Access-based intuitionistic knowledge

Abstract

We introduce the concept of $\textit{access-based}$ intuitionistic knowledge which relies on the intuition that agent $i$ knows $\varphi$ if $i$ has found $\textit{access to a proof}$ 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 (Lewitzka, 2017, Journal of Logic and Computation, 27, 201--212) designed as combinations of $\mathit{IPC}$ and $\mathit{CPC}$ and as systems for the reasoning about proof, i.e. intuitionistic truth. We adopt a formalization of common knowledge from (Lewitzka, 2011, Studia Logica, 97, 233--264) and interpret it here as access-based common knowledge. We compare our proposal with recent approaches to intuitionistic knowledge (Artemov and Protopopescu, 2016, The Review of Symbolic Logic, 9, 266--298; Lewitzka, 2019, Annals of Pure and Applied Logic, 170, 218--250) and bring together these different concepts in a unifying semantic framework based on Heyting algebra expansions.