A New Road Towards Universal Logic?

Guillaume Aucher 1, 2
2 LIS - Logical Information Systems
IRISA-D7 - GESTION DES DONNÉES ET DE LA CONNAISSANCE
Abstract : A generic logic called ‘Gaggle logic’ is introduced. It is based on Gaggle theory and deals with connectives of arbitrary arity that are related to each other by abstract laws of residuation. We list the 96 binary connectives and the 16 unary connectives of Gaggle logic. We provide a sound and complete calculus for Gaggle logic which enjoys strong cut elimination and the display property. We show that Gaggle logic is decidable and satisfies the properties of conservativity and interpolation. We also introduce specific inference rules called ‘protoanalytic’ inference rules. These rules are such that, when added to the calculus of Gaggle logic, we obtain a calculus which still enjoys strong cut elimination and the display property. If the language considered contains conjunction and disjunction, then the interpolation theorem also transfers to these extensions of Gaggle logic. In a second part of the report, we generalize the Kracht’s correspondence results established for the basic tense logic to Gaggle logic. We prove that a logic extending Gaggle logic is axiomatizable by means of so-called ‘protoanalytic’ inference rules if, and only if, the class of frames on which such a logic is based is definable by specific first-order frame conditions, also called ‘protoanalytic’. We provide algorithms that compute the corresponding protoanalytic inference rules from the protoanalytic first-order frame conditions, and vice versa. We illustrate these algorithms on well-known structural inference rules and we show in particular how we can recover classical logic from Gaggle logic by the addition of protoanalytic inference rules that refine the standard classical inference rules.
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Guillaume Aucher. A New Road Towards Universal Logic?. [Research Report] Université de Rennes 1. 2017. ⟨hal-01483342v3⟩

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