David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Applied Non-Classical Logics 13 (1):9-19 (2003)
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.
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References found in this work BETA
Laurence Horn (1989). A Natural History of Negation. University of Chicago Press.
Jean Van Heijenoort (ed.) (1967). From Frege to Gödel. Cambridge, Harvard University Press.
Jaakko Hintikka (1973). Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Oxford,Clarendon Press.
Nicholas Rescher (1969). Many-Valued Logic. New York, Mcgraw-Hill.
F. P. Ramsey (1927). Facts and Propositions. Proceedings of the Aristotelian Society 7 (1):153-170.
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