论文标题
跨足的模态逻辑：贝叶斯推理的半定量解释
Transfinite Modal Logic: a Semi-quantitative Explanation for Bayesian Reasoning
论文作者
论文摘要
贝叶斯推理在人类理性和机器学习中都起着重要作用。在本文中，我们引入了传播模态逻辑，该逻辑将模态逻辑与序数算术相结合，以使贝叶斯推理半优化。从技术上讲，我们首先研究了序数算术的一些非平凡特性，然后使我们能够自然而优雅地将正常的模态逻辑的语义扩展到新颖的转移模态逻辑上，同时仍然使Kripke模型的普通定义完全完整。尽管有所有的直接数学定义，我们认为在实践中，这种逻辑实际上也可以适合完全有限的解释。我们建议，传播模态逻辑以一种相当清晰，简单的形式捕获了贝叶斯推理的本质，它为夏洛克·福尔摩斯（Sherlock Holmes）的著名说法提供了一个完美的解释：“当您消除了不可能的事情时，无论剩下的一切，无论多么不可能，都必须是真理。”我们还证明了我们逻辑的有限模型属性定理的对应物。
Bayesian reasoning plays a significant role both in human rationality and in machine learning. In this paper, we introduce transfinite modal logic, which combines modal logic with ordinal arithmetic, in order to formalize Bayesian reasoning semi-quantitatively. Technically, we first investigate some nontrivial properties of ordinal arithmetic, which then enable us to expand normal modal logic's semantics naturally and elegantly onto the novel transfinite modal logic, while still keeping the ordinary definition of Kripke models totally intact. Despite all the transfinite mathematical definition, we argue that in practice, this logic can actually fit into a completely finite interpretation as well. We suggest that transfinite modal logic captures the essence of Bayesian reasoning in a rather clear and simple form, in particular, it provides a perfect explanation for Sherlock Holmes' famous saying, "When you have eliminated the impossible, whatever remains, however improbable, must be the truth." We also prove a counterpart of finite model property theorem for our logic.
