Abstract
We study the expansion of the four-valued Belnap–Dunn logic by a pair of constants 0 and 1 which express respectively the weakest inconsistent proposition and the strongest complete proposition. We then further expand this logic by the intuitionistic implication and use this expansion to introduce a logic which is a conservative extension of both classical and intuitionistic logic. The key idea behind this way of combining classical and intuitionistic logic is that classical negation is nothing but the De Morgan negation of the Belnap–Dunn logic restricted to classical contexts, i.e. contexts in which completeness and consistency is assumed. Finally, sequent calculi are provided for the logics introduced in the paper.
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