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Full Idea
A weak completeness theorem shows that a sentence is provable whenever it is valid; a strong theorem, that a sentence is provable from a set of sentences whenever it is a logical consequence of the set.
Gist of Idea
Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
Source
George Boolos (On Second-Order Logic [1975], p.52)
Book Ref
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.525
A Reaction
So the weak version says |- φ → |= φ, and the strong versions says Γ |- φ → Γ |= φ. Presumably it is stronger if it can specify the source of the inference.
| 14249 | Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley] |
| 13841 | Why should compactness be definitive of logic? [Boolos, by Hacking] |
| 10829 | A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos] |
| 10830 | Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos] |
| 10832 | '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos] |
| 10833 | Many concepts can only be expressed by second-order logic [Boolos] |
| 10834 | Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos] |