Abstract It is well-known that the paraconsistent logic LP is not functionally complete: the set of LP propositional connectives is not sufficient to express all possible LP truth functions. In this note, we revisit this simple result, from a fresh philosophical and technical perspective. We investigate whether LP is ‘expressively complete’ after all—when the key definitions and proofs are re-situated in a non-classical metatheory. This is done by using a relational semantics rather than a functional semantics, and without classical negation in the metalanguage. We show that any connective with a truth table can be expressed in LP (suitably understood). More generally, we arrive at a more abstract view about the definability of connectives in an important family of non-classical logics.
Omori et al. (2025) studied this question.