Meyer, R. K.; Slaney, J. K. Abelian logic (from A to Z). (English) Zbl 0694.03019 Paraconsistent logic, Essays on the inconsistent, 245-288 (1989). [For the entire collection see Zbl 0678.00003.] This paper describes a propositional logic A, which corresponds to Abelian \(\ell\)-groups. A’s properties, as might be expected, are somewhat bizarre. A is stricter than the relevance logic R in requiring us to distinguish, not merely whether, but also how many times, a premiss is used; and resists very strongly the paradoxes of implication \(P\to (Q\to P)\) and ex falso quodlibet (P&\(\sim P)\to Q\), and also formulas related to W(P\(\to (P\to Q))\to (P\to Q)\). A is inconsistent but not trivial, and also has the infinite model property. The paper raises open problems about A and its extensions to quantified and propositionally quantified systems, and is written in high spirits. Reviewer: J.Mackenzie Cited in 6 ReviewsCited in 18 Documents MSC: 03B60 Other nonclassical logic 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 03B45 Modal logic (including the logic of norms) Keywords:paraconsistent logic; Abelian \(\ell \)-groups; relevance logic; paradoxes; infinite model property Citations:Zbl 0678.00003 PDFBibTeX XML