Hyper Natural Deduction

Paper introduces a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system, by adding additional rules providing means for communication between derivations. It is shown that the Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron's Hyper sequent Calculus. The paper also provides conversions for normalisation and prove the existence of normal forms for the Hyper Natural Deduction system

Hyper Natural Deduction for Gödel Logic—A natural deduction system for parallel reasoning

We introduce a system of Hyper Natural Deduction for Gödel Logic as an extension of Gentzen’s system of Natural Deduction. A deduction in this system consists of a finite set of derivations which uses the typical rules of Natural Deduction, plus additional rules providing means for communication between derivations. We show that our system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron’s Hypersequent Calculus. We provide conversions for normalization extending usual conversions for Natural Deduction and prove the existence of normal forms for Hyper Natural Deduction for Gödel Logic. We show that normal deductions satisfy the subformula property