We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent's knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to the notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz's solution.

中文翻译：

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认知和规范论证的时间逻辑，适用于 Protagoras 悖论

我们将线性时间逻辑（具有过去和未来的模态）与辩护逻辑的道义版本相结合，以提供一个框架来推理时间以及认知和规范原因。除了时间模态之外，由此产生的逻辑包含两种证成断言：认知证成断言和道义证成断言。前者为代理人的知识提供了理由，后者给出了为什么一个命题是强制性的理由。我们为逻辑提供了两种语义：一种基于拟合模型，另一种基于邻域模型。邻域语义的使用使我们能够正确定义道义辩护断言的对偶，这对应于道义逻辑中的许可概念。然后，我们根据这些语义建立逻辑公理系统的健全性和完整性。此外，我们在这个逻辑中形式化了 Protagoras 与 Euat​​hlus 悖论，并对悖论进行了精确分析，并简要讨论了莱布尼茨的解决方案。