If we reject the ontological [Pg 203] argument, we seem driven to conclude that the existence of a world is an accident—i.e. it is not logically necessary. If that