Skip to main content
SpringerLink
Log in

Conditioning for Boolean Subsets, Indicator Functions and Fuzzy Subsets

  • Chapter
  • First Online:
On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 336))

  • 805 Accesses

Abstract

This chapter deals with measure-free conditioning. It starts with the mean value based definition of conditional fuzzy subsets which again gives a fuzzy subset. Applying this general construction to indicator functions, it is proved that these conditionals form an MV-algebra and that this is isomorphic to the already known MV-algebra of the interval based conditional Boolean subsets. In the following, the problem of iteration is completely solved with the result that there are exactly two types of iteration, called the blurred resp. the sharper one, which remain in the corresponding MV-algebras. Moreover, the general concept of conditional operators plays a significant role. Finally, the problem of extending an uncertainty measure is discussed.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. Birkhoff, G.: Lattice Theory. Providence, RI (1960)

    Google Scholar 

  2. Calabrese, P.: An algebraic synthesis of the foundations of logic and probability. Inf. Sci. 42, 187–237 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)

    Article  MATH  Google Scholar 

  4. Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  5. Coletti, G., Scozzafava, R.: From conditional events to conditional measures: a new axiomatic approach. Ann. Math. Artif. Intell. 32, 373–392 (2001)

    Article  MathSciNet  Google Scholar 

  6. Di Nola, A., Scozzafava, R.: Partial algebraic conditional spaces. Int. J. Uncertainty, Fuzziness Knowl.-Based Syst. 12, 781–789 (2004)

    Article  MATH  Google Scholar 

  7. Dubois, D., Prade, H.: Conditioning, non-monotonic logic and non-standard uncertainty models. In: Goodman, I.R., Gupta, M.M., Nguyen, H.T., Rogers, G.S. (eds.) Conditional Logic in Expert Systems, pp. 115–158. North-Holland, Amsterdam (1991)

    Google Scholar 

  8. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics. Studies in Logic and the Foundations of Mathematics 151, Elsevier (2007)

    Google Scholar 

  9. Goodman, I.R., Nguyen, H.T., Walker, E.A.: Conditional Inference and Logic for Intelligent Systems—A Theory of Measure-free Conditioning. North-Holland, Amsterdam (1991)

    Google Scholar 

  10. Höhle, U., Weber, S.: Uncertainty measures, realizations and entropies. In: Goutsias, J., Mahler, R.P.S., Nguyen, H.T. (eds.) Random Sets: Theory and Applications, pp. 259–295. Springer, Berlin (1997)

    Chapter  Google Scholar 

  11. Höhle, U., Weber, S.: On conditioning operators. In: Höhle, U., Rodabaugh, S. (eds.) Mathematics of Fuzzy Sets—Logic, Topology and Measure Theory, pp. 653–673. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  12. Nguyen, H.T.: On representation and combinability of uncertainty. In: Abstracts of the 2nd IFSA Congress, Tokyo (1987)

    Google Scholar 

  13. Pfanzagl, J.: Theory of Measurement, 2nd edn. Physica, Würzburg, Wien (1971)

    Book  MATH  Google Scholar 

  14. Rie\(\check{c}\)an, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, vol. 2, pp. 869–909. North-Holland, Amsterdam (2002)

    Google Scholar 

  15. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.E.: A characterization of interval-valued residuated lattices. Int. J. Approximate Reasoning 49, 478–487 (2008)

    Article  MATH  Google Scholar 

  16. Weber, S.: On conditional measures and events. In: Abstracts of the 9th Linz Seminar on Fuzzy Set Theory (1987)

    Google Scholar 

  17. Weber, S.: Conditioning on MV-algebras and additive measures. I. Fuzzy Sets Syst. 92, 241–250 (1997)

    Article  MATH  Google Scholar 

  18. Weber, S.: Conditioning on MV-algebras and additive measures, further results. In: Dubois, D., Prade, H., Klement, E.P. (eds.) Fuzzy Sets, Logics and Reasoning about Knowledge, pp. 175–199. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  19. Weber, S.: Uncertainty measures—problems concerning additivity. Fuzzy Sets Syst. 160, 371–383 (2009)

    Article  MATH  Google Scholar 

  20. Weber, S.: A complete characterization of all weakly additive measures and of all valuations on the canonical extension of any finite MV-chain. Fuzzy Sets Syst. 161, 1350–1367 (2010)

    Article  MATH  Google Scholar 

  21. Weber, S.: Measure-free conditioning and extensions of additive measures on finite MV-algebras. Fuzzy Sets Syst. 161, 2479–2504 (2010)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

I am very grateful to Peter Klement, as organizator and motor of his Linz Seminars where I received a lot of stimulations since my first participation in 1983, and as friend from our common time working together and in private occations.

Author information

Authors and Affiliations

  1. Institut Für Mathematik, Universität Mainz, FB 08, Mainz, Germany

    Siegfried Weber

Authors
  1. Siegfried Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siegfried Weber .

Editor information

Editors and Affiliations

  1. Dept of Knowledge-Based Math. Systems, Johannes Kepler University Linz, Linz, Austria

    Susanne Saminger-Platz

  2. Dept. of Mathematics & Descriptive Geo., Slovak University of Technology, BRATISLAVA, Slovakia

    Radko Mesiar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Weber, S. (2016). Conditioning for Boolean Subsets, Indicator Functions and Fuzzy Subsets. In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-28808-6_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28807-9

  • Online ISBN: 978-3-319-28808-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation