The logic of equilibrium and abelian lattice ordered groups (Q1879318)

The authors introduce a quasivariety, called BAL, and its corresponding logic Bal. They prove that BAL is definitionally equivalent to abelian lattice-ordered groups and that BAL models a logic of balance of opposing pieces of evidence. Bal is specified by five axioms and four inference rules. Truth values are thought of as deviations from a state of equilibrium. It is proved that the tautology problem for Bal is co-NP-complete. A deduction theorem is established, and various possible notions of Bal-consequence are discussed. BAL (resp. Bal) is strongly related to MV-algebras (resp., infinite-valued propositional Łukasiewicz logic). As proved by the present reviewer, MV-algebras are categorically equivalent to lattice-ordered abelian groups with strong unit. Further, the tautology problem of infinite-valued Łukasiewicz propositional logic is co-NP-complete. For further information see the monograph [\textit{R. L. O. Cignoli}, \textit{I. M. L. D'Ottaviano} and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht (2000; Zbl 0937.06009)].