The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G¹Ł∀ and a noncumulative hypersequent calculus G²Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G¹Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G²Ł∀-provable. For a tableau version of the calculus G²Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G²Ł∀-provable sentence in the prenex form.

中文翻译：

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无限值一阶Łukasiewicz逻辑：超序列计算，无结构规则，无需对Prenex形式的语句进行证明搜索

有理一阶Pavelka逻辑是无穷大一阶Łukasiewicz逻辑truth由真常数的扩展。对于这样的逻辑，我们引入一个累积hypersequent演算ģ ¹ Ł∀和非累积hypersequent演算ģ ² Ł∀没有结构的推理规则。我们将这些结石与具有结构规则的Baaz-Metcalfe继发性牙结石GŁ∀进行比较。特别是，我们表明，每GŁ∀-证明的句子为G ¹ Ł∀-可证，并在prenex形式Ł∀一句话是GŁ∀，可证明当且仅当它为G ² Ł∀-证明的。对于微积分G ² table的表格版本，我们描述了一系列证明搜索算法，该算法允许我们构造每个G ²的证明。ne可证明的句子，采用后缀形式。