Strong completeness requires that every semantically valid inference over all many-sorted structures is provable, but Lindström's theorem shows any logic stronger than first-order that gains expressive power loses completeness or compactness.

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Lindström's theorem(mathematical logic): A famous mathematical result (named after Swedish logician Per Lindström) proving that first-order logic is the strongest logic that can keep certain desirable properties—if you make it more powerful, you have to give something up.
Many-sorted structures(mathematical logic): Mathematical systems that deal with multiple different types or categories of objects at once, rather than treating everything the same way.
Semantically valid inference(logic): A logical argument where the conclusion must be true whenever all the premises (starting assumptions) are true, based on what the words actually mean.
Strong completeness(Distinguished from weak completeness, which only concerns tautologies): If φ is a semantic consequence of Γ (Γ ⊨ φ), then φ is provable from Γ (Γ ⊢ φ)

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