Hyper Natural Deduction

Abstract

We introduce a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system. A Hyper Natural Deduction consists of a finite set of derivations which may use, beside typical Natural Deduction rules, additional rules providing means for communication between derivations. We show that our Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron's Hypersequent Calculus. We also provide conversions for normalisation and prove the existence of normal forms for our Hyper Natural Deduction system.

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

“…Observe that for a different cut (indicated by double backslashes) the deduction in Fig. 1 would not be correct according to [BP15]. We conjecture that the version presented in [BP15] is correct if we demand that its conditions are satisfied for any possible cut operation.…”

“…The results in this paper were first announced in [BP15]. However, there are several changes: First of all, we have switched to an inductive definition of HNGL, and we introduce the previous explicit definition as an equivalent alternative definition needed to prove normalization -in this paper we denote the alternative, explicit definition with X .…”

“…Second, we improved the mathematical presentation of X using notions from graph theory. Finally, we made one essential change to correct an error in the definition given in [BP15] -without that change the system defined there is not sound. In the version given in [BP15] we only demanded the existence of a total order on labels for communication and split rules, which is compatible with their occurrences on branches through derivations.…”

“…We now add additional requirements for labels of contraction rules. The reason for this is that without considering contraction labels, the definition in [BP15] would not be sound for Gödel Logic. The following two prederivations form a set which is a valid deduction according to [BP15] of the formula ((A ∧ C → B) ∧ (C → D)) ∨ ((B ∧ D → C) ∧ (B → A)).…”

“…There is a cut (indicated with single slashes) which deletes edges in the graph, such that the above is a correct HNGL derivation according to [BP15]. Observe that for a different cut (indicated by double backslashes) the deduction in Fig.…”