Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Knowledge Without Complete Certainty

  • Conference paper
  • First Online:
Logic, Language, Information, and Computation (WoLLIC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11541))

  • 653 Accesses

Abstract

We present an epistemic logic ELF (Epistemic Logic with Filters) where knowledge does not require complete certainty. In this logic, instead of saying that an agent knows a particular fact if it is true in every accessible world, we say that it knows the fact if it is true in a sufficiently large set accessible worlds. On a technical level, we do this by enriching the standard Kripke models of epistemic logic with a set of filters: a sufficiently large set of worlds is one that is in the filter. We introduce semantics for ELF, and give a sound and complete proof system.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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.

    An epistemic modal use of that is the Majority Logic of [24].

  2. 2.

    This is similar to the approach advocated in [21], see also Remark 3.

  3. 3.

    But, unless one uses a very strong notion of justification, not sufficient [13].

  4. 4.

    Because we require W to be a set, as opposed to a class, we may have to restrict ourselves to the models of \(\mathfrak {T}\) in some set-theoretic universe \(\mathcal {U}\), where \(W\not \in \mathcal {U}\).

  5. 5.

    The axiom L is, using the other axioms and rules, interderivable with the axiom D, given by \(\square \varphi \rightarrow \lozenge \varphi \). One could, therefore, think of \(\mathbf {WKL}\) as “weak KD” instead of “weak KL”. Our reason for preferring L over D in this context is that L more closely follows the semantical constraint that \(\emptyset \not \in \mathcal {F}(w)\).

  6. 6.

    Note that \(\not \models \square \top \) in ELF, since \(\mathcal {M},w\not \models \square \top \) when \(R(w)\not \in \mathcal {F}(w)\).

References

  1. Aucher, G.: How our beliefs contribute to interpret actions. In: Pěchouček, M., Petta, P., Varga, L.Z. (eds.) CEEMAS 2005. LNCS (LNAI), vol. 3690, pp. 276–285. Springer, Heidelberg (2005). https://doi.org/10.1007/11559221_28

    Chapter  Google Scholar 

  2. Baltag, A., Bezhanishvili, N., Özgün, A., Smets, S.: Justified belief and the topology of evidence. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds.) WoLLIC 2016. LNCS, vol. 9803, pp. 83–103. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-52921-8_6

    Chapter  MATH  Google Scholar 

  3. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. In: Proceedings of 7th LOFT. Texts in Logic and Games, vol. 3, pp. 13–60. Amsterdam University Press (2008)

    Google Scholar 

  4. Ben-David, S., Ben-Eliyahu, R.: A modal logic for subjective default reasoning. Artif. Intell. 116, 217–236 (2000). https://doi.org/10.1016/S0004-3702(99)00081-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Blackburn, P., van Benthem, J., Wolter, F. (eds.): Handbook of Modal Logic. Elsevier (2006)

    Google Scholar 

  6. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  7. Cartan, H.: Théorie des filtres. Comptes Rendus de l’Académie des Sciences de Paris 205, 595–598 (1937)

    MATH  Google Scholar 

  8. Chellas, B.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)

    Book  Google Scholar 

  9. Chisholm, R.: Perceiving: A Philosophical Study. Cornell University Press, Ithaca (1957)

    Google Scholar 

  10. Conitzer, V.: A puzzle about further facts. Erkenntnis (2018).https://doi.org/10.1007/s10670-018-9979-6

    Article  Google Scholar 

  11. DeRose, K.: Contextualism and knowledge attributions. Philos. Phenomenol. Res. 52, 913–929 (1992). https://doi.org/10.2307/2107917

    Article  Google Scholar 

  12. van Ditmarsch, H., Halpern, J., van der Hoek, W., Kooi, B. (eds.): Handbook of Epistemic Logic. College Publications (2015)

    Google Scholar 

  13. Gettier, E.: Is justified true belief knowledge? Analysis 23, 121–123 (1963). https://doi.org/10.2307/3326922

    Article  Google Scholar 

  14. Harman, G.: Thought. Princeton University Press, Princeton (1973)

    Google Scholar 

  15. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca (1962)

    MATH  Google Scholar 

  16. van der Hoek, W.: Systems for knowledge and belief. J. Log. Comput. 3(2), 173–195 (1993). https://doi.org/10.1093/logcom/3.2.173

    Article  MathSciNet  MATH  Google Scholar 

  17. Keisler, H.: Elementary Calculus: An Approach Using Infinitesimals. Prindle Weber & Schmidt (1986)

    Google Scholar 

  18. Kraus, S., Lehmann, D.: Knowledge, belief and time. Theor. Comput. Sci. 58, 155–174 (1988). https://doi.org/10.1016/0304-3975(88)90024-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44, 167–207 (1990). https://doi.org/10.1016/0004-3702(90)90101-5

    Article  MathSciNet  MATH  Google Scholar 

  20. Kyburg, H.: Probability and the Logic of Rational Belief. Wesleyan University Press, Middletown (1961)

    Google Scholar 

  21. Lewis, D.: Elusive knowledge. Australas. J. Philos. 74, 549–567 (1996). https://doi.org/10.1080/00048409612347521

    Article  Google Scholar 

  22. Nozick, R.: Philosophical Explanations. Harvard University Press, Cambridge (1981)

    Google Scholar 

  23. Pacuit, E.: Neighborhood Semantics for Modal Logic. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-67149-9

    Book  MATH  Google Scholar 

  24. Pacuit, E., Salame, S.: Majority logic. In: Proceedings of Ninth KR, pp. 598–605 (2004)

    Google Scholar 

  25. Putnam, H.: Reason, Truth and History. Cambridge University Press, Cambridge (1982)

    Google Scholar 

  26. Segerberg, K.: Irrevocable belief revision in dynamic doxastic logic. Notre Dame J. Formal Log. 39(3), 287–306 (1998). https://doi.org/10.1305/ndjfl/1039182247

    Article  MathSciNet  MATH  Google Scholar 

  27. Stalnaker, R.: On logics of knowledge and belief. Philos. Stud. 128(1), 169–199 (2005). https://doi.org/10.1007/s11098-005-4062-y

    Article  MathSciNet  Google Scholar 

  28. Unger, P.: Philosophical Relativity. Oxford University Press, New York (1984)

    Google Scholar 

Download references

Acknowledgements

We are grateful to Ramanujam, who suggested the idea of defining knowledge through filters to us. We also thank two anonymous reviewers for their helpful comments. Hans van Ditmarsch is also affiliated to IMSc, Chennai, as associate researcher.

Author information

Authors and Affiliations

  1. CNRS, LORIA, Vandœuvre-lès-Nancy, France

    Hans van Ditmarsch

  2. University of Liverpool, Liverpool, UK

    Louwe B. Kuijer

Authors
  1. Hans van Ditmarsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louwe B. Kuijer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louwe B. Kuijer .

Editor information

Editors and Affiliations

  1. Department of Philosophy and Religious Studies, Utrecht University, Utrecht, The Netherlands

    Rosalie Iemhoff

  2. Utrecht Institute of Linguistics OTS, Utrecht University, Utrecht, The Netherlands

    Michael Moortgat

  3. Centro de Informática, Universidade Federal de Pernambuco (UFPE), Recife, Brazil

    Ruy de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer-Verlag GmbH Germany, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

van Ditmarsch, H., Kuijer, L.B. (2019). Knowledge Without Complete Certainty. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_38

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-59533-6_38

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-59532-9

  • Online ISBN: 978-3-662-59533-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation