Superabelian logics

This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat $\Ab$ as the base system and refer to its expansions as \emph {superabelian logics}. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist $2^{2^ω}$ distinct logics in this family. Second, we introduce \emph{pointed Abelian logic} (pAb), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes \emph{Łukasiewicz unbound logic}. We provide axiomatizations for its finitary and infinitary versions as extensions of pAb and establish their precise relationship with standard Łukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.

💡 Research Summary

The paper undertakes a comprehensive algebraic investigation of “superabelian logics”, which are expansions of Abelian logic (Ab) – the logic of Abelian lattice‑ordered groups (ℓ‑groups). The authors treat Ab as the base system and explore two principal families of extensions.

The first family consists of infinitary extensions of Ab, i.e., logics that allow inference rules with infinitely many premises while keeping the original language (→, &, ∨, ∧, t). By focusing on the real additive group ℝ, the authors define an infinitary consequence relation ⊨_ℝ and provide a single infinitary rule that axiomatizes it. This rule essentially permits any real‑valued assignment to be used as a premise, yielding a logic strictly stronger than the analogous infinitary logic based on the rational group ℚ (⊨_ℝ ⊂ ⊨_ℚ). Moreover, by varying the infinitary rule they obtain a rich hierarchy: there are 2^{2^{ℵ₀}} distinct infinitary extensions of Ab, each with its own completeness theorem. This demonstrates that, unlike the finitary case where Ab has no non‑trivial extensions (the only finitary extension is the inconsistent logic), the infinitary landscape is extraordinarily rich.

The second family introduces a new constant f to the language, giving rise to pointed Abelian logic (pAb), the logic of pointed Abelian ℓ‑groups. The constant is interpreted as the element –1, i.e., f(x)=x−1. With this addition, the authors show that pAb captures precisely the “Łukasiewicz unbound logic” (Lu), a variant of Łukasiewicz logic that uses the whole real line as the set of truth values (instead of the unit interval). They provide axiomatizations for both the finitary and infinitary versions of Lu as extensions of pAb, and they construct a formal translation that establishes an exact correspondence between pAb (and its extensions) and standard Łukasiewicz logic. In particular, pAb is shown to be strongly complete with respect to the four real‑valued models ℝ_{−1}, ℝ_{0}, ℝ_{1}, and ℝ itself.

Beyond these two main lines, the paper demonstrates that the same methodology can be applied to other pointed groups such as the integers ℤ and the rationals ℚ. For each, analogous pointed logics are defined, axiomatized, and proved complete with respect to appropriate algebraic semantics.

Methodologically, the work relies heavily on abstract algebraic logic (AAL) and universal algebra. Ab is identified as a weakly implicative logic, algebraizable in the sense of Blok and Pigozzi, with its equivalent algebraic semantics given by the variety AL of Abelian ℓ‑groups. The authors prove that Ab has no non‑trivial finitary extensions because every non‑trivial Abelian ℓ‑group generates the whole quasivariety, leaving only the inconsistent finitary extension. This justifies the shift to infinitary rules for richer extensions.

The paper’s contributions can be summarized as follows:

1. Infinitary Landscape of Ab – Introduction of a canonical infinitary rule for ℝ, proof of completeness, and demonstration of a continuum of distinct infinitary extensions (2^{2^{ℵ₀}} in total).

2. Pointed Extension (pAb) and Łukasiewicz Unbound Logic – Definition of pAb via a new constant, axiomatization of its finitary and infinitary versions, and a precise translation establishing equivalence with Łukasiewicz logic.

3. Generalization to Other Pointed Groups – Extension of the above framework to groups such as ℤ and ℚ, yielding new logics with analogous completeness results.

4. Algebraic Foundations – Clarification of the relationship between logical extensions, subvarieties, quasivarieties, and generalized quasivarieties, together with a new syntactic proof of the semilinearity of Ab.

Overall, the paper deepens our understanding of how Abelian lattice‑ordered groups can serve as a unifying algebraic semantics for a wide spectrum of many‑valued logics, and it opens avenues for further research into infinitary proof systems, pointed algebraic structures, and their applications in areas like fuzzy logic, continuous model theory, and error‑correcting codes.