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 ‘protologic’ 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 protologic. We provide a sound and complete calculus for protologic which enjoys strong cut elimination and the display property. We show that protologic 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 protologic, 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 protologic. In a second part of the report, we develop a correspondence theory for protologic using the well– known correspondence results for basic tense logic. We prove that a logic extending protologic is axiomatizable with 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.
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Rapport
[Research Report] Université de Rennes 1. 2017
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https://hal.inria.fr/hal-01483342
Contributeur : Guillaume Aucher <>
Soumis le : lundi 20 mars 2017 - 09:49:52
Dernière modification le : jeudi 23 mars 2017 - 01:07:53

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  • HAL Id : hal-01483342, version 2

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Guillaume Aucher. A New Road Towards Universal Logic. [Research Report] Université de Rennes 1. 2017. <hal-01483342>

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