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Tableaux for Free Logics with Descriptions

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

Abstract

The paper provides a tableau approach to definite descriptions. We focus on several formalizations of the so-called minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic. We consider five analytic tableau systems corresponding to different kinds of free logic, including the logic of definedness applied in computer science and constructive mathematics for dealing with partial functions (here called negative quasi-free logic). The tableau systems formalise MFD based on PFL (positive free logic), NFL (negative free logic), PQFL and NQFL (the quasi-free counterparts of the former ones). Also the logic \(\textsf {NQFL}^{-}\) is taken into account, which is equivalent to NQFL, but whose language does not comprise the existence predicate. It is shown that all tableaux are sound and complete with respect to the semantics of these logics.

Both authors are supported by the National Science Centre, Poland (grant number: DEC- 2017/25/B/HS1/01268). The second author is supported by the EPSRC projects OASIS (EP/S032347/1), AnaLOG (EP/P025943/1), and UK FIRES (EP/S019111/1), the SIRIUS Centre for Scalable Data Access, and Samsung Research UK.

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Notes

  1. 1.

    The reader may find a more fine-grained presentation of MFD and its extensions in Lambert’s [21], Bencivenga’s [4] or Lehmann’s [23] works.

References

  1. Baaz, M., Iemhoff, R.: Gentzen calculi for the existence predicate. Stud. Logica. 82(1), 7–23 (2006). https://doi.org/10.1007/s11225-006-6603-6

    Article  MathSciNet  MATH  Google Scholar 

  2. Beeson, M.J.: Foundations of Constructive Mathematics. Metamathematical Studies, Springer, Heidelberg (1985)

    Book  Google Scholar 

  3. Bencivenga, E., Lambert, K., van Fraasen, B.: Logic, Bivalence and Denotation. Ridgeview, Atascadero (1991)

    Google Scholar 

  4. Bencivenga, E.: Free logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, pp. 147–196. Springer, Dordrecht (2002). https://doi.org/10.1007/978-94-017-0458-8_3

    Chapter  Google Scholar 

  5. Bostock, D.: Intermediate Logic. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  6. Ebbinghaus, H.D., Flum, J., Thomas, W.: Mathematical Logic. Undergraduate Texts in Mathematics, Springer, New York (1994). https://doi.org/10.1007/978-1-4757-2355-7

    Book  MATH  Google Scholar 

  7. Feferman, S.: Definedness. Erkenntnis 43, 295–320 (1995). https://doi.org/10.1007/BF01135376

    Article  MathSciNet  Google Scholar 

  8. Fitting, M., Mendelsohn, R.: First-Order Modal Logic. Kluwer, Dordrecht (1998). https://doi.org/10.1007/978-94-011-5292-1

  9. Garson, J.W.: Modal Logic for Philosophers. Cambridge University Press, Cambridge (2006). https://doi.org/10.1017/CBO9780511617737

  10. Gumb, R.: An extended joint consistency theorem for a nonconstructive logic of partial terms with definite descriptions. Stud. Logica. 69(2), 279–292 (2001). https://doi.org/10.1023/A:1013822008159

    Article  MathSciNet  MATH  Google Scholar 

  11. Indrzejczak, A.: Cut-free modal theory of definite descriptions. In: Bezhanishvili, G., D’Agostino, G., Metcalfe, G., Studer, T. (eds.) Advances in Modal Logic 12, pp. 387–406. College Publications, London (2018)

    MATH  Google Scholar 

  12. Indrzejczak, A.: Fregean description theory in proof-theoretical setting. Log. Logical Philos. 28(1), 137–155 (2019). https://doi.org/10.12775/LLP.2018.008

    Article  MathSciNet  MATH  Google Scholar 

  13. Indrzejczak, A.: Existence, definedness and definite descriptions in hybrid modal logic. In: Olivetti, N., Verbrugge, R., Negri, S., Sandu, G. (eds.) Advances in Modal Logic 13, pp. 349–368. College Publications, London (2020)

    Google Scholar 

  14. Indrzejczak, A.: Free definite description theory - sequent calculi and cut elimination. Log. Logical Philos. 29(4), 505–539 (2020). https://doi.org/10.12775/LLP.2018.008

    Article  MathSciNet  Google Scholar 

  15. Indrzejczak, A.: Free logics are cut-free. Stud. Logica. 109(4), 859–886 (2020). https://doi.org/10.1007/s11225-020-09929-8

    Article  MathSciNet  Google Scholar 

  16. Indrzejczak, A.: Russellian definite description theory - a proof theoretic approach. Rev. Symb. Logic 1–26 (2021). https://doi.org/10.1017/S1755020321000289

  17. Indrzejczak, A., Zawidzki, M.: Tableaux for free logics with descriptions (2021). arXiv:2107.07228

  18. Kalish, D., Montague, R., Mar, G.: Logic. Techniques of Formal Reasoning, 2nd edn. Oxford University Press, New York (1980)

    Google Scholar 

  19. Kürbis, N.: A binary quantifier for definite descriptions in intuitionist negative free logic: natural deduction and normalization. Bull. Section Log. 48(2), 81–97 (2019). https://doi.org/10.18778/0138-0680.48.2.01

    Article  MATH  Google Scholar 

  20. Kürbis, N.: Two treatments of definite descriptions in intuitionist negative free logic. Bull. Section Log. 48(4), 299–317 (2019). https://doi.org/10.18778/0138-0680.48.4.04

    Article  MathSciNet  MATH  Google Scholar 

  21. Lambert, K.: A theory of definite descriptions. In: Lambert, K. (ed.) Philosophical Applications of Free Logic, pp. 17–27. Kluwer (1962)

    Google Scholar 

  22. Lambert, K.: Free logic and definite descriptions. In: Lambert, K. (ed.) New Essays in Free Logic, pp. 37–48. Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-015-9761-6-2

    Chapter  MATH  Google Scholar 

  23. Lehmann, S.: More free logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 5, 2nd edn., pp. 197–259. Springer, Dordrecht (2002). https://doi.org/10.1007/978-94-017-0458-8-4

    Chapter  Google Scholar 

  24. Maffezioli, P., Orlandelli, E.: Full cut elimination and interpolation for intuitionistic logic with existence predicate. Bull. Section Log. 48(2), 137–158 (2019). https://doi.org/10.18778/0138-0680.48.2.04

    Article  MathSciNet  MATH  Google Scholar 

  25. Orlandelli, E.: Labelled calculi for quantified modal logics with definite descriptions. J. Logic Comput. (2021). https://doi.org/10.1093/logcom/exab018, exab018

  26. Pavlović, E., Gratzl, N.: A more unified approach to free logics. J. Philos. Log. 50(1), 117–148 (2020). https://doi.org/10.1007/s10992-020-09564-7

    Article  MathSciNet  MATH  Google Scholar 

  27. Pelletier, F.J., Linsky, B.: What is Frege’s theory of descriptions? In: Linsky, B., Imaguire, G. (eds.) On Denoting: 1905–2005, pp. 195–250. Philosophia Verlag, Munich (2005)

    Google Scholar 

  28. Scott, D.: Identity and existence in intuitionistic logic. In: Fourman, M., Mulvey, C., Scott, D. (eds.) Applications of Sheaves, pp. 660–696. Springer, Heidelberg (1979). https://doi.org/10.1007/BFb0061839

    Chapter  Google Scholar 

  29. Tennant, N.: Natural Logic. Edinburgh University Press, Edinburgh (1978)

    MATH  Google Scholar 

  30. Tennant, N.: A general theory of abstraction operators. Philos. Q. 54(214), 105–133 (2004). https://doi.org/10.1111/j.0031-8094.2004.00344.x

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Michał Zawidzki

  2. Department of Logic, University of Łódź, Łódź, Poland

    Andrzej Indrzejczak & Michał Zawidzki

Authors
  1. Andrzej Indrzejczak
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  2. Michał Zawidzki
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Correspondence to Michał Zawidzki .

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Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

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Indrzejczak, A., Zawidzki, M. (2021). Tableaux for Free Logics with Descriptions. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_4

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  • DOI: https://doi.org/10.1007/978-3-030-86059-2_4

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  • Online ISBN: 978-3-030-86059-2

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