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MINIMAL MODAL LOGICS, CONSTRUCTIVE MODAL LOGICS AND THEIR RELATIONS

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  27 March 2025

TIZIANO DALMONTE Open the ORCID record for TIZIANO DALMONTE [Opens in a new window]
TIZIANO DALMONTE*
Affiliation:
FREE UNIVERSITY OF BOZEN-BOLZANO FACULTY OF ENGINEERING NOI TECHPARK - BRUNO-BUOZZI-STRAßE 1 - VIA BRUNO BUOZZI, 1, 39100 BOZEN-BOLZANO ITALY
*
E-mail: tiziano.dalmonte@unibz.it
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Abstract

We present a family of minimal modal logics (namely, modal logics based on minimal propositional logic) corresponding each to a different classical modal logic. The minimal modal logics are defined based on their classical counterparts in two distinct ways: (1) via embedding into fusions of classical modal logics through a natural extension of the Gödel–Johansson translation of minimal logic into modal logic S4; (2) via extension to modal logics of the multi- vs. single-succedent correspondence of sequent calculi for classical and minimal logic. We show that, despite being mutually independent, the two methods turn out to be equivalent for a wide class of modal systems. Moreover, we compare the resulting minimal version of K with the constructive modal logic CK studied in the literature, displaying tight relations among the two systems. Based on these relations, we also define a constructive correspondent for each minimal system, thus obtaining a family of constructive modal logics which includes CK as well as other constructive modal logics studied in the literature.

Keywords

constructive modal logicminimal modal logicmodal companionsequent calculusneighbourhood semantics

MSC classification

Primary: 03B45: Modal logic (including the logic of norms) 03F03: Proof theory, general
Secondary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 42
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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