2

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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.