“Two Negations are More than One,” p. 455–468 in Graham Priest on Dialetheism and Paraconsistency, edited by Can Başkent, Thomas Macaulay Ferguson, Springer.

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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 \({\mathord\sim}p\) is given by the falsity of \(p\), and the falsity of \({\mathord\sim}p\) is given by the truth of \(p\). Since truth and falsity are only loosely connected, \(p\) and \({\mathord\sim}p\) 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 \({\mathord\sim}p\) in a situation amounts to \(p\) failing in certain other situations. Since those situations can be different from this one, \(p\) and \({\mathord\sim}p\) 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.

This paper is dedicated to my friend and mentor, Professor Graham Priest.


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I’m Greg Restall, and this is my personal website. I teach philosophy and logic as Professor of Philosophy at the University of Melbourne. ¶ Start at the home page of this site—a compendium of recent additions around here—and go from there to learn more about who I am and what I do. ¶ This is my personal site on the web. Nothing here is in any way endorsed by the University of Melbourne.

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