Full Idea
A Löwenheim-Skolem theorem holds for anything which, on my delineation, is a logic.
Gist of Idea
If it is a logic, the Löwenheim-Skolem theorem holds for it
Source
Ian Hacking (What is Logic? [1979], §13)
Book Reference
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.246
A Reaction
I take this to be an unusually conservative view. Shapiro is the chap who can give you an alternative view of these things, or Boolos.