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Privations, Negations and the Square: Basic Elements of a Logic of Privations

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Book cover Around and Beyond the Square of Opposition

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

I try to explain the difference between three kinds of negation: external negation, negation of the predicate and privation. Further I use polygons of opposition as heuristic devices to show that a logic which contains all three mentioned kinds of negation must be a fragment of a Łukasiewicz-four-valued predicate logic. I show, further, that, this analysis can be elaborated so as to comprise additional kinds of privation. This would increase the truth-values in question and bring fragments of (more generally speaking) Łukasiewicz-n-valued predicate logics into the scene.

For discussions concerning privation and amendments in the first drafts of this paper, I would like to thank Prof. Dr. Hans Burkhardt, Georgios Karageorgoudis (both Munich) and, last but not least, two anonymous referees.

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Notes

  1. 1.

    Some contemporary authors like Blau [5, p. 49] and Horn [10, pp. 122–132], call the negation of the predicate: ‘presupposing’ negation, because, unlike the external negation it presupposes the existence of the entity expressed in the subject. Especially Blau (loc. cit.) calls it ‘strong negation’ because it entails the external negation but is not entailed by it. Since in Blau’s three-valued logic there are exactly two negations: (a) a formal construction, which formalizes the negation of the predicate/by transposition, and (b) a weak negation which formalizes the external negation of the natural language, the former is the only strong negation in Blau’s three-valued logic. But, if I am allowed to anticipate one of my results, the negation of the predicate is not the strongest kind of negation in the logic of privation.

  2. 2.

    Gottwald [8, pp. 55–57] gives a brief exposition of the construction of the Łukasiewicz-four-valued predicate logic; he [8, pp. 45–47] also gives the axioms and the deduction rules of the Łukasiewicz-four-valued propositional logic. For a brief but full account of the semantics of the Łukasiewicz-four-valued propositional logic cf. [13, pp. 29–32].

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Authors and Affiliations

  1. University of Erfurt, Hauspostfach 64, PSF 900221, 99105, Erfurt, Germany

    Stamatios Gerogiorgakis

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  1. Stamatios Gerogiorgakis
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Correspondence to Stamatios Gerogiorgakis .

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  1. , Institute of Philosophy, Federal University of Rio de Janeiro, Largo São Francisco de Paula 1, Rio de Janeiro, RJ 20051-0, Brazil

    Jean-Yves Béziau

  2. , Institute of Philosophy, University of Bern, Länggassstrasse 49a, Bern, 3000, Switzerland

    Dale Jacquette

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Gerogiorgakis, S. (2012). Privations, Negations and the Square: Basic Elements of a Logic of Privations. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_16

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