According to Bertrand Russell's Theory of Descriptions, the negation operator in a singular sentence takes wide and narrow scope: we distinguish between "some S is not P" (where negation takes "narrow scope") and "it is not the case that "some S is P" (where negation takes "wide scope"). The problem with this view is that there appears to be no such scope distinction in the case of proper names. The sentences "Socrates is not bald" and "it is not the case that Socrates is bald" both appear to have the same meaning, and they both appear to assert or presuppose the existence of someone (Socrates) who is not bald, so that negation takes narrow scope. The theory of descriptions has generally fallen into disrepute, though there have been recent attempts to revive it by Stephen Neale and Frank Jackson. According to the Direct reference view, an early version of which was originally proposed by Peter Strawson, and (some have argued) even earlier by Gottlob Frege, a proper name strictly has no meaning when there is no object to which it refers. This view is sometimes justified by the argument that the semantic function of a proper name is to tell us which object bears the name, and thus to identify some object. But no object can be identified if none exists. Thus, a proper name must have a bearer if it is to be meaningful. To adapt an argument of Strawson's, someone who points to an apparently empty space, uttering "that's a fine red one" communicates nothing to someone who cannot see or understand what he is pointing to. Variants of the Direct reference view have been proposed by Saul Kripke, Gareth Evans, Scott Soames and others.
According to the "two sense" view of existence, existential statements fall into two classes.
1. Those asserting existence in a wide sense. These are typically of the form "N is P" for singular N, or "some S is P".
2. Those asserting existence in a narrow sense. These are typically of the form "N exists" or "S's exist".
The problem is then evaded as follows. "Pegasus flies" implies existence in the wide sense, for it implies that something flies. But it does not imply existence in the narrow sense, for we deny existence in this sense by saying that Pegasus does not exist. In effect, the world of all things divides, on this view, into those (like Socrates, Venus the planet, New York) that have existence in the narrow sense, and those (like Sherlock Holmes, Venus the goddess, Minas Tirith) that do not.
Supporters of this view (which derives from Alexius Meinong) include Terence Parsons and Edward Zalta.
The difficulty with this view is (a) that common sense suggests that there are no such things as fictional characters, places, (b) there is no strong evidence for two kinds of existential sentence as used in ordinary language.
Relevant also to the topic are views posed by Moore, which were then, perhaps elaborated by Wittgenstein. GE Moore used to walk into a room full of students and then raise his right hand, and say "I am raising my right hand," "Now, someone disprove me!" The truth is that it is so common sensical, so obvious that he is raising his right hand, that no one could consistently assert that he was not holding up his hand. In "A Defence of Common Sense" Moore basically turns the traditional Cartesian view on its head because instead of saying that everything must be doubted and one must go back to that one essential thing that cannot be doubted, and then you can believe; he says that first we must believe, then we can question. Because he can raise his hand and know its there, because it's obvious "Nothing is knowable" is a contradiction. ("Nothing is knowable" is also a paradox because how is it even conceivable that "nothing" could be known.)
Another problem that stops people defining existence is that a definition/explanation is meant to put something into simpler terms (make it easier to understand,) existence itself is too simple and basic to be explained (cannot by made simpler) and therefore a possible explanation is difficult to come up with.