Singular negative existential propositions such as 'Santa Claus does not exist' can be made to look puzzling without any explicit theoretical view on board, with Socrates-style questions such as: how can anything not exist? How can one ever truly say that something doesn't exist? For if one is right, there is no such thing as what one is talking about, and therefore one is talking about nothing.
Or: 'Santa Claus doesn't exist' – what is this thing which doesn't exist? How could there be such a thing?
For those who have a Millian conception of naming (and associated views of propositions), the problem of negative existentials assumes a particularly acute form. If all there is to the meaning of a name is its bearer, and if substituting co-referring names does not affect the 'proposition' (not my usage) expressed by the sentence, then how can a statement like 'Santa Claus does not exist' mean anything at all? Furthermore, how can it be true? And how can different true negative existentials have different meanings, as they seem to do?
All these questions push the Millian toward analysing existence statements – giving an account of what they really mean. (Witness for example Kripke's tortured discussion in Reference and Existence. Of course, he never officially and unequivocally endorses Millianism, but he's flirted outrageously with it in public and finds it intuitive.)
Having a non-Millian conception of naming such as the one I propose, on which names are recognized as being tied to individual concepts (or having uses - roles in the systems of language they occupy - which are semantically relevant), makes it a lot easier to answer these questions. But this does not mean that we are using individual concepts (or name-uses) as elements in an analysis of existence statements.
'N does not exist' does not, for instance, mean exactly the same thing as'“N” has no bearer' or 'The “N”-concept has empty extension' (that is, on a natural and sufficiently fine-grained conception of proposition-meanings): the existential proposition is not about a name or a concept.
We can say it is about N, if we understand 'about' as not having existential import, or we can say that it is not about any real thing, but it would muddy the waters intolerably to say that it is about a name or a concept.
And yet we can see what would make a person want to say that it is about a name or a concept. What we can truly and properly say is that, in 'N does not exist', the function of the name 'N' is not to pick out an object – rather, this name (rather than some other name) is used in order to bring a particular individual concept (or name-use) into the act (though I would not want to say 'under consideration', for it need not become an object of thought).
But then what do we say about the function of a name in a proposition like 'John is tall'? It is no less true to say that 'John' functions to bring a particular individual concept or name-use into the act, but here we can also say that it functions to pick out an object. But some have found it intuitive to say that a name functions purely to pick out an object. Given a certain very narrow concept of 'function', this is fine too, although it could be misleading – it could lead to the troubles of Millianism.
Let us now take a representative selection of the questions raised at the beginning of this section, showing how they can be answered with the conceptions of propositions and naming I favour. This is done without giving an analysis of existence statements (in the classical sense of giving something else which they are then said to mean). As with identity statements, the trick is to treat existence statements on their own terms, and to recognize that they occupy a special role for us, and work in a quite particular way.
How can anything not exist?
Just as we sometimes use names in a way which carries existential import, as in 'John is tall', and sometimes use them in a way which does not, as in 'Santa Claus does not exist', 'Some children believe in Santa Claus', we use terms like 'there is', 'something' and 'anything' in two different ways: with or without existential import. A clear example of the latter sort of use would be a kid saying 'I don't believe in Santa Claus, the Easter Bunny, or anything like that'.
The difficulty and puzzlingness of the above question derives from this ambiguity. The thorough answer is: in the sense of 'anything' etc. in which they carry existential import, nothing can fail to exist – i.e. it is not the case that it is possible for anything to not exist, but in the sense of 'anything' etc. in which they do not carry existential import, there are indeed things which do not exist.
At this point, individual concepts (notions of things), or name-uses can be brought into consideration, to help us make sense of the fact that we talk like this. What gives rise to it is that we sometimes have individual concepts without objects, name uses where the name lacks a bearer. We then formulate propositions which, if we treat them by analogy with propositions like 'John is tall' and 'Someone is in this room', look as though they would have to involve (existing) things in order to be true, in the way that these propositions would have to involve John and a person in the room – but in fact they face no such requirement. We use them in connection with objectless concepts, bearerless names etc. (and this connection is quite different from that which holds between 'John is tall' and John himself).
How can one ever truly say that something doesn't exist? For if one is right, there is no such thing as what one is talking about, and therefore one is talking about nothing.
In light of the above, this question can be disposed of quickly. We can truly say that something doesn't exist by using 'something' in the sense in which it doesn't carry existential import, and in virtue of the fact that we have objectless individual concepts and involve them in our talk. In the sentence after the question ('For if one ...'), 'there is no such thing as' and 'nothing' are used in their existential-import-having senses, and so there is no real conflict in what is being said here. It is just being said in a potentially misleading way.
How can a statement like 'Santa Claus does not exist' mean anything at all?
The proposition works by means of the fact that the name 'Santa Claus' brings an individual concept (or a way of using a name) into the act – not by referring to it, but because that is the concept tied to that name (or that is the way that name is used). The proposition is true iff the concept (or name-use) of 'Santa Claus' has an object.
This is not to say that the proposition means the same as any proposition about concepts or name-uses, or that the proposition holds of just the same possible situations as those of which what is said on the right hand side of the 'iff' holds. We are using the biconditional here not to give an analysis but to give a necessary and sufficient condition for the proposition in question, which we have before us, actually being true.
How can different true negative existentials have different meanings, as they seem to do?
By bringing different individual concepts (or name-uses) into the act.
Kripke, Saul A. (2013). Reference and Existence. The John Locke Lectures. Oxford University Press.