Nearly every normal modal logic is paranormal

Logique Et Analyse 48 (189-192):279-300 (2005)
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Abstract

An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences hold independently of the nature of the statements involved. A negation-inconsistent logic is a logic having at least one model that satisfies both some statement and its negation. A negation-incomplete logic has at least one model according to which neither some statement nor its negation are satisfied. Paraconsistent logics are negation-inconsistent yet non-overcomplete; paracomplete logics are negation-incomplete yet non-overcomplete. A paranormal logic is simply a logic that is both paraconsistent and paracomplete. Despite being perfectly consistent and complete with respect to classical negation, nearly every normal modal logic, in its ordinary language and interpretation, admits to some latent paranormality: It is paracomplete with respect to a negation defined as an impossibility operator, and paraconsistent with respect to a negation defined as non-necessity. In fact, as it will be shown here, even in languages without a primitive classical negation, normal modal logics can often be alternatively characterized directly by way of their paranormal negations and related operators. So, instead of talking about ‘necessity’, ‘possibility’, and so on, modal logics could be seen just as devices tailored for the study of (modal) negation. This paper shows how and to what extent this alternative characterization of modal logics can be realized.

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Joao Marcos
Universidade Federal do Rio Grande do Norte

References found in this work

Modal Logic: An Introduction.Brian F. Chellas - 1980 - New York: Cambridge University Press.
A Natural History of Negation.Laurence R. Horn - 1989 - University of Chicago Press.
An introduction to modal logic.G. E. Hughes - 1968 - London,: Methuen. Edited by M. J. Cresswell.
A Natural History of Negation.Laurence R. Horn - 1989 - Philosophy and Rhetoric 24 (2):164-168.
An Introduction to Modal Logic.George Edward Hughes & M. J. Cresswell - 1968 - London, England: Methuen. Edited by M. J. Cresswell.

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