Linear Abelian Modal Logic

Hamzeh Mohammadi

doi:10.18778/0138-0680.2023.30

Authors

Hamzeh Mohammadi Isfahan University of Technology, Department of Mathematical Sciences Author https://orcid.org/0000-0002-7726-3074

DOI:

https://doi.org/10.18778/0138-0680.2023.30

Keywords:

many-valued logic, modal logic, abelian logic, hypersequent calculus, cut-elimination

Abstract

A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization is investigated.

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