The Geometry of Negation

Abstract
There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power.
Keywords Negation
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DOI 10.3166/jancl.13.9-19
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References found in this work BETA
A Natural History of Negation.Laurence Horn - 1989 - University of Chicago Press.
From Frege to Gödel.Jean Van Heijenoort (ed.) - 1967 - Cambridge: Harvard University Press.
Many-Valued Logic.Nicholas Rescher - 1969 - New York: Mcgraw-Hill.
Facts and Propositions.F. P. Ramsey - 1927 - Proceedings of the Aristotelian Society 7 (1):153-170.

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