What question did this study set out to answer?

This research aims to explore how interpretations of modal logics can be constructed using additive and normal formulas.

July 1, 2026Open Access

On Interpretations of Normal Modal Logics

Key Points

This research aims to explore how interpretations of modal logics can be constructed using additive and normal formulas.
Studied interpretations of modal logics where Boolean connectives maintain identical interpretations.
Defined the modal operator diamond through arbitrary formulas α(p) that are both additive and normal.
Provided comprehensive descriptions of all additive and normal formulas in five modal logics: K, GL, Grz, S4, and S5.
Described all additive and normal formulas for K, GL, and S5, including those with parameters.
Demonstrated that α(p) must be additive and normal for valid interpretations in the target logic.

Abstract

We study interpretations of modal logics in one another where the Boolean connectives are interpreted identically and the modal operator diamond is interpreted by an arbitrary formula α(p).Clearly, such a formula α(p) defines an interpretation of a normal modal logic whenever α(p) is additive (that is, preserves disjunction) and normal (that is, preserves bottom) in the target logic.In the present paper, we provide a complete description of all additive and normal formulas in five prominent modal logics: K, GL, Grz, S4, and S5.For K, GL, and S5, we also describe all additive and normal formulas with parameters.

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Cite This Study

Lev V. Dvorkin (Sun,) studied this question.

synapsesocial.com/papers/6a44ad765cd2549c8bc430f3 https://doi.org/https://doi.org/10.4204/eptcs.447.18

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