The logic of awareness, first proposed by Fagin and Halpern, addressed the problem of logical omniscience by introducing the notion of awareness and distinguishing explicit knowledge from implicit knowledge. In their framework, explicit knowledge was defined as the conjunction of implicit knowledge and awareness, each of which was represented by modal operators. Their definition, however, may derive undesirable propositions that cannot be considered explicit knowledge when Modus Ponens is applied within implicit knowledge. Hence, focusing on indistinguishability among possible worlds, dependent on awareness, we refine the definition of explicit knowledge. In our semantics, we require that the aware implicit knowledge is not necessarily explicit knowledge, though explicit knowledge must be aware as well as implicit. We employ an example of elementary geometry, where different students may or may not reach the final answer, depending on whether they are aware of learned mathematical facts. Thereafter, we formally present the syntax and the semantics of our language, named Awareness-Based Indistinguishability Logic ($\mathrm{AIL}$). We prove that $\mathrm{AIL}$ has more expressive power than the logic of Fagin and Halpern, and show that the latter is embeddable in $\mathrm{AIL}$. Furthermore, we provide an axiomatic system of $\mathrm{AIL}$ and prove its soundness and completeness.

翻译：意识逻辑由Fagin与Halpern首次提出，通过引入意识概念并区分显性知识与隐性知识，以解决逻辑全知问题。在其框架中，显性知识被定义为隐性知识与意识的合取，二者分别由模态算子表示。然而，当在隐性知识内部应用假言推理时，该定义可能推导出不应被视为显性知识的命题。因此，我们聚焦于依赖意识的可能世界间的不可区分性，对显性知识的定义进行精炼。在我们的语义框架中，我们要求具有意识的隐性知识不必然成为显性知识，尽管显性知识必须同时具备意识性与隐性。我们采用初等几何学的示例进行说明：不同学生能否得出最终答案，取决于他们是否意识到已学习的数学事实。随后，我们正式提出我们语言——基于意识的不可区分性逻辑（$\mathrm{AIL}$）的语法与语义。我们证明$\mathrm{AIL}$比Fagin与Halpern的逻辑具有更强的表达能力，并证明后者可嵌入$\mathrm{AIL}$中。此外，我们构建了$\mathrm{AIL}$的公理系统，并证明了其可靠性与完备性。