My lecturer defines logical validity (in the English language) like so:
'An argument is logically valid if and only if there is no (uniform) interpretation (of subject-specific expressions) under which the premises are all true, and the conclusion is false.'
He contrasts subject-specific expressions (e.g., Donald Trump, Aristotle, chemical element, London), with logical expressions (i.e., if, not, if and only if, every, some). Logical expressions are not subject to re-interpretation; they keep their standard English meanings all the time.
My question is this: is the following argument valid?
P1: Santa Claus does not exist. C: Something does not exist.
Now, on the one hand, I'm inclined to say yes: if I replace 'Santa Clause' with any other noun, or I replace the property of does not exist with any other property, the resulting argument is such that: if the premises are true, so too is the conclusion.
On the other hand, I'm hesistant to say yes: if I replace 'something' with, for example, 'a car', then the resulting argument seems to involve a true premise, and a false conclusion.
Moreover, I'd unhesitatingly say that the following argument is valid:
P1: Santa Claus does not exist. P2: Santa Claus is something. C: Something does not exist.
Could Santa Clause not be 'something'? From another angle: is 'something' a 'subject-specific expression'? I'm inclined to think it is not, but I'm not sure how to justify this thought. (My inkling is that it has something (lol) to do with the fact that 'something' is a pronoun, whilst 'a car' is a noun? Also, I'm aware that the argument in question involves a valid rule of inference in FOL. But I wonder if this is one of those cases where validity in FOL comes apart from more informal characterisations of validity in the English language (e.g., http://www.jimpryor.net/teaching/courses/intro/notes/leibniz-epist.html).)
Alex Kruckman is right, this is more philosophy than maths. Let me give a very brief answer that may be enough for your purposes.
There is the mainstream (kantian) answer to this kind of antinomy: existence is not a predicate. A Russell-Frege analysis of "Santa Claus does not exist" is
$\neg\exists x; x$ is Santa Claus.
The lesson here is that logical form is not necessarily unique and equal to the grammatical form (Frege). The (logical) subject of the sentence is not Santa Claus, but the property of being Santa Claus, and the sentence says that the extension of this property is empty.
This is the basis of Quine's meta-ontology: existence is not a predicate, it is univocal, it is the same as being and it is adequately captured by the existential quantifier. "Something does not exist" does not correspond to a logical sentence in this account.
Of course, there are alternative views on this topic. There is Meinong's meta-ontology, for example, in which there are things that subsist but do not exist (being is not the same as existence). Santa Claus could be an example. In this account, the argument that you have given is taken to be valid. I shall stop here.