Francesco Berto’s ‘not’s

Whatever does Francesco Berto mean when he uses the word ‘not’, its allies and cognates in his paper ‘A Modality Called ‘Negation’’ (Mind, forthcoming)?  The article is a most intricate and thorough mathematical systematisation of a variety of ‘negations’ that have appeared in the philosophical and logical literature over the last century or so.  But Berto does not tell us which one he is using, despite the fact the (unquoted) word ‘not’ and its associates are thick on every page.  Either he means what we all mean or he means nothing, and neither alternative sits at all happily with his project.  The matter brings up a very basic question about Logic: is it concerned to discriminate correct uses of 'not' from incorrect ones, or is its object just to set out formal systems embracing different uses without any discrimination?