Metatheory and dialetheism

Given a formal language, a metalanguage is a language which can express — amongst other things — statements about it and its properties. And a metatheory is a theory couched in that language concerning how some of those notions behave. Two such notions that have been of particular interest to modern logicians — for obvious reasons — are truth and validity. These notions are, however, notoriously deeply entangled in paradox. A standard move is to take the metalanguage to be distinct from the language in question, and so avoid the paradoxes. One of the attractions of a dialetheic approach to the paradoxes of self-reference is that this move may be avoided. One may have a language with the expressive power to talk about — among other things — itself, and a theory in that language about how notions such as truth and validity for that language behave. The contradictions delivered by these notions are forthcoming, but they are quarantined by the use of a paraconsistent logic. The point of this paper is to discuss this project, the extent to which it has been successful, and the places where issues still remain.