Two negations are more than one

In models for paraconsistent logics, the semantic values of sentences and their negations are less tightly connected than in classical logic. In ‘American Plan’ logics for negation, truth and falsity are, to some degree, independent. The truth of ~ p is given by the falsity of p, and the falsity of ~ p is given by the truth of p. Since truth and falsity are only loosely connected, p and ~ p can both hold, or both fail to hold. In ‘Australian Plan’ logics for negation, negation is treated rather like a modal operator, where the truth of ~ in a situation amounts to p failing in certain other situations. Since those situations can be different from this one, p and ~ p might both hold here, or might both fail here. So much is well known in the semantics for paraconsistent logics, and for first-degree entailment and logics like it, it is relatively easy to translate between the American Plan and the Australian Plan. It seems that the choice between them seems to be a matter of taste, or of preference for one kind of semantic treatment or another. This paper explores some of the differences between the American Plan and the Australian Plan by exploring the tools they have for modelling a language in which we have two negations.