Graded Structures of Opposition in Fuzzy Natural Logic

Abstract

The main objective of this paper is devoted to two main parts. First, the paper introduces logical interpretations of classical structures of opposition that are constructed as extensions of the square of opposition. Blanché’s hexagon as well as two cubes of opposition proposed by Morreti and pairs Keynes–Johnson will be introduced. The second part of this paper is dedicated to a graded extension of the Aristotle’s square and Peterson’s square of opposition with intermediate quantifiers. These quantifiers are linguistic expressions such as “most”, “many”, “a few”, and “almost all”, and they correspond to what are often called “fuzzy quantifiers” in the literature. The graded Peterson’s cube of opposition, which describes properties between two graded squares, will be discussed at the end of this paper.

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Notes

  1. 1.

    The quantifier “most” is understood in this paper as “close to all”. Another meaning considered in the literature is “simple most”, which is true if more than 50% of cases are positive. This meaning is standardly used in voting or elections.

  2. 2.

    Recall that, in general, a presupposition is a background belief or something that you assume to be true, in order to continue with what you are saying or thinking.

  3. 3.

    Blanché introduced \({\mathbf {Y}}\) in [37], before completing it with \({\mathbf {U}}\) in [38].

  4. 4.

    It is necessary that the universal quantifiers carry a presupposition of existential import for the entailments to their respective particular forms to hold.

  5. 5.

    Compare, e.g., somebody smoking 10 cigarettes per day with somebody who smokes 50 cigarettes per day.

  6. 6.

    It is necessary that the universal quantifiers carry a presupposition of existential import for the entailments to their respective particular forms to hold.

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Acknowledgements

The paper has been supported from the project “LQ1602 IT4Innovations excellence in science”. The paper relates to the project COST CA17124 Methods of fuzzy modeling in the analysis of forensic digital data (10/2018-12/2020).

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Correspondence to Petra Murinová.

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The work was supported by the MŠMT project NPU II project LQ1602 “IT4Innovations excellence in science”.

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Murinová, P. Graded Structures of Opposition in Fuzzy Natural Logic. Log. Univers. 14, 495–522 (2020). https://doi.org/10.1007/s11787-020-00265-y

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Mathematics Subject Classification

  • Primary 68T30
  • Secondary 03A05
  • 03B05
  • 68T37

Keywords

  • Graded square of opposition
  • Graded hexagon of opposition
  • Graded cube of opposition
  • Fuzzy natural logic
  • Generalized intermediate quantifiers
  • Generalized syllogisms