Let L be a ground language, and L+ the extension of L with additional signature terms and axioms. We say L+ is syntactically conservative over L if every sentence in L that is provable in L+ is already provable in L; we say L+ satisfies the syntactic eliminability criterion if for every sentence in L+, there is a sentence in L such that these two sentences are provably equivalent in L+.

These two definitions have semantic analogues: L+ is semantically conservative over L if for every formula in L and every interpretation M, if the formula is valid in L+ for M, then it is valid in L for M; the definition of semantic eliminativeness follows analogously.

My question is: in https://plato.stanford.edu/entries/definitions/#ConEli, it is claimed that ``even if we suppose that strong completeness theorems hold for L and L+, the two formulations are not equivalent. Indeed, several different, non-equivalent formulations of the two criteria are possible within each framework, the syntactic and the semantic.''

I couldn't see how, with strong completeness, these two definitions could come apart (aren't they only differ in terms of whether we are talking about proof-theoretic entailment or semantic entailment?) Any hint or reference is greatly appreciated.


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