Hostname: page-component-6766d58669-7fx5l Total loading time: 0 Render date: 2026-05-18T10:22:22.902Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIVOCITY OF INTUITIONISTIC AND CLASSICAL CONNECTIVES

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  26 December 2024

RODOLFO C. ERTOLA-BIRABEN Open the ORCID record for RODOLFO C. ERTOLA-BIRABEN [Opens in a new window]  and
BRANDEN FITELSON Open the ORCID record for BRANDEN FITELSON [Opens in a new window]
RODOLFO C. ERTOLA-BIRABEN
Affiliation:
CENTRE FOR LOGIC EPISTEMOLOGY AND THE HISTORY OF SCIENCE (CLE) UNIVERSIDADE ESTADUAL DE CAMPINAS (UNICAMP) SÉRGIO BUARQUE DE HOLANDA 251 PO BOX 13083-859 CAMPINAS, SP BRAZIL E-mail: rcertola@unicamp.br
BRANDEN FITELSON*
Affiliation:
DEPARTMENT OF PHILOSOPHY AND RELIGION NORTHEASTERN UNIVERSITY 360 HUNTINGTON AVENUE BOSTON, MA 02115 USA
*
E-mail: b.fitelson@northeastern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the univocity (or uniqueness) of connectives in intuitionistic and classical sentential logic. Specifically, unlike Gentzen systems, Hilbert systems for (various fragments of) intuitionistic and classical logic do not always determine univocal (or unique) conditional connectives. This paper explains when univocal conditional connectives are achieved in Hilbert systems for intuitionistic and classical sentential logic (and when they are not). In the final section, we discuss the (non-)univocity of the Sheffer stroke in Hilbert vs. Gentzen systems for classical sentential logic.

Keywords

univocal connectivesintuitionistic logicclassical logic

MSC classification

Primary: 03F03: Proof theory, general 03B05: Classical propositional logic 03B20: Subsystems of classical logic (including intuitionistic logic)

Information

Type
Article
Information
Bulletin of Symbolic Logic , Volume 31 , Issue 3 , September 2025 , pp. 488 - 497
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Belnap, N., Tonk, plonk and plink . Analysis , vol. 22 (1962), no. 6, pp. 130–134.Google Scholar
Caicedo, X. and Cignoli, R., An algebraic approach to intuitionistic connectives . The Journal of Symbolic Logic , vol. 66 (2001), no. 4 , pp. 1620–1636.Google Scholar
Ertola-Biraben, R., On univocal connectives . Logic and Logical Philosophy , vol. 19 (2009), no. 1, pp. 5–13.Google Scholar
Frege, G., Begriffsschrift, eine der Arithmetischen Nachgebildete Formelsprache des Reinen Denkens , Nebert, Halle, 1879.Google Scholar
Gentzen, G., Investigations into logical deduction , The Collected Papers of Gerhard Gentzen , North Holland, Amsterdam, 1969, pp. 68–131.Google Scholar
Humberstone, L., The Connectives , MIT Press, Cambridge, MA, 2011.Google Scholar
Kleene, S., Introduction to Metamathematics , D. Van Nostrand Co., New York, 1952.Google Scholar
Kovács, L. and Voronkov, A., First-order theorem proving and vampire , Computer Aided Verification (Sharygina, N. and Veith, H. editors), Springer, Heidelberg, pp. 1–35.Google Scholar
McCune, W., Prover9 and Mace4, 2009. http://www.cs.unm.edu/~mccune/prover9/.Google Scholar
McCune, W., Otter 3.3 Reference Manual. Technical Report ANL/MSC-TM-263 , Argonne National Laboratory, Lemont, 2003.Google Scholar
Nicod, J., A reduction in the number of primitive propositions of logic . Proceedings of the Cambridge Philosophical Society , vol. 19 (1917), no. 1917, pp. 32–41.Google Scholar
Paoli, F., Substructural Logics: A Primer , Springer, Dordrecht, 2002.Google Scholar
Read, S., Sheffer’s stroke: a study in proof-theoretic harmony . Danish Yearbook of Philosophy , vol. 34 (1999), no. 1, pp. 7–23.Google Scholar
Scharle, T., Axiomatization of propositional calculus with Sheffer functors . Notre Dame Journal of Formal Logic , vol. 6 (1965), no. 3, pp. 209–217.Google Scholar
Smiley, T., The independence of connectives . The Journal of Symbolic Logic , vol. 27 (1962), no. 4, pp. 426–436.Google Scholar
Zach, R., Natural deduction for the Sheffer stroke and Peirce’s arrow (and any other truth-functional connective) . Journal of Philosophical Logic , vol. 45 (2016), no. 2, pp. 183–197.Google Scholar