I am not quite sure I interpret the following sentence correctly in Bertrand Russell's paper on existential import:
and among classes there is just one which does not exist, namely, the class having no members, which is called the null-class.
This seems to be saying that the empty set does not exist, and although I would agree with that, it seems to me that it very unlikely to be what he meant.
So my question is:
What did Russell mean when he wrote that the null-class, the class he defined as having no members, did not exist?
I put here the relevant passage which is on his distinction between what he sees as the two meanings of the word "existence":
The first point in regard to which clearness is essential concerns the meaning of the word “existence.” There are two meanings of this word, as distinct as stocks in a flower-garden and stocks on the Stock Exchange, which yet are continually being confused, or at least supposed somehow connected. Of these meanings, only one occurs in philosophy or in common parlance, and only the other occurs in mathematics or in symbolic logic. Until it is realised that they have absolutely nothing to do with each other, it is quite impossible to have clear ideas on our present topic.
(a) The meaning of existence which occurs in philosophy and in daily life is the meaning which can be predicated of an individual, the meaning in which we inquire whether God exists, in which we affirm that Socrates existed, and deny that Hamlet existed. The entities dealt with in mathematics do not exist in this sense: the number 2, or the principle of the syllogism, or multiplication, are objects which mathematics considers, but which certainly form no part of the world of existent things. This sense of existence lies wholly outside Symbolic Logic, which does not care a pin whether its entities exist in this sense or not.
(b) The sense in which existence is used in symbolic logic is a definable and purely technical sense, namely this: To say that A exists means that A is a class which has at least one member. Thus whatever is not a class (e.g., Socrates) does not exist in this sense; and among classes there is just one which does not exist, namely, the class having no members, which is called the null-class. In this sense, the class of numbers (e.g.) exists, because 1, 2, 3, etc., are members of it; but in sense (a) the class and its members alike do not exist: they do not stand out in a part of space and time, nor do they have that kind of super-sensible existence which is attributed to the Deity.
Bertrand Russell, The Existential Import of Propositions (1905)