The Boolean logic of subsets, usually presented as `propositional logic,’ is considered as being “classical” while intuitionistic logic and the many sublogics and off-shoots are “non-classical.” But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.