Skip to main content
SpringerLink
Log in

Free Quantification in Four-Valued and Fuzzy Bilattice-Valued Logics

  • Conference paper
  • First Online:
Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14375))

  • 257 Accesses

Abstract

We introduce a variant of free logic (i.e., a logic admitting terms with nonexistent referents) that accommodates truth-value gluts as well as gaps. Employing a suitable expansion of the Belnap–Dunn four-valued logic, we specify a dual-domain semantics for free logic, in which propositions containing non-denoting terms can be true, false, neither true nor false, or both true and false. In each model, the dual domain semantics separates existing and non-existing objects into two subdomains, making it possible to quantify either over all objects or existing objects only. We also outline a fuzzy variant of the dual-domain semantics, accommodating non-denoting terms in fuzzy contexts that can be partially indeterminate or inconsistent.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
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

References

  1. Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inform. 5, 25–63 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Běhounek, L., Dvořák, A.: Non-denoting terms in fuzzy logic: an initial exploration. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K.T., Krawczak, M. (eds.) IWIFSGN/EUSFLAT -2017. AISC, vol. 641, pp. 148–158. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66830-7_14

    Chapter  Google Scholar 

  3. Běhounek, L., Novák, V.: Towards fuzzy partial logic. In: Proceedings of the IEEE 45th International Symposium on Multiple-Valued Logics (ISMVL 2015), pp. 139–144. IEEE (2015)

    Google Scholar 

  4. Belnap, N.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 5–37. D. Reidel Publishing Company, Dordrecht (1977)

    Chapter  Google Scholar 

  5. Bencivenga, E.: Free logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn., vol. 5, pp. 147–196. Kluwer (2002)

    Google Scholar 

  6. Bílková, M., Frittella, S., Kozhemiachenko, D., Majer, O.: Qualitative reasoning in a two-layered framework. Int. J. Approximate Reasoning 154, 84–108 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bílková, M., Frittella, S., Kozhemiachenko, D., Majer, O.: Two-layered logics for paraconsistent probabilities (2023). https://doi.org/10.48550/arXiv.2303.04565

  8. Carnielli, W., Antunes, H.: An objectual semantics for first-order LFI1 with an application to free logics. In: Haeusler, E.H., Pereira, L.C.P.D., Viana, J.P. (eds.) A Question is More Illuminating than an Answer. A Festschrift for Paulo A.S. Veloso, pp. 58–81. College Publications, London (2020)

    Google Scholar 

  9. Dunn, J.M.: Intuitive semantics for first-degree entailment and “coupled trees.’’. Philos. Stud. 29, 149–168 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Genito, D., Gerla, G.: Connecting bilattice theory with multivalued logic. Logic Log. Philos. 23, 15–45 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Kleene, S.C.: On notation for ordinal numbers. J. Symb. Log. 3, 150–155 (1938)

    Article  MATH  Google Scholar 

  12. Klein, D., Majer, O., Rafiee Rad, S.: Probabilities with gaps and gluts. J. Philos. Log. 50, 1107–1141 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nolt, J.: Free logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Fall 2021 edn. (2021). https://plato.stanford.edu/archives/fall2021/entries/logic-free/

  14. Omori, H., Wansing, H.: 40 years of FDE: an introductory overview. Stud. Logica. 105(6), 1021–1049 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sano, K., Omori, H.: An expansion of first-order Belnap-Dunn logic. Logic J. IGPL 22(3), 458–481 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Turunen, E., Öztürk, M., Tsoukiás, A.: Paraconsistent semantics for Pavelka style fuzzy sentential logic. Fuzzy Sets Syst. 161, 1926–1940 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

L. Běhounek and A. Dvořák were supported by project No. 22-01137S of the Czech Science Foundation.

Author information

Authors and Affiliations

  1. University of Ostrava, IRAFM, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Libor Běhounek, Martina Daňková & Antonín Dvořák

Authors
  1. Libor Běhounek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martina Daňková
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonín Dvořák
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Libor Běhounek .

Editor information

Editors and Affiliations

  1. Japan Advanced Inst of Science and Technology, Nomi, Ishikawa, Japan

    Van-Nam Huynh

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Bac Le

  3. Osaka Metropolitan University, Sakai, Osaka, Japan

    Katsuhiro Honda

  4. Osaka University, Toyonaka, Osaka, Japan

    Masahiro Inuiguchi

  5. Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan

    Youji Kohda

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

Běhounek, L., Daňková, M., Dvořák, A. (2023). Free Quantification in Four-Valued and Fuzzy Bilattice-Valued Logics. In: Huynh, VN., Le, B., Honda, K., Inuiguchi, M., Kohda, Y. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2023. Lecture Notes in Computer Science(), vol 14375. Springer, Cham. https://doi.org/10.1007/978-3-031-46775-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-46775-2_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-46774-5

  • Online ISBN: 978-3-031-46775-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation