Discussions of linguistic reference rarely define what reference is. An example is the SEP article here, which as far as I can see does not give any definition, merely stating that linguistic reference is ‘a relation that obtains between certain sorts of linguistic expressions and what speakers use those expressions to talk about’. And what relation is that? Slightly more helpfully it characterises it ‘metaphorically’ as a mechanism that enables us to talk about the world – say about George Bush – through referring terms which hook on to the world. But only slightly. What are these hooks?
Let’s try this. If Socrates is not running and Plato is running, ‘Socrates is running’ is false, and ‘Plato is running’ is true. But ‘a man is running’ is true, because Plato is running. The non-referring, i.e. indefinite, statement is true or false regardless of who is running. But the truth of falsity of the singular statements depends on who is running.
So a singular or referring statement is true when there is some particular person such that the predicate is satisfied by that person, and is false when that person fails to satisfy the predicate, even if the predicate is satisfied by other people. But a
non-referring or existential statement is true when any individual, no matter who, satisfies the predicate.
Does that work?