Modal correspondence for models (Q1267083)

Correspondence theory in modal logic provides results of the following form: a modal formula \(\varphi\) is valid in a frame \(F\) iff a first-order formula \(c_\varphi\) is true in \(F\) regarded as a first-order model. The paper considers a possibility of obtaining similar correspondence results for models. It is shown, in particular, that each frame correspondence result can be extended to a model correspondence result: a model \(M\) satisfies all substitution instances of \(\varphi\) in infinitary modal logic iff \(c_\varphi\) is true in a certain frame uniquely determined by \(M\).