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Analyticity, balance and non-admissibility of Cut in Stoic Logic

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Date
04/2019
Author
Bobzien, Susanne
Dyckhoff, Roy
Keywords
Sequent
Analyticity
Stoic logic
Proof theory
Decidability
Relevance
Balance
Cut-admissibility
BC Logic
QA75 Electronic computers. Computer science
T-NDAS
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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.
Citation
Bobzien , S & Dyckhoff , R 2019 , ' Analyticity, balance and non-admissibility of Cut in Stoic Logic ' , Studia Logica , vol. 107 , no. 2 , pp. 375–397 . https://doi.org/10.1007/s11225-018-9797-5
Publication
Studia Logica
Status
Peer reviewed
DOI
https://doi.org/10.1007/s11225-018-9797-5
ISSN
0039-3215
Type
Journal article
Rights
Copyright The Author(s) 2018. This 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.
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/13269

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