A monadic logic of ordered abelian groups

Metcalfe, George; Tuyt, Olim (2020). A monadic logic of ordered abelian groups. In: Olivetti, Nicola; Verbrugge, Rineke; Negri, Sara; Sandu, Gabriel (eds.) Proceedings of AiML 2020. Advances in Modal Logic: Vol. 13 (pp. 441-457). College Publications

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A many-valued modal logic with connectives interpreted in the ordered additive group of real numbers is introduced as a modal counterpart of the one-variable fragment of a (monadic) first-order real-valued logic. It is shown that the logic is decidable and admits an interpretation of the one-variable fragment of first-order Lukasiewicz logic. Completeness of an axiom system for the modal-multiplicative fragment is established via a Herbrand theorem for its first-order counterpart. A functional representation theorem is then proved for a class of monadic lattice-ordered abelian groups and used to establish completeness of an axiom system for the full logic.

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