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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 Ref
'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.
| 13833 | 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking] |
| 13834 | Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking] |
| 13835 | Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking] |
| 13837 | With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking] |
| 13838 | A decent modern definition should always imply a semantics [Hacking] |
| 13839 | Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking] |
| 13843 | If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking] |
| 13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
| 13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
| 13842 | Second-order completeness seems to need intensional entities and possible worlds [Hacking] |
| 13845 | The various logics are abstractions made from terms like 'if...then' in English [Hacking] |