ENPages
36
32
a quality common and peculiar to all the members of a gath, then there certainly is such a quality; and you yourself have in this very same breath described one such quality, in saying that they are all members of the gath in question. So for every gath there is a corresponding sam. But it is not true that for every sam there is a corresponding gath. Since there is the sam of the phenix, although it happens not to exist up to date. But there is no such gath since there is no phenix. Another point which I observe puzzles the Hon. Bertrand Russell in his ‘Principles of Mathematics’ is whether a collection which has but a single individual member is identical with that individual or not. The proper answer is
37
33
that if by a collection you mean a sam, the sam of the sun is not the sun, since it is an ens rationis having an essence, while the individual has no essence and is not an ens rationis. But if you mean the gath, the gath of the sun has no being at all except the existence of the sun which is all the being the individual existent sun has. Therefore, having precisely the same being they are identical and no distinction except a grammatical or linguistic one can be drawn between them. Mr. Russell's being puzzled by this is a good illustration of how impossible it is to treat of philosophy without making a special vocabulary such as all other sciences make. It is, however, far more needed in philosophy
38
36
than in any other science, for the reason that the words of ordinary speech are needed by philosophy for its raw material.
What has been said of qualities is equally true of relations, which may be regarded as the qualities of sets of individuals. That is to say, if any form of relation is logically possible between the members of two given gaths, a relation of that form actually exists between them.
In order to illustrate this principle, I will consider a proposition which the ablest mathematicians have been endeavoring for half a century to demonstrate to beeither true or false. The proposition assumes that there are two gaths, which we may call the A's and the B's. And now if we use the phrase 'a "one to one
39
38
relation" to denote such a relation that no relate stands in that relation to two different correlates nor any two relates to the same correlate; like the relation of husband to wife in a monogamous country, or better of father to eldest child, then it is assumed that there is no one-to-one relation in which every B stands to an A. Then the question is whether there is necessarily any one to one relation in which every A stands to a B.
Since there is no one-to-one relation in which whatever B there may be stands to an A, it must be logically impossible that there should be any such relation. But since the form relation in itself is not absurd, and since in point of fact every B does stand in such a relation to some member of a gath, for every B stands in the
40
40
relation of identity to some B, it follows that the logical impossibility lies in some existential limitation of the As. However, dropping at the outset the assumption about the one to one relation considering any two gaths, the As and the Bs, there is a relation in which no relate stands to two different correlates (though two relates may stand in it to the same correlate) in which every B stands to an A; for the Bs may so stand to any one A. Now setting out with such a relation, let us ask whether by a change of it so as to change each correlate into some other individual of the universe, it is possible so to change it as to make every A a correlate of it. Let us trace out the consequences of supposing this not to be possible. Let us under this hypothesis change our relation by a change of correlates so as to render it a one to one relation, and let us suppose this to be done in every possible