 Finally, it is noteworthy to observe that algebraic fibring does not intrinsically forbid the emergence of interactions between connectives of the logics being combined. In fact, the notion of morphism in the category of logic systems being employed is the key to creating or blocking interactions. In order to exemplify this assertion, consider the case of the failure to recover classical logic from its \(\{\neg\}\)-fragment and \(\{\vee\}\)-fragment by algebraic fibring. From a proof-theoretic perspective, the key reason for this failure is that the rule   \[ \tag{* } \frac{\Gamma, \varphi\vdash \psi \;\quad\; \Delta, \neg\varphi\vdash \psi} {\Gamma, \Delta\vdash \psi} \]   of the logic of classical negation is not preserved by algebraic fibring in categories of logic systems having translations between logics as morphisms (recall  subsection 4.4),  such as the category of Hilbert calculi or consequence relations.