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**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 |