We introduce the concept of access-based intuitionistic knowledge which relies on the intuition that agent $i$ knows $\varphi$ if $i$ has found access to a proof of $\varphi$. Basic principles are distribution and factivity of knowledge as well as $\square\varphi\rightarrow K_i\varphi$ and $K_i(\varphi\vee\psi) \rightarrow (K_i\varphi\vee K_i\psi)$, where $\square\varphi$ reads `$\varphi$ is proved'. The formalization extends a family of classical modal logics designed in [Lewitzka 2015, 2017, 2019] as combinations of $IPC$ and $CPC$ and as systems for the reasoning about proof, i.e. intuitionistic truth. We adopt a formalization of common knowledge from [Lewitzka 2011] and interpret it here as access-based common knowledge. We compare our proposal with recent approaches to intuitionistic knowledge [Artemov and Protopopescu 2016; Lewitzka 2017, 2019] and bring together these different concepts in a unifying semantic framework based on Heyting algebra expansions.

中文翻译：

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基于访问的直觉知识

我们引入了基于访问的直觉知识的概念，它依赖于这样一种直觉，即如果 $i$ 已经找到了 $\varphi$ 的证明，则代理 $i$ 知道 $\varphi$。基本原则是知识的分布和事实性以及 $\square\varphi\rightarrow K_i\varphi$ 和 $K_i(\varphi\vee\psi) \rightarrow (K_i\varphi\vee K_i\psi)$，其中 $\ square\varphi$ 读作“$\varphi$ 被证明”。形式化扩展了在 [Lewitzka 2015, 2017, 2019] 中设计的一系列经典模态逻辑，作为 $IPC$ 和 $CPC$ 的组合以及用于证明推理的系统，即直觉真理。我们采用了 [Lewitzka 2011] 中的公共知识的形式化，并将其解释为基于访问的公共知识。我们将我们的提议与最近的直觉知识方法进行比较 [Artemov and Protopopescu 2016;