Summary: It is well known that there is a categorical equivalence between lattice-ordered abelian groups (or \(\ell\)-groups) and conical BCK-algebras. The aim of this paper is to study this equivalence from the perspective of logic, in particular, to study the relationship between two deductive systems: conical logic \({\mathcal C}o\) and a logic of \(\ell\)-groups, \({\mathcal B}al^\circ\). In [Arch. Math. Logic 43, No. 2, 141–158 (2004; Zbl 1060.03086)], A. Galli, R. A. Lewin and M. Sagastume introduce a system \({\mathcal B}al\) which models the logic of balance of opposing forces with a single distinguished truth value that represents equilibrium. Its equivalent algebraic semantics BAL (via Blok-Pigozzi’s construction) is definitionally equivalent to the variety of \(\ell\)-groups. In this paper we define the system \({\mathcal B}al^\circ\) which is equivalent to \({\mathcal B}al\) and whose Lindenbaum-Tarski algebra is an \(\ell\)-group. On the other hand, we define the conic logic \({\mathcal C}o\) and we prove that it can be naturally merged in the system \({\mathcal B}al^\circ\). Also, we prove that every formula of \({\mathcal B}al^\circ\) has a normal form that depends on translations of formulas of \({\mathcal C}o\).