Skip to main content
SpringerLink
Log in

Minimal Complete Propositional Natural Deduction Systems

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

For each truth-functionally complete set of connectives, we construct a sound and complete natural deduction system containing no axioms and the smallest possible number of inference rules, namely one.

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

Access this article

Log in via an institution

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. Bernstein, B.A. (1932). On Nicod’s reduction in the number of primitives of logic. Mathematical Proceedings of the Cambridge Philosophical Society, 28(4), 427–432.

    Article  Google Scholar 

  2. Gagnon, L.S. (1976). Nor logic: a system of natural deduction. Notre Dame Journal of Formal Logic, XVII(2), 293–294.

    Article  Google Scholar 

  3. Nicod, J. (1917). A reduction in the number of the primitive propositions of logic. Proceedings of the Cambridge Philosophical Society, 19, 32–41.

    Google Scholar 

  4. Post, E.L. (1921). Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43(3), 163–185.

    Article  Google Scholar 

  5. Price, R. (1961). The stroke function in natural deduction. Mathematical Logic Quarterly, 7(7), 117–123.

    Article  Google Scholar 

  6. Rose, A. (1949). A reduction in the number of the axioms of the propositional calculus. Norsk matematisk tidsskrift, 31, 113–115. [Reviewed by Quine, W. V., The Journal of Symbolic Logic, 1950, Vol.15(2), p.139.]

    Google Scholar 

  7. Sheffer, H.M. (1913). A set of five independent postulates for Boolean algebras, with application to logical constants. Transactions of the American Mathematical Society, 14, 481–488.

    Article  Google Scholar 

  8. Tennant, N. (1978). Natural logic. Edinburgh: Edinburgh University Press.

    Google Scholar 

  9. Tennant, N. (1979). La barre de Sheffer dans la logique des sequents et des syllogismes. Logique et Analyse, 88, 503–514.

    Google Scholar 

  10. Whitehead, A.N., & Russell, B. (1910). Principia mathematica, First Edition Vol. I. New York: Cambridge University Press.

  11. Whitehead, A.N., & Russell, B. (1927). Principia mathematica, Second Edition Vol. I. New York: Cambridge University Press.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Actuarial Science, American University in Cairo, AUC Avenue, P.O. Box 74, New Cairo, 11835, Egypt

    Amr Elnashar & Wafik Boulos Lotfallah

Authors
  1. Amr Elnashar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wafik Boulos Lotfallah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wafik Boulos Lotfallah.

Additional information

This paper is mainly the work of the first author, an undergraduate student in the American University in Cairo, New Cairo, Egypt. It extends a project in a logic course taught by the second author (the corresponding author), a professor in the Department of Mathematics and Actuarial Science, who only set the problem, guided the student, and shaped the final paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elnashar, A., Lotfallah, W.B. Minimal Complete Propositional Natural Deduction Systems. J Philos Logic 47, 803–815 (2018). https://doi.org/10.1007/s10992-017-9450-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-017-9450-1

Keywords

Search

Navigation