 From the technical point of view there is an important problem concerning composition of logics known as collapsing problem, firstly identified in Gabbay 1996b, and later formalized in del Cerro and Herzig 1996. In the latter paper, it is shown that, by freely combining classical propositional logic and intuitionistic propositional logic at the semantical level (technically: by constructing their unconstrained fibring in the category of interpretation systems, recall  subsection 4.3)  the resulting logic collapses to classical logic. More precisely, the resulting logic will consist of two twin copies of classical propositional logic, having two negations, two implications and so on. Clearly, the respective copies of each connectives will be proved to be inter-derivable in the resulting logic: \(\neg_1 \varphi\) will be equivalent to \(\neg_2\varphi\), \((\varphi\to_1 \psi)\) will be equivalent to \((\varphi\to_2 \psi)\) and so on. The collapse only happens when considering the algebraic fibring at the semantical level: in Caleiro and Ramos 2007 was shown that the collapse does not occur when computing the algebraic fibring of the respective Hilbert calculi.