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

Studia Logica 107 (2):375-397 (2019)
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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

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Susanne Bobzien
University of Oxford

Citations of this work

Stoic Sequent Logic and Proof Theory.Susanne Bobzien - 2019 - History and Philosophy of Logic 40 (3):234-265.
Sextus Empiricus' Fourth Conditional and Containment Logic.Yale Weiss - 2019 - History and Philosophy of Logic 40 (4):307-322.

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References found in this work

Relevant Logic : a Philosophical Examination of Inference.Stephen Read - 1988 - Revue Philosophique de la France Et de l'Etranger 179 (4):656-656.
Untersuchungen über das logische Schließen. II.Gerhard Gentzen - 1935 - Mathematische Zeitschrift 39:405–431.
Contraction-free sequent calculi for intuitionistic logic.Roy Dyckhoff - 1992 - Journal of Symbolic Logic 57 (3):795-807.
Early Structural Reasoning. Gentzen 1932.Enrico Moriconi - 2015 - Review of Symbolic Logic 8 (4):662-679.

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