Algebraizing A→

Abelian Logic is a paraconsistent logic discovered independently by Meyer and Slaney [10] and Casari [2]. This logic is also referred to as Abelian Group Logic (AGL) [12] since its set of theorems is sound and complete with respect to the class of Abelian groups. In this paper we investigate the pure implication fragment A→ of Abelian logic. This is an extension of the implication fragment of linear logic, BCI. A Hilbert style axiomatic system for A→ can obtained by adding the axiom A (dubbed the ‘axiom of relativity’ by Meyer and Slaney) to BCI, as follows: B (α → β) → ((γ → α) → (γ → β)) C (α → (β → γ)) → ((β → (α → γ)) I α → α A ((α → β) → β) → α MP α, α → β ⇒ β.
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