Skip to main content
SpringerLink
Log in

Analytic Calculi for Monadic PNmatrices

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

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

Abstract

Analytic calculi are a valuable tool for a logic, as they allow for effective proof-search and decidability results. We study the axiomatization of generalized consequence relations determined by monadic partial non-deterministic matrices (PNmatrices). We show that simple axiomatizations can always be obtained, using inference rules which can have more than one conclusion. Further, we prove that these axiomatizations are always analytic, which seems to raise a contrast with recent non-analyticity results for sequent-calculi with PNmatrix semantics.

This research was done under the scope of Project UID/EEA/50008/2019 of Instituto de Telecomunicações (IT, financed by the applicable framework (FCT/MEC through national funds and cofunded by FEDER-PT2020), and is part of the MoLC project of SQIG at IT. Thanks are due to two anonymous referees for their valuable feedback, which helped improving an earlier version of this paper.

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.

    Cut for formulas demands, for every \(\varGamma ,\varDelta ,\{A\}\subseteq L_\varSigma (P)\): (C\(^{\text {F}}\)) if \(\varGamma ,A\vartriangleright \varDelta \) and \(\varGamma \vartriangleright A,\varDelta \) then \(\varGamma \vartriangleright \varDelta \).

  2. 2.

    Note that in general \(\vartriangleright ^\varLambda _R\) is not a generalized consequence relation. It still satisfies properties (D) and (C), but only weaker versions of (O) and (S).

References

  1. Avron, A., Lev, I.: Non-deterministic multiple-valued structures. J. Log. Comput. 15(3), 241–261 (2005)

    Article  MathSciNet  Google Scholar 

  2. Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic. HALO, vol. 16, pp. 227–304. Springer, Heidelberg (2011). https://doi.org/10.1007/978-94-007-0479-4_4

    Chapter  Google Scholar 

  3. Avron, A., Ben-Naim, J., Konikowska, B.: Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics. Logica Universalis 1(1), 41–70 (2007)

    Article  MathSciNet  Google Scholar 

  4. Avron, A., Konikowska, B., Zamansky, A.: Cut-free sequent calculi for C-systems with generalized finite-valued semantics. J. Log. Comput. 23(3), 517–540 (2013)

    Article  MathSciNet  Google Scholar 

  5. Baaz, M., Lahav, O., Zamansky, A.: Finite-valued semantics for canonical labelled calculi. J. Autom. Reason. 51(4), 401–430 (2013)

    Article  MathSciNet  Google Scholar 

  6. Caleiro, C., Marcelino, S., Marcos, J.: Combining fragments of classical logic: when are interaction principles needed? Soft Comput. 23(7), 2213–2231 (2019)

    Article  Google Scholar 

  7. Caleiro, C., Marcelino, S., Rivieccio, U.: Characterizing finite-valuedness. Fuzzy Sets Syst. 345, 113–125 (2018)

    Article  MathSciNet  Google Scholar 

  8. Caleiro, C., Marcos, J., Volpe, M.: Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics. Theor. Comput. Sci. 603, 84–110 (2015)

    Article  MathSciNet  Google Scholar 

  9. Carnielli, W., Coniglio, M., Marcos, J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 14. Kluwer (2007)

    Google Scholar 

  10. Carnielli, W., Marcos, J.: A taxonomy of C-systems. In: Carnielli, W., Coniglio, M., D’Ottaviano, I. (eds.) Paraconsistency: The Logical Way to the Inconsistent. Lecture Notes in Pure and Applied Mathematics, vol. 228, pp. 1–94. Marcel Dekker (2002)

    Google Scholar 

  11. Ciabattoni, A., Lahav, O., Spendier, L., Zamansky, A.: Taming paraconsistent (and other) logics: an algorithmic approach. ACM Trans. Comput. Log. 16(1), 5:1–5:23 (2014)

    Article  MathSciNet  Google Scholar 

  12. Font, J.: Belnap’s Four-valued logic and De Morgan lattices. Log. J. IGPL 5(3), 1–29 (1997)

    Article  MathSciNet  Google Scholar 

  13. Marcelino, S., Caleiro, C.: Disjoint fibring of non-deterministic matrices. In: Kennedy, J., de Queiroz, R.J.G.B. (eds.) WoLLIC 2017. LNCS, vol. 10388, pp. 242–255. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55386-2_17

    Chapter  MATH  Google Scholar 

  14. Marcelino, S., Caleiro, C.: Axiomatizing non-deterministic many-valued generalized consequence relations. Synthese (to appear). https://doi.org/10.1007/s11229-019-02142-8

  15. Scott, D.: Completeness and axiomatizability in many-valued logic. In: Henkin, L., Addison, J., Chang, C., Craig, W., Scott, D., Vaught, R. (eds.) Proceedings of the Tarski Symposium, volume XXV of Proceedings of Symposia in Pure Mathematics, pp. 411–435. American Mathematical Society (1974)

    Google Scholar 

  16. Shoesmith, D., Smiley, T.: Multiple-Conclusion Logic. Cambridge University Press, Cambridge (1978)

    Book  Google Scholar 

  17. Wójcicki, R.: Theory of Logical Calculi, Synthese Library, vol. 199. Kluwer (1998)

    Google Scholar 

  18. Wroński, A.: A three element matrix whose consequence operation is not finitely based. Bull. Sect. Log. 2(8), 68–70 (1979)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SQIG - Instituto de Telecomunicações, Dep. Matemática - Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Carlos Caleiro & Sérgio Marcelino

Authors
  1. Carlos Caleiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sérgio Marcelino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sérgio Marcelino .

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

Caleiro, C., Marcelino, S. (2019). Analytic Calculi for Monadic PNmatrices. 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_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-59533-6_6

  • 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