In this talk I will introduce the ongoing debate on grounding negation. Two main-views will be put forward, the "American Plan", which considers negation as a switch between truth and falsity ( where no restraints are made whether this are exclusive or exhaustive ) and the "Australian Plan", which argues in favor of an inferentialist conception of negation grounded in the idea of compatibility. We will briefly discuss the upshots and disadvantages of this approaches, coming to the conclusion that both the difficulty when dealing with paraconsistent negation within mathematical expressive theories, such as generalized algebra-valued models for Set theory. Therefore, we present three different operators defined on linear algebras which we will evaluate for the negation-like properties demanded by the Australian and American Plan. We will show that there exists a delicate trade-off between the regularity properties of a negation and the expressive power of the algebra-valued models, that neither plans are sensitive too. In particular we show that the amount of non-elementary equivalent models, obtained by adjoining a respective operator, is inverse to the to the amount of properties that a negation manages to satisfy! We conclude by introducing our own account of negation based on an algebraic perspective: the minimalist account of negation. We will argue that this account is broad enough to accept a wide amount of non-classical negations, but at the same allows us to distinguish classical and intutionistic-like negations. Finally, we argue that a consequence of our minimalist account is that paraconsistency is not ascribable to a negation, but rather to a whole logical calculus.
[Recommended lectures: A modality called negation, Berto (2015) and There is more to modality then negation, Omori and De (2018) ]