An intuitionistic characterization of classical logic
It is common to analyse an extension of logical system by restricting the semantical framework characterizing the basic logic. The author points out that such a restriction does not mean a semantical explanation for ``extending logical system'' because the meanings of logical constants may be changed by the restriction, or more frankly, by another semantics. From this viewpoint the author proposes an approach to comparative analysis of logical systems, which is developed in this paper for propositional intermediate logics with respect to Kripke semantics. Given a Kripke frame, the validity of a formula is defined by means of both valuations and transformations of truth values (i.e., open sets of possible worlds), which are called intentional maps here, as a formula is valid if for any valuation and for any intentional map the value is transformed to the universe (i.e., the whole space of possible worlds). Thus, we can consider the classes of intentional maps, namely their properties, instead of those of frames without restricting the whole class of frames, which then raises another semantical characterization problem. Under this perspective it is observed that any Kripke-complete intermediate logic can thus be charaterized enjoying its (unique) ``weakest characterization property'' of intentional maps. In particular, the weakest characterization property for classical logic is shown to be those transforming dense sets to the universe, from which it follows that a syntactical translation is an embedding of classical logic into intuitionistic logic iff its semantical counterpart has the property.
scientific article; zbMATH DE number 5314495