Analyticity, Balance and Non-admissibility of Cut in Stoic Logic

Studia Logica:1-23 (2018)
Authors
Susanne Bobzien
Oxford University
Abstract
This paper shows that, for the Hertz–Gentzen Systems of 1933, extended by a classical rule T1 and using certain axioms, 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 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. OPEN ACCESS
Keywords Sequent Stoic Logic Decidability  Analyticity   Stoic Logic  Proof Theory  Decidability   Relevance  Gentzen
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Reprint years 2018
DOI 10.1007/s11225-018-9797-5
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References found in this work BETA

Early Structural Reasoning. Gentzen 1932.Enrico Moriconi - 2015 - Review of Symbolic Logic 8 (4):662-679.
Contraction-Free Sequent Calculi for Intuitionistic Logic.Roy Dyckhoff - 1992 - Journal of Symbolic Logic 57 (3):795-807.
On the Completeness of Non-Philonian Stoic Logic.Peter Milne - 1995 - History and Philosophy of Logic 16 (1):39-64.

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