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

  • Susanne Bobzien
  • Roy DyckhoffEmail author
Open Access
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Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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

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