Abelian Logic is a paraconsistent logic discovered independently by Meyer and Slaney  and Casari . This logic is also referred to as Abelian Group Logic (AGL)  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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