Skip to main content
Springer Nature Link
Log in

Pawlak, Belnap and the Magical Number Seven

  • 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:

  • 485 Accesses

Abstract

We are considering the algebraic structure of the Pawlak-Brouwer-Zadeh lattice to distinguish vagueness due to imprecision from ambiguity due to coarseness. We show that a general class of many-valued logics useful for reasoning about data emerges from this context. All these logics can be obtained from a very general seven-valued logic which, interestingly enough, corresponds to a reasoning system developed by Jaina philosophers four centuries BC. In particular, we show how the celebrated Belnap four-valued logic can be obtained from the very general seven-valued logic based on the Pawlak-Brouwer-Zadeh lattice.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Belnap, N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Boston (1976)

    Google Scholar 

  2. Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern uses of multiple-valued logic, vol. 2, pp. 5–37. Springer, Dordrecht (1977). https://doi.org/10.1007/978-94-010-1161-7_2

    Chapter  Google Scholar 

  3. Burch, G.B.: Seven-valued logic in Jain philosophy. Int. Phil. Q. 4(1), 68–93 (1964)

    Article  Google Scholar 

  4. Cattaneo, G.: Generalized rough sets (preclusivity fuzzy-intuitionistic (BZ) lattices). Stud. Logica. 58, 47–77 (1997)

    Article  MathSciNet  Google Scholar 

  5. Cattaneo, G., Algebraic methods for rough approximation spaces by lattice interior–osure operations. In: Mani, A., Cattaneo, G., Düntsch, I. (eds.) Algebraic Methods in General Rough Sets, pp. 13–56. Birkhäuser, Cham (2018)

    Google Scholar 

  6. Cattaneo, G., Ciucci, D.: Algebraic structures for rough sets. Trans. Rough Sets II, LNCS 3135, 208–252 (2004)

    Article  Google Scholar 

  7. Cattaneo, G., Nisticò, G.: Brouwer-Zadeh posets and three-valued Łukasiewicz posets. Fuzzy Sets Syst. 33, 165–190 (1989)

    Article  Google Scholar 

  8. Greco, S., Matarazzo, B., Słowiński, R.: Rough set theory for multicriteria decision analysis. Eur. J. Oper. Res. 129, 1–47 (2001)

    Article  Google Scholar 

  9. Greco, S., Matarazzo, B., Słowiński, R.: The bipolar complemented de morgan brouwer-zadeh distributive lattice as an algebraic structure for the dominance-based rough set approach. Fund. Inf. 115, 25–56 (2012)

    MathSciNet  Google Scholar 

  10. Greco, S., Matarazzo, B., Słowiński, R.: Distinguishing vagueness from ambiguity by means of Pawlak-Brouwer-Zadeh lattices. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 297, pp. 624–632. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31709-5_63

    Chapter  Google Scholar 

  11. Greco, S., Matarazzo, B., Słowiński, R.: Distinguishing vagueness from ambiguity in dominance-based rough set approach by means of a bipolar pawlak-brouwer-zadeh lattice. In: Polkowski, L., et al. (eds.) IJCRS 2017. LNCS (LNAI), vol. 10314, pp. 81–93. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60840-2_6

    Chapter  Google Scholar 

  12. Greco, S., Matarazzo, B., Słowiński, R.: Distinguishing vagueness from ambiguity in rough set approximations. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 26(Suppl. 2), 89–125 (2018)

    Article  MathSciNet  Google Scholar 

  13. Miller, G.A.: The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol. Rev. 63(2), 81 (1956)

    Article  Google Scholar 

  14. Nakayama, Y., Akama, S., Murai, T.: Four-valued tableau calculi for decision logic of rough set. Procedia Comput. Sci. 126, 383–392 (2018)

    Article  Google Scholar 

  15. Pawlak, Z.: Rough Sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)

    Article  Google Scholar 

  16. Pawlak, Z.: Rough Sets. Kluwer, Dordrecht (1991)

    Book  Google Scholar 

  17. Polkowski, L.: Rough Sets. Physica-Verlag, Heidelberg (2002)

    Book  Google Scholar 

  18. Priest, G.: Jaina logic: a contemporary perspective. Hist. Phil. Logic 29(3), 263–278 (2008)

    Article  MathSciNet  Google Scholar 

  19. Yao, Y.: The geometry of three-way decision. Appl. Intell. 51(9), 6298–6325 (2021)

    Article  Google Scholar 

  20. Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46, 39–59 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

Salvatore Greco wishes to acknowledge the support of the Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR) - PRIN 2017, project “Multiple Criteria Decision Analysis and Multiple Criteria Decision Theory”, grant 2017CY2NCA. The research of Roman Słowiński was supported by the SBAD funding from the Polish Ministry of Education and Science. This research also contributes to the PNRR GRInS Project.

Author information

Authors and Affiliations

  1. Department of Economics and Business, University of Catania, Corso Italia, 55, 95129, Catania, Italy

    Salvatore Greco

  2. Institute of Computing Science, Poznań University of Technology, 60-965, Poznań, Poland

    Roman Słowiński

  3. Institute for Systems Research, Polish Academy of Sciences, 01-447, Warsaw, Italy

    Roman Słowiński

Authors
  1. Salvatore Greco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Słowiński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Salvatore Greco .

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

Greco, S., Słowiński, R. (2023). Pawlak, Belnap and the Magical Number Seven. 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_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-50959-9_15

  • 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