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Modern Versus Classical Structures of Opposition: A Discussion

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Abstract

The aim of this work is to revisit the proposal made by Dag Westerståhl a decade ago when he provided a modern reading of the traditional square of opposition and of related structures. We propose a formalization of this modern view and contrast it with the classical one. We discuss what may be a modern hexagon of opposition and a modern cube, and show their interest in particular for relating quantitative expressions.

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Notes

  1. We do not consider other forms of parameterization in this article.

  2. If we call \(\alpha \), \(\epsilon \), o, \(\iota \), the numbers in [0, 1] that are the values of the expressions associated with vertices \({\textbf {A}}\), \({\textbf {E}}\), \({\textbf {O}}\), \({\textbf {I}}\) respectively, then contradictions is ensured by the relations \(o= 1 -\alpha \) and \(\iota = 1- \epsilon \), contrariety by \(\alpha + \epsilon \le 1\), sub-contrariety by \(\iota + o \ge 1\), and implications by \(\alpha \le \iota \) and \(\epsilon \le o\). This extends the logical relations in the classical square of opposition if we choose the connectives of Łukasiewicz logic. Namely, \(o = \lnot \alpha = 1- \alpha \), \(\iota = \lnot \epsilon = 1- \epsilon \), \(\alpha \wedge \epsilon = \max (0, \alpha + \epsilon -1) =0\), \(\iota \vee o= \min (1,\iota + o) = 1\), \(\alpha \rightarrow \iota = \min (1, 1 - \alpha + \iota ) = 1\) and \(\epsilon \rightarrow o = \min (1, 1 - \epsilon + o) = 1\) [5].

  3. This is one of the possible expressions of a L-valued Sugeno integral, which is an important family of aggregation functions [19, 21, 39].

  4. Indeed the integral can be equivalently written \(S(\mu ,x)= \max _{T\in 2^\mathcal {X}} \min (\mu (T), x_{\inf }(T))\).

  5. Nelson, a post Kantian philosopher, indeed in his 1921 lectures [29], seems to be the first to use hexagonal diagrams for discussing opposition between abstract notions, in some places. In his hexagons, the opposite vertices appear to be contradictories. In the introduction of his translation of Nelson lectures, Leal has offered an abstract rendering of the diagrams, which appear to be topologically equivalent to Blanché’s hexagon [2]. Yet, in Leal version of the diagram, it does not seem that the vertices corresponding to A and E should be contraries.

  6. For modern hexagons, we continue to use the same names for the vertices, but we use calligraphic letters instead of standard capitals, since we are no longer expecting the same logical relations between the vertices as in the classical hexagon. In fact, as we shall see these letters are now just a matter of convenience for naming the vertices.

  7. The contradictory, contrariety, sub-contrariety, and implication relations between the graded expressions associated to the vertices of the hexagon are defined with the same multiple-valued logic connectives as the ones used for the graded square, recalled in footnote 2.

  8. However, an isomorphic cube already appeared in the middle of last century in a discussion of syllogisms [33], see [16] for details.

  9. What we call here “JK cube” was in fact presented as an octagon by Johnson [23] and Keynes [24]. It has been rediscovered several times, e.g., [9, 40].

  10. This latter point is a matter of convenience and readibility, and not a distinctive feature with respect to Moretti’s cube. Indeed we may choose to use the front and back facets of the JK cube as the diagonal plans of a cube, just leading to a new isomorphic cube the diagonals of which relate contradictories.

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Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse (IRIT), Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

  2. Equipe de Recherche en Ingénierie des Connaissances (ERIC), Université Claude Bernard Lyon 1, Lyon, France

    Agnès Rico

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Correspondence to Henri Prade.

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Dubois, D., Prade, H. & Rico, A. Modern Versus Classical Structures of Opposition: A Discussion. Log. Univers. (2024). https://doi.org/10.1007/s11787-024-00347-1

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