Linear Abelian Modal Logic - Mohammadi, Hamzeh (Bulletin of the Section of Logic, 2023)
A many-valued modal logic, called linear abelian modal logic \(\rm {LK(A)}\) is introduced as an extension of the abelian modal logic \(\rm K(A)\). Abelian modal logic \(\rm K(A)\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm LK(A)\) is axiomatized by extending \(\rm K(A)\) with the modal axiom schemas \(\Box(φ\veeψ)→(\Boxφ\vee\Boxψ)\) and \((\Boxφ\wedge\Boxψ)→\Box(φ\wedgeψ)\). 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.