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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

MSC:

03B60 Other nonclassical logic
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
03B45 Modal logic (including the logic of norms)

Citations:

Zbl 0678.00003