Normality operators and classical recapture in many-valued logic

Logic Journal of the IGPL 28 (5):657-683 (2020)
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In this paper, we use a ‘normality operator’ in order to generate logics of formal inconsistency and logics of formal undeterminedness from any subclassical many-valued logic that enjoys a truth-functional semantics. Normality operators express, in any many-valued logic, that a given formula has a classical truth value. In the first part of the paper we provide some setup and focus on many-valued logics that satisfy some of the three properties, namely subclassicality and two properties that we call fixed-point negation property and conservativeness. In the second part of the paper, we introduce normality operators and explore their formal behaviour. In the third and final part of the paper, we establish a number of classical recapture results for systems of formal inconsistency and formal undeterminedness that satisfy some or all the properties above. These are the main formal results of the paper. Also, we illustrate concrete cases of recapture by discussing the logics $\mathsf{K}^{\circledast }_{3}$, $\mathsf{LP}^{\circledast }$, $\mathsf{K}^{w\circledast }_{3}$, $\mathsf{PWK}^{\circledast }$ and $\mathsf{E_{fde}}^{\circledast }$, that are in turn extensions of $\mathsf{{K}_{3}}$, $\mathsf{LP}$, $\mathsf{K}^{w}_{3}$, $\mathsf{PWK}$ and $\mathsf{E_{fde}}$, respectively.



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Massimiliano Carrara
University of Padua

References found in this work

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A useful four-valued logic.N. D. Belnap - 1977 - In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel.

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