Characterizing negation to face dialetheism

It has been said that, when some paraconsistent logicians supporting dialetheism assert: "For some sentence α, both α and not-α are true", therefore claiming that the Law of Non-Contradiction (LNC) fails, we should wonder what "true" and "not" mean here. After surveying two classical paraconsistent approaches to negation (provided by da Costa's positive-plus systems and Graham Priest's Logic of Paradox), I describe a negation with the following features: (1) its definition does not make reference to the controversial concept truth; (2) it has strong pre-theoretical appeal and motivation, because it performs an indispensable expressive function in language and communication; and (3) it is accepted by dialetheists, too, since it is based on a very deep metaphysical intuition they also show to fully share: this intuition I call the one of material exclusion. If my characterization is sufficient to confer a determinate meaning to the negation in question, we can conveniently formulate via this negation a version of the LNC which I take, therefore, to be indisputable also from the dialetheist's point of view. Such a result, however, does not constitute a quick and easy success against supporters of true contradictions. It may simply show that the versions of the LNC dialetheists most convincingly attack are those that were not to be defended, and that supporters of consistency have been historically confused in assimilating them to the indisputable one. Quine's famous argument, that to change the logic is to change the subject, may be right to this extent: classical negation and non-classical negations have different meanings. But the substantial issue that Quine never addressed is why we should suppose that the meaning of the vernacular negation is classical.