Access-based Intuitionistic Knowledge

Abstract

We introduce the concept of access-based intuitionistic knowledge which relies on the intuition that agent $i$ knows $\phi$ if $i$ has found access to a proof of $\phi$. Basic principles are distribution and factivity of knowledge as well as $\square \phi \to K_i\phi$ and $K_i(\phi \vee \psi )\to (K_i\phi \vee K_i\psi )$, where $\square \phi$ reads `$\phi$ 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.

28 pages