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A Formal Model of the Intermediate Quantifiers “A Few”, “Several” and “A Little”

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Fuzzy Techniques: Theory and Applications (IFSA/NAFIPS 2019 2019)

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Abstract

In this paper, we extend the model of intermediate quantifiers by three new ones, namely “a few, a little” and “several”. We proved some of the fundamental properties of these quantifiers and relations to the other ones. We also demonstrate that they naturally fall in the generalized square of opposition.

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Notes

  1. 1.

    To improve readability of formulas, we quite often write the type only once in the beginning of the formula and then omit it. Alternatively, we write \(A\in { Form}_{\alpha }\) to emphasize that A is a formula of type \(\alpha \) and do not repeat its type again.

  2. 2.

    The special (derived) formula \(\varUpsilon _{oo} A_o\) says that \(A_o\) in every model has a non-zero truth value and \(\hat{\varUpsilon }_{oo} A_o\) that \(A_o\) has a general truth value (i.e., neither false 0, nor true 1).

  3. 3.

    Recall that \( \mathop {\pmb { \& }}\nolimits \) is interpreted by Łukasiewicz conjunction \(\otimes \) and \(\mathop {\pmb {\nabla }}\) is interpreted by Łukasiewicz disjunction \(\oplus \).

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Acknowledgements

The work was supported from ERDF/ESF by the project “Centre for the development of Artificial Intelligence Methods for the Automotive Industry of the region” No. CZ.02.1.01/0.0/0.0/17-049/0008414.

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, NSC IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Vilém Novák & Petra Murinová

Authors
  1. Vilém Novák
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  2. Petra Murinová
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Corresponding author

Correspondence to Petra Murinová .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA

    Ralph Baker Kearfott

  2. Center for Computing Research (CIC), Instituto Politecnico Nacional, Ciudad de Mexico, Mexico

    Ildar Batyrshin

  3. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

    Marek Reformat

  4. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Martine Ceberio

  5. Department of Computer Science, University of Texas at El Paso , El Paso, TX, USA

    Vladik Kreinovich

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Novák, V., Murinová, P. (2019). A Formal Model of the Intermediate Quantifiers “A Few”, “Several” and “A Little”. In: Kearfott, R., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds) Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS 2019 2019. Advances in Intelligent Systems and Computing, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-030-21920-8_39

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