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A Diagrammatic Bridge Between Standard and Non-standard Logics: The Numerical Segment

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Visual Reasoning with Diagrams

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

Abstract

The system of ‘Distinctive’ Predicate Calculus (DPC) used here is part of a more general enterprise to bring logic closer to natural language and natural intuitions. This system works on the central concept of “middle” or “intermediate” and finds a synthesis in the diagram of the ‘Numerical Segment’ (NS). We intend to show how this diagrammatic tool and its logic can be theoretically used as an interesting link up with non-standard logics (fuzzy, polyvalent, paraconsistent) and applied to new form of diagrammatic reasoning.

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Notes

  1. 1.

    Blanché tells the story of a “mortal combat” between the partial and the particular in the evolution of Aristotle’s thinking [5, p. 39, note 27]. A forerunner of Blanché was Sesmat [21].

  2. 2.

    See Lenzen [16]. J.H. Lambert writes: “…we have these three cases: 1. Every b is a, 2. Only some b is a, 3. No b is a. Of the three propositions only one can be true at any given time” [15, pp. 76–77] (translation in English by the author). The falsity of one does not allow one to conclude to the truth of either other.

  3. 3.

    All sets in question are distinct from the universe of objects u and from the null set [3, pp. 53–55]. In the cases 5 and 7, u coincides with the union of the two sets b and a, the difference being that, in case 7, b is the complement of a in u. In the other five cases, u is represented by the rectangular frame. One notes that such a mutually exclusive distribution over the proposition types is not possible in the classic square of opposition. The 7 types of propositions can already be found in De Morgan [8, pp. 65–67] that called them complex propositions and symbolized by D−D, −D′−P−C′−C, −C (respectively for our cases from 1 to 7). In the same work, De Morgan makes use of segments (or letters arranged in rows) to illustrate propositions [8, pp. 61, 79, 81–82, etc.]. After De Morgan, Keynes and other logicians identified the 7 cases. None of them (De Morgan included) used the “partial” quantificator.

  4. 4.

    The same goes for the next Figs. 2 and 3, while in Figs. 4, 5 and 6, the colors have the function of underscoring.

  5. 5.

    On this basis we have constructed the Distinctive Triangular and Hexagonal Calculi [7, pp. 242–246]. The known laws of immediate inference have now been extended with the equivalence of the two Y-obverses: Yba=Yba′ (where a′ stands for the complement of a in u). The Russian logician N.A. Vasil’ev developed a truly tripartite logic like the one just sketched. In 1910, he called it the Logic of Notions. Besides the proposition types that he called ‘general’ (that is, our ‘universal’), he posited the ‘accidental’ type, and the ‘judgments’ could be not only affirmative or (internally) negative, but also indifferent. See Seuren [22] and Suchon [23] about triangular schemata, Blanché [4] and Beziau [2] for hexagonal schemata developments.

  6. 6.

    The acronym “alo” is taken from a talk of J.-Y. Béziau (at the Congress “Logic Now and Then”, Brussels, November 5–7, 2008).

  7. 7.

    The square does not exhaust the possible combinations. For example, it does not contain those with discontinuous quantifiers/intervals, or those that lack extremes, such as have various intermediate quantifiers between them.

  8. 8.

    Translation in English by the author.

  9. 9.

    N/M is an intermediate numerical quantifier.

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Authors and Affiliations

  1. V. della Libertà 5, 47042, Cesenatico, FC, Italy

    Ferdinando Cavaliere

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  1. Ferdinando Cavaliere
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Correspondence to Ferdinando Cavaliere .

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  1. IRIST, Université de Strasbourg, Rue de l'Université 7, Strasbourg, 67000, France

    Amirouche Moktefi

  2. Department of Philosophy, Yale University, Hillhouse Avenue 38, New Haven, 06511, Connecticut, USA

    Sun-Joo Shin

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Cavaliere, F. (2013). A Diagrammatic Bridge Between Standard and Non-standard Logics: The Numerical Segment. In: Moktefi, A., Shin, SJ. (eds) Visual Reasoning with Diagrams. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0600-8_5

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