In this paper, the correspondence between derivability (syntactic consequences obtained from ⊢) and convertibility in rewriting , the so-called logicality, is studied in a generic way (i.e. logic-independent). This is achieved by giving simple conditions to characterise logics where (bidirectional) rewriting can be applied. These conditions are based on a property defined on proof trees, that we call semi-commutation. Then, we show that the convertibility relation obtained via semi-commutation is equivalent to the inference relation ⊢ of the logic under consideration.