论文标题
通过实价的一阶逻辑的推理的新基础
New foundations of reasoning via real-valued first-order logics
论文作者
论文摘要
一般而言，多个价值的逻辑，尤其是模糊的逻辑，通常集中于基于保存完整真理的后果概念，鉴于实际单位间隔[0,1]，语义中的值1表示。在最近的一篇论文（\ emph {通过真实价值逻辑的不确定性基础}，arxiv：2008.02429v2，2021），罗纳德·法金（Ronald Fagin），瑞安·里格尔（Ryan Riegel）和亚历山大·灰色真实价值，不仅是1，在给定的范围内的前提和结论。在本文中，我们将其工作扩展到多维句子的一阶（以及模态）逻辑。我们给出了公理系统并证明了相应的完整定理，首先假设结构是在固定域上定义的，后来是在变化域的逻辑上定义的。作为副产品，我们还获得了这些逻辑有限价值版本的0-1定律。
Many-valued logics in general, and fuzzy logics in particular, usually focus on a notion of consequence based on preservation of full truth, typical represented by the value 1 in the semantics given the real unit interval [0,1]. In a recent paper (\emph{Foundations of Reasoning with Uncertainty via Real-valued Logics}, arXiv:2008.02429v2, 2021), Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on multi-dimensional sentences that allow to prescribe any truth-values, not just 1, for the premises and conclusion of a given entailment. In this paper, we extend their work to the first-order (as well as modal) logic of multi-dimensional sentences. We give axiomatic systems and prove corresponding completeness theorems, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a 0-1 law for finitely-valued versions of these logics.
