Many valued paraconsistent logic

In contrast to most logics, in paraconsistent logic it is not true that everything followed from a contradiction. The semantics for one of the best known paraconsistent logics, LP, permits sentences to be both true and false; but at the same time, the semantic characterization of the logical particles is classical. We define the notion of “molecular logic”, of which all finite valued variants of LP are a type. Generally paraconsistent logics do not contain extensional conditionals. Molecular logics of n values may be conservatively extended to standard many valued logics of 2ⁿ-1 values, in which it is easy to define extensional conditionals with the usual detachment rules. The extension, while paraconsistent relative to a negation satisfying standard conditions on the original n values, is not paraconsistent relative to a negation satisfying standard conditions on the 2ⁿ-1 values of the extension. We conclude that the logic LP and its many valued generalizations are paraconsistent because of expressive incompleteness