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Refinements of subatomic natural deduction

机译：完善亚原子自然演绎

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Subatomic natural deduction combines natural deduction rules with subatomic systems [21]. The latter make proofs of atomic sentences and the study of their component structure accessible to methods of structural proof theory and, thereby, admit a proof-theoretic account of the semantics of atomic sentences and their components. The article proposes two refinements of subatomic natural deduction. The first combines it with algebraic degree functions into systems of graded natural deduction which allow us to model, in a proof-theoretic manner, how the degree of assent to a conclusion may depend on the degrees of assent to its premisses. The second refinement consists in the introduction of term assumption rules which allow us to add (resp. subtract) contents to (from) term assumptions, thereby allowing us to represent aspects of dynamic reasoning in a natural deduction setting. Normalization is established for graded, and with certain limitations, for dynamic, as well as for dynamic graded systems of minimal and intuitionistic first-order logic.
机译：亚原子自然演绎结合了自然演绎规则与亚原子系统[21]。后者使原子句子的证明和它们的组成结构的研究可通过结构证明理论的方法来进行，从而承认原子句子及其组成部分的语义的证明理论解释。本文提出了亚原子自然演绎的两个改进。第一种方法将其与代数度函数结合到分级的自然演绎系统中，这使我们能够以证明理论的方式对结论的认可程度如何取决于其前提的认可程度进行建模。第二个改进之处在于引入了术语假设规则，该规则使我们能够向（从）术语假设中添加（减去）内容，从而使我们能够在自然演绎条件下表示动态推理的各个方面。建立归一化用于动态分级系统，并且具有一定的局限性，适用于动态以及最小和直觉一阶逻辑的动态分级系统。

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