This paper establishes a dual theory about knowledge and argumentation. Our
idea is rooted at both epistemic logic and argumentation theory, and we aim to
merge these two fields, not just in a superficial way but to thoroughly
disclose the intrinsic relevance between knowledge and argumentation.
Specifically, we define epistemic Kripke models and argument Kripke models as a
dual pair, and then work out a two-way generation method between these two
types of Kripke models. Such generation is rigorously justified by a duality
theorem on modal formulae's invariance. We also provide realistic examples to
demonstrate our generation, through which our framework's practical utility
gets strongly advocated. We finally propose a philosophical thesis that
knowledge is essentially dynamic, and we draw certain connection to Maxwell's
demon as well as the well-known proverb "knowledge is power".

该论文建立了有关知识和论证的双重理论，在表面上融合了认识逻辑和论证理论，通过使用双重的 Krinke 模型来证明这种联系，并提出了关于知识本质上是动态的哲学命题。

静态知识与动态论证：基于克里普克语义的双重理论

Static Knowledge vs. Dynamic Argumentation: A Dual Theory Based on Kripke Semantics

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.

本文介绍了一个半定量的方法 —— 用阶数算术来将模态逻辑系统和贝叶斯推理结合起来。最终的系统可以将传统模态逻辑语义自然地，优雅地映射到新的系统中。我们声称该逻辑系统可以被完全有限地表述，并提供了对 “除去不可能的，剩下的，无论多么不可能都是真的” 这一句话的完美解释。此外，还证明了该逻辑系统的一个有限模型特性定理。

无限模态逻辑：贝叶斯推理的半定量解释

Transfinite Modal Logic: a Semi-quantitative Explanation for Bayesian Reasoning

Recent years witness a growing interest in nonstandard epistemic logics of
"knowing whether", "knowing what", "knowing how", and so on. These logics are
usually not normal, i.e., the standard axioms and reasoning rules for modal
logic may be invalid. In this paper, we show that the conditional "knowing
value" logic proposed by Wang and Fan \cite{WF13} can be viewed as a disguised
normal modal logic by treating the negation of the Kv operator as a special
diamond. Under this perspective, it turns out that the original first-order
Kripke semantics can be greatly simplified by introducing a ternary relation
$R_i^c$ in standard Kripke models, which associates one world with two
$i$-accessible worlds that do not agree on the value of constant $c$. Under
intuitive constraints, the modal logic based on such Kripke models is exactly
the one studied by Wang and Fan (2013,2014}. Moreover, there is a very natural
binary generalization of the "knowing value" diamond, which, surprisingly, does
not increase the expressive power of the logic. The resulting logic with the
binary diamond has a transparent normal modal system, which sharpens our
understanding of the "knowing value" logic and simplifies some previously hard
problems.

本文研究了非标准知识逻辑及其关联的 Kripke 模型，并通过引进一个三元关系 $R_i^c$，将一阶 Kripke 模型简化，使得基于这些模型的模态逻辑正常化，并发现了 “knowing value” 钻石运算符的自然二元推广，从而更好地理解了这种逻辑。