Abstract
We say that a semantical function Г is correlated with a syntactical function F iff for any structure A and any sentence ϕ we have A ⊧ Fϕ ↔ Γ A ⊧ ϕ.
It is proved that for a syntactical function F there is a semantical function Г correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function Г there is a syntactical function F correlated with Г iff for any finitely axiomatizable class X the class Г −1X is also finitely axiomatizable (i.e. iff Г is continuous in model class topology).
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Gajda, A., Krynicki, M. & Szczerba, L. A note on syntactical and semantical functions. Stud Logica 46, 177–185 (1987). https://doi.org/10.1007/BF00370379
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DOI: https://doi.org/10.1007/BF00370379