Skip to main content
SpringerLink
Log in

Proof-Theory and Semantics for a Theory of Definite Descriptions

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

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

Abstract

This paper presents a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier I. I forms a formula from two formulas. Ix[F, G] means ‘The F is G’. This approach has the advantage of incorporating scope distinctions directly into the notation. Cut elimination is proved for a system of classical positive free logic with I and it is shown to be sound and complete for the semantics. The system has a number of novel features and is briefly compared to the usual approach of formalising ‘the F’ by a term forming operator. It does not coincide with Hintikka’s and Lambert’s preferred theories, but the divergence is well-motivated and attractive.

The research in this paper was funded by the Alexander von Humboldt Foundation.

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

Similar content being viewed by others

Notes

  1. 1.

    Frege’s treatment of the function that is a ‘substitute for the definite article’ is different. Frege’s operator \(\backslash \) applies to names of objects, not to (simple or complex) predicates or function symbols. Typically these names refer to the extensions of concepts, but this is not necessary. \(\backslash \xi \) returns the unique object that falls under a concept, if \(\xi \) is a name of the extension a concept under which a unique object falls, and its argument in all other cases. See [9, §11].

  2. 2.

    This axiom bears some resemblance to Frege’s Basic Law VI, the sole axiom for his operator \(\backslash \), which is \(a=\backslash \acute{\varepsilon }(a=\varepsilon )\) [9, §18]. But see footnote 1.

  3. 3.

    For a survey of various theories and their axioms, see [1, 8, 27, 30].

  4. 4.

    An earlier approach is by Czermak [4]. Gratzl provides a cut free proof system for Russell’s theory of definite descriptions, including his method for marking scope [11].

  5. 5.

    This paper also briefly considers rules for classical non-free and negative free logic.

  6. 6.

    The rules are, in fact, those given for non-free classical logic at the end of [22]: it is a noteworthy result that, whereas in the context of this logic these rules are redundant and Ix[F, G] definable in Russellian fashion as \(\exists x(F\wedge \forall y(F_y^x\rightarrow x=y)\wedge G)\), added to classical positive free logic, the outcome is a theory of considerable logical and philosophical interest.

  7. 7.

    It would be possible to define \(\exists !t\) as \(\exists x \ x=t\), where \(\exists \) may in turn be defined in terms of \(\forall \) and \(\lnot \). However, treating it as primitive is formally and philosophically preferable: formally, it lends itself more easily to cut elimination, and philosophically, it permits to take existence as conceptually basic, with the quantifiers explained in terms of it: the attempted definition of \(\exists !\) is arguably circular, as the rules of inference governing \(\forall \), which explain its meaning, appeal to \(\exists !\). The semantic clause for \(\forall \), too, implicitly appeals to the concept of existence, as it ranges only over objects in the domain of the model which are considered to exist, that is, those of which \(\exists !\) is true.

References

  1. Bencivenga, E.: Free logics. In: Gabbay, D., Guenther, F. (eds.) Handbook of Philosophical Logic. Volume III: Alternatives to Classical Logic, pp. 373–426. Springer, Dortrecht (1986)

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  3. Cocchiarella, N.: A logic of actual and possible objects. J. Symb. Log. 31(4), 668–689 (1966)

    Google Scholar 

  4. Czermak, J.: A logical calculus with definite descriptions. J. Philos. Log. 3(3), 211–228 (1974)

    Article  MathSciNet  Google Scholar 

  5. Dummett, M.: Frege. Philosophy of Language, 2nd edn. Duckworth, London (1981)

    Google Scholar 

  6. Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Harcourt Academic Press, San Diego (2000)

    Google Scholar 

  7. Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Kluwer, Dordrecht (1998)

    Book  Google Scholar 

  8. van Fraassen, B.C.: On (the x) (x = lambert). In: Spohn, W., van Fraassen, B.C. (ed.) Existence and Explanation. Essays Presented in Honor of Karel Lambert. Kluwer, Dordrecht (1991)

    Google Scholar 

  9. Frege, G.: Grundgesetze der Arithmetik. Begriffsschriftlich abgeleited. I. Band. Jena: Hermann Pohle (1893)

    Google Scholar 

  10. Garson, J.W.: Modal Logic for Philosophers, 2nd edn. Cambridge University Press (2013)

    Google Scholar 

  11. Gratzl, N.: Incomplete symbols - definite descriptions revisited. J. Philos. Log. 44(5), 489–506 (2015)

    Article  MathSciNet  Google Scholar 

  12. Hintikka, J.: Towards a theory of definite descriptions. Analysis 19(4), 79–85 (1959)

    Article  Google Scholar 

  13. Hughes, G., Cresswell, M.: A New Introduction to Modal Logic. Routledge (1996)

    Google Scholar 

  14. Indrzejczak, A.: Cut-free modal theory of definite descriptions. In: Bezhanishvili, G., D’Agostino, G.M., Studer, T. (eds.) Advances in Modal Logic, vol. 12, pp. 359–378. College Publications, London (2018)

    Google Scholar 

  15. Indrzejczak, A.: Fregean description theory in proof-theoretical setting. Logic Log. Philos. 28(1), 137–155 (2018)

    MathSciNet  MATH  Google Scholar 

  16. 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. College Publications, Rickmansworth (2020)

    Google Scholar 

  17. Indrzejczak, A.: Free definite description theory - sequent calculi and cut elimination. Log. Logical Philos. 29(4), 505–539 (2020)

    Google Scholar 

  18. Indrzejczak, A.: Free logics are cut free. Stud. Logica. 109(4), 859–886 (2020)

    Article  MathSciNet  Google Scholar 

  19. Indrzejczak, A.: Russellian definite description theory - a proof-theoretic approach. Forthcoming in the Review of Symbolic Logic (2021)

    Google Scholar 

  20. Kürbis, N.: A binary quantifier for definite descriptions in intuitionist negative free logic: Natural deduction and normalisation. Bull. Section Log. 48(2), 81–97 (2019)

    Article  MathSciNet  Google Scholar 

  21. Kürbis, N.: Two treatments of definite descriptions in intuitionist negative free logic. Bull. Section Log. 48(4), 299–318 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Kürbis, N.: A binary quantifier for definite descriptions for cut free free logics. Forthcoming in Studia Logica (2021)

    Google Scholar 

  23. Kürbis, N.: Definite descriptions in intuitionist positive free logic. Logic Log. Philos. 30(2), 327–358 (2021)

    Google Scholar 

  24. Kürbis, N.: Proof-Theory and Semantics for a Theory of Definite Descriptions. Online version on arXiv arXiv:2108.03944 (2021)

  25. Lambert, K.: Notes on “E!”: II. Philos. Stud. 12(1/2), 1–5 (1961)

    Google Scholar 

  26. Lambert, K.: Notes on “E!”: III. Philos. Stud. 13(4), 51–59 (1961)

    Google Scholar 

  27. Lambert, K.: Foundations of the hierarchy of positive free definite description theories. In: Free Logic. Selected Essays. Cambridge University Press (2004)

    Google Scholar 

  28. Leblanc, H., Thomason, R.: Completeness theorems for some presupposition-free logics. Fundam. Math. 62, 125–164 (1968)

    Article  MathSciNet  Google Scholar 

  29. Meyer, R.K., Lambert, K.: Universally free logic and standard quantification theory. J. Symb. Log. 33(1), 8–26 (1968)

    Article  MathSciNet  Google Scholar 

  30. Morscher, E., Simons, P.: Free logic: a fifty-year past and an open future. In: Morscher, E., Hieke, A. (eds.) New Essays in Free Logic in Honour of Karel Lambert. Kluwer, Dortrecht (2001)

    Chapter  Google Scholar 

  31. Neale, S.: Descriptions. MIT Press, Cambridge (1990)

    Google Scholar 

  32. Priest, G.: An Introduction to Non-Classical Logic, 2nd edn. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  33. Smullyan, A.: Modality and description. J. Symb. Log. 13, 31–7 (1948)

    Article  MathSciNet  Google Scholar 

  34. Whitehead, A., Russell, B.: Principia Mathematica, vol. 1. Cambridge University Press, Cambridge (1910)

    MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank Andrzej Indrzejczak for comments on this paper and discussions of the proof-theory of definite descriptions in general. Some of this material was presented at Heinrich Wansing’s and Hitoshi Omori’s Work in Progress Seminar at the University of Bochum, to whom many thanks are due for support and insightful comments. Last but not least I must thank the referees for Tableaux 2021 for their thoughtful and considerate reports on this paper.

Author information

Authors and Affiliations

  1. Department of Logic and Methodology of Science, University of Łódź, Łódź, Poland

    Nils Kürbis

  2. Department of Philosophie I, University of Bochum, Bochum, Germany

    Nils Kürbis

Authors
  1. Nils Kürbis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nils Kürbis .

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kürbis, N. (2021). Proof-Theory and Semantics for a Theory of Definite 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_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-86059-2_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86058-5

  • Online ISBN: 978-3-030-86059-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation