Skip to main content
SpringerLink
Log in

Hexagons of Opposition in Linguistic Three-Way Decisions

  • Conference paper
  • First Online:
Rough Sets (IJCRS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14481))

Included in the following conference series:

  • 165 Accesses

Abstract

In three-way decision theory, three disjoint sets covering a given universe, are determined: the positive, negative, and boundary regions. They correspond to three types of decisions on their objects: acceptance, rejection, and abstention or non-commitment. A linguistic approach for identifying the three regions relies on specific evaluative linguistic expressions, such as “very big”, “roughly small”, “not small”, “medium”, and so forth.

In this article, we construct hexagons of opposition using the regions generated by different evaluative linguistic expressions. Then, we explore the logical relations between the vertices of different hexagons.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The function such that \(X \mapsto \dfrac{|X|}{|U|}\) for each \(X \subseteq U\) is understood as a normalized fuzzy measure [17].

  2. 2.

    This is a fuzzy quantifier \(\mathcal {S}_{many}\) assigning a value of [0,1] to each pair of fuzzy sets. In [6], it has been proven that \(\mathcal {S}_{many}(A,B)=\lnot Sm\left( \dfrac{|A \cap B|}{|A|}\right) \), when A and B are classical set of the given universe.

  3. 3.

    By a tri-partition of U we mean a collection of three mutually disjoint subsets covering U. So, notice that the limit cases where one or two sets are empty are included.

References

  1. Abbruzzese, R., Gaeta, A., Loia, V., Lomasto, L., Orciuoli, F.: Detecting influential news in online communities: an approach based on hexagons of opposition generated by three-way decisions and probabilistic rough sets. Inf. Sci. 578, 364–377 (2021)

    Article  MathSciNet  Google Scholar 

  2. Béziau, J.Y.: The power of the hexagon. Log. Univers. 6(1–2), 1–43 (2012)

    Article  MathSciNet  Google Scholar 

  3. Boffa, S., Ciucci, D.: Three-way decisions with evaluative linguistic expressions. arXiv preprint arXiv:2307.11766 (2023)

  4. Boffa, S., Ciucci, D., Murinová, P.: Comparing hexagons of opposition in probabilistic rough set theory. In: Ciucci, D., et al. (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. CCIS, vol. 1601, pp. 622–633. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-08971-8_51

  5. Boffa, S., Murinová, P., Novák, V.: Graded polygons of opposition in fuzzy formal concept analysis. Int. J. Approx. Reason. 132, 128–153 (2021)

    Article  MathSciNet  Google Scholar 

  6. Boffa, S., Murinová, P., Novák, V.: A proposal to extend relational concept analysis with fuzzy scaling quantifiers. Knowl.-Based Syst. 231, 107452 (2021)

    Article  Google Scholar 

  7. Ciucci, D., Dubois, D., Prade, H.: Oppositions in rough set theory. In: Li, T., et al. (eds.) RSKT 2012. LNCS (LNAI), vol. 7414, pp. 504–513. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31900-6_62

    Chapter  Google Scholar 

  8. Dubois, D., Prade, H.: From blanché’s hexagonal organization of concepts to formal concept analysis and possibility theory. Log. Univers. 6, 149–169 (2012)

    Article  MathSciNet  Google Scholar 

  9. Gaeta, A.: Evaluation of emotional dynamics in social media conversations: an approach based on structures of opposition and set-theoretic measures. Soft Comput. 27(15), 10893–10903 (2023)

    Google Scholar 

  10. Moral, A., Castiello, C., Magdalena, L., Mencar, C.: Explainable Fuzzy Systems. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-71098-9

  11. Moretti, A.: Why the logical hexagon? Log. Univers. 6(1), 69–107 (2012)

    Article  MathSciNet  Google Scholar 

  12. Novák, V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets Syst. 159(22), 2939–2969 (2008)

    Article  MathSciNet  Google Scholar 

  13. Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets Syst. 159(10), 1229–1246 (2008)

    Article  MathSciNet  Google Scholar 

  14. Novák, V., Perfilieva, I., Dvorak, A.: Insight into Fuzzy Modeling. John Wiley & Sons, Hoboken (2016)

    Google Scholar 

  15. Parsons, T.: The traditional square of opposition (1997)

    Google Scholar 

  16. Pfeifer, N., Sanfilippo, G.: Probabilistic squares and hexagons of opposition under coherence. Int. J. Approx. Reason. 88, 282–294 (2017)

    Article  MathSciNet  Google Scholar 

  17. Sugeno, M.: Theory of fuzzy integrals and its applications. Doctoral Thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  18. Yao, Y.: Three-way decision: an interpretation of rules in rough set theory. In: Wen, P., Li, Y., Polkowski, L., Yao, Y., Tsumoto, S., Wang, G. (eds.) RSKT 2009. LNCS (LNAI), vol. 5589, pp. 642–649. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02962-2_81

    Chapter  Google Scholar 

  19. Yao, Y.: Three-way decisions with probabilistic rough sets. Inf. Sci. 180(3), 341–353 (2010)

    Article  MathSciNet  Google Scholar 

  20. Yao, Y.: Three-way granular computing, rough sets, and formal concept analysis. Int. J. Approx. Reason. 116, 106–125 (2020)

    Article  MathSciNet  Google Scholar 

  21. Yao, Y., et al.: An outline of a theory of three-way decisions. In: RSCTC, vol. 7413, pp. 1–17 (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Milano-Bicocca, DISCo, Viale Sarca 336, 20126, Milan, Italy

    Stefania Boffa & Davide Ciucci

Authors
  1. Stefania Boffa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide Ciucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefania Boffa .

Editor information

Editors and Affiliations

  1. IRCCS Istituto Ortopedico Galeazzi, Milano, Italy

    Andrea Campagner

  2. Ghent University, Ghent, Belgium

    Oliver Urs Lenz

  3. Chongqing University of Posts and Telecommunications, Chongqing, China

    Shuyin Xia

  4. University of Warsaw, Warsaw, Poland

    Dominik Ślęzak

  5. AGH University of Science and Technology, Kraków, Poland

    Jarosław Wąs

  6. University of Regina, Regina, SK, Canada

    JingTao Yao

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Boffa, S., Ciucci, D. (2023). Hexagons of Opposition in Linguistic Three-Way Decisions. In: Campagner, A., Urs Lenz, O., Xia, S., Ślęzak, D., Wąs, J., Yao, J. (eds) Rough Sets. IJCRS 2023. Lecture Notes in Computer Science(), vol 14481. Springer, Cham. https://doi.org/10.1007/978-3-031-50959-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-50959-9_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-50958-2

  • Online ISBN: 978-3-031-50959-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation