Skip to main content
Log in

Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Goldfarb, W., On Dummett’s “Proof-theoretic Justifications of Logical Laws”, in T. Piecha and P. Schroeder-Heister [6], pp. 195–210.

  2. Harrop, R., Concerning formulas of the types \(A \rightarrow B \vee C\), \(A \rightarrow (Ex)B(x)\) in intuitionistic formal systems, Journal of Symbolic Logic 25:27–32, 1960.

    Article  Google Scholar 

  3. Iemhoff, R., On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic 66:281–294, 2001.

    Article  Google Scholar 

  4. Piecha, T., Completeness in proof-theoretic semantics, in T. Piecha and P. Schroeder-Heister [6], pp. 231–251.

  5. Piecha, T., W. de Campos Sanz, and P. Schroeder-Heister, Failure of completeness in proof-theoretic semantics, Journal of Philosophical Logic 44:321–335, 2015. First published online August 1, 2014.

  6. Piecha, T., and P. Schroeder-Heister, (eds.), Advances in Proof-Theoretic Semantics, vol. 43 of Trends in Logic, Springer, 2016.

  7. Piecha, T., and P. Schroeder-Heister, Atomic systems in proof-theoretic semantics: two approaches, in J. Redmond, O. Pombo Martins, and Á. N. Fernández, (eds.), Epistemology, Knowledge and the Impact of Interaction, Springer, Cham, 2016, pp. 47–62.

  8. Piecha, T., and P. Schroeder-Heister, The definitional view of atomic systems in proof-theoretic semantics, in P. Arazim, and T. Lávička, (eds.), The Logica Yearbook 2016, College Publications, London, 2017, pp. 185–200.

    Google Scholar 

  9. Prawitz, D., Natural Deduction: A Proof-Theoretical Study, Almqvist & Wiksell, Stockholm, 1965. Reprinted by Dover Publications, Mineola, N.Y., 2006.

  10. Prawitz, D., Ideas and results in proof theory, in J. E. Fenstad, (ed.), Proceedings of the Second Scandinavian Logic Symposium, vol. 63 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971, pp. 235–307.

  11. Prawitz, D., Towards a foundation of a general proof theory, in P. Suppes, et al., (eds.), Logic, Methodology and Philosophy of Science IV, North-Holland, Amsterdam, 1973, pp. 225–250.

    Google Scholar 

  12. Prawitz, D., An approach to general proof theory and a conjecture of a kind of completeness of intuitionistic logic revisited, in L. C. Pereira, E. H. Haeusler, and V. de Paiva, (eds.), Advances in Natural Deduction, vol. 39 of Trends in Logic, Springer, Berlin, 2014, pp. 269–279.

    Chapter  Google Scholar 

Download references

Acknowledgements

This work was carried out within the French-German ANR-DFG project “Beyond Logic: Hypothetical Reasoning in Philosophy of Science, Informatics, and Law”, DFG Grant Schr 275/17-1. We are grateful to two referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thomas Piecha.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Piecha, T., Schroeder-Heister, P. Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics. Stud Logica 107, 233–246 (2019). https://doi.org/10.1007/s11225-018-9823-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-018-9823-7

Keywords

Navigation