Studia Logica

, Volume 107, Issue 2, pp 375–397 | Cite as

Analyticity, Balance and Non-admissibility of \(\varvec{Cut}\) in Stoic Logic

  • Susanne Bobzien
  • Roy DyckhoffEmail author
Open Access


This paper shows that, for the Hertz–Gentzen Systems of 1933 (without Thinning), extended by a classical rule T1 (from the Stoics) and using certain axioms (also from the Stoics), all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence (in contrast to one of Gentzen’s 1933 results) that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss.


Sequent Analyticity Stoic logic Proof theory Decidability Relevance Balance Cut-admissibility 



Susanne Bobzien thanks All Souls College, Oxford, for support; Roy Dyckhoff thanks the University of St Andrews for an Honorary position. Both are grateful to Alex Leitsch and Stefan Hetzl for their contribution [7], and to two anonymous referees for their helpful remarks.


  1. 1.
    Bobzien, S., Stoic syllogistic. In Oxford Studies in Ancient Philosophy, volume 14, OUP, Oxford, 1996, pp. 133–192.Google Scholar
  2. 2.
    Bobzien, S., Stoic logic. In B. Inwood, (ed.), The Cambridge Companion to Stoic Philosophy, CUP, 2003, pp. 85–123.Google Scholar
  3. 3.
    Dyckhoff, R., Contraction-free sequent calculi for intuitionistic logic, Journal of Symbolic Logic 57:795–807, 1992.CrossRefGoogle Scholar
  4. 4.
    Dyckhoff, R., Implementation of Stoic propositional logic, 2017.Google Scholar
  5. 5.
    Gentzen, G., Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen, Mathematische Annalen 107:329–350, 1933.CrossRefGoogle Scholar
  6. 6.
    Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift 39:176–210, 405–431, 1935.CrossRefGoogle Scholar
  7. 7.
    Leitsch, A., and S. Hetzl, Personal communication, January 2018.Google Scholar
  8. 8.
    Milne, P., On the completeness of non-Philonian Stoic logic, History and Philosophy of Logic 16:39–64, 1995.CrossRefGoogle Scholar
  9. 9.
    Milne, P., Unpublished MS, 2012.Google Scholar
  10. 10.
    Moriconi, E., Early structural reasoning. Gentzen 1932, Review of Symbolic Logic 8:662–679, 2015.CrossRefGoogle Scholar
  11. 11.
    Read, S., Relevant Logic, Blackwell, 1988.Google Scholar
  12. 12.
    Schroeder-Heister, P., Resolution and the origins of structural reasoning: early proof-theoretic ideas of Hertz and Gentzen, Bull. Symbolic Logic 8:246–265, 2001.CrossRefGoogle Scholar
  13. 13.
    Sextus Empiricus, Against the Logicians, Translated by Richard Bett, Cambridge Texts in the History of Philosophy, Cambridge Univ. Press, 2005.Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of OxfordOxfordUK
  2. 2.University of St AndrewsSt AndrewsUK

Personalised recommendations