The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

中文翻译：

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使用模态运算符测量某些逻辑中的不一致性

1978 年第一次提到一阶逻辑中公式集的不一致测度概念，但那篇论文只给出了它们的分类。2002 年针对命题逻辑中的公式集给出了第一个实际的数值不一致性度量。从那时起，逻辑学和人工智能的研究人员开发了大量的不一致性度量理论。虽然从逻辑的角度来看这是一个有趣的话题，但这项工作的一个重要动机也是一些智能系统可能会在其操作中遇到不一致。这项研究主要涉及命题知识库，即命题逻辑公式的有限集。本文的目标是以形式化的方式将不一致度量的概念扩展到公式集，其中模态运算符“必然”和“可能”应用于命题逻辑公式。我们将框架用于语义，但与模态逻辑中常用的框架不同，以便于测量不一致。由于一组公式对于不同的框架可能有不同的不一致度量，我们定义了一个标准框架的概念，它可以用于语言中所有有限的公式集。我们为两种语言执行此操作。第一种语言 AMPL 包含将运算符前缀应用于命题逻辑公式的公式。第二种语言 CMPL 添加了可以有限方式应用于 AMPL 公式的连接词。我们展示了如何将命题逻辑不一致度量扩展到这些公式集。最后，我们定义了一个新概念，弱不一致性度量，并展示了如何计算它。