## Modal logic and classical logic.(English)Zbl 0639.03014

Indices. Monographs in Philosophical Logic and Formal Linguistics, III. Napoli: Bibliopolis. 234 p. (1985).
The work is a rewritten version of the author’s dissertation: “Modal correspondence theory”, and a supplementary report: “Modal logic as second-order logic”. Its leading theme is the interplay between the modal (intensional) and classical (extensional) perspective, made possible by the fact that possible worlds models for modal systems, structured by a relation of alternativity or accessibility, can also be viewed as semantic structures for the standard languages of classical logic. The book begins with a survey of the model theory, algebra and proof theory of modal logic, and then raises the question just when a given modal principle expresses a standard first-order constraint on the alternativity relation. Well-known examples of such correspondences are Lp$$\to p$$ expressing reflexivity, Lp$$\to LLp$$ transitivity, and Mp$$\to LMp$$ symmetry of the relation. Next, the author raises the converse question: he starts from first-order relational conditions and asks for modal axioms matching them. He also studies many remaining irreducibly higher-order modal principles. Finally, the results and techniques developed are used to explore the area of higher-order logic itself.
Reviewer: I.Gullvåg

### MSC:

 03B45 Modal logic (including the logic of norms) 03B10 Classical first-order logic 03B15 Higher-order logic; type theory (MSC2010) 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations