Why do we make statements like Vulcan doesn't exist or Pegasus never existed or there's no such person as Santa Claus?
One usage is in proofs of non-existence by reductio ad absurdum where such statements form the concluding line. Such proofs start with an hypothesis that an object satisfying a certain condition exists, ∃x.Px, in first order logic. For convenience in the subsequent argument we give this object a name, N, say. We then proceed to deduce a falsehood. This often takes the form of first deducing p and then deducing ~p. This is a classic contradiction. We infer from this that the original hypothesis is false. That is, that no object satisfies the condition. And we usually express this by asserting N does not exist, using the name we attached to our hypothetical object. Note that we are not obliged to use this form of words. The formal conclusion of the argument is ~∃x.Px. We do not need to say N does not exist at all. It is merely a convenient informal shorthand.
Now, my contention is that all use of singular negative existentials is really shorthand for a general existential denial. Sometime in the past the name was introduced to us as a label for some object satisfying a condition. This is a case of existential instantiation. If ∃x.Px is true then introduce a new logical constant, c, say, and by instantiation we have Pc. Informally, c is a name for the object that witnesses the truth of ∃x.Px. Of course, c may only be used within the scope of the hypothetical ∃x.Px, and this is made very clear in the 'proof box' method of presenting an argument. In effect, until we arrive at the contradiction that denies us ∃x.Px, the proof box remains open. This is how we live our lives. We are in the midst of hundreds, maybe thousands of proof boxes opened with general existential hypotheses. There was an ancient Greek thinker who wrote several texts highly influential on medieval and subsequent European philosophy. Call him Aristotle. Aristotle was an ancient Greek thinker who wrote several texts highly influential on medieval and subsequent European philosophy. If it ever turns out that there was no such man we simply say Aristotle never existed. Close proof box.
Why do I question whether these denials are indicative? Well, a consequence of closing the proof box that allows us to infer ~∃x.Px is that we must discard all the formulae deduced inside the box. Some of these may be re-deducible outside the box, but certainly all those involving the constant c have to go. c no longer has a referent. So we have to throw away everything we thought we knew about this object. Aristotle never existed is actually an imperative rather than an indicative.
So I throw out a challenge: find a use of a negative singular existential that does not fit this pattern.