Stoic Sequent Logic and Proof Theory

History and Philosophy of Logic:1-32 (forthcoming)

Authors
Susanne Bobzien
Oxford University
Abstract
This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.
Keywords Stoic logic  Sequent calculus  Proof theory  Substructural logic  Connexive logic  Decidability  Gentzen  Propositional logic  Ancient logic
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DOI 10.1080/01445340.2019.1579624
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Proof Analysis in Intermediate Logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1-2):71-92.
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Stoic Syllogistic.Susanne Bobzien - 1996 - Oxford Studies in Ancient Philosophy 14:133-92.
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