Access-based Intuitionistic Knowledge

TL;DR

This paper introduces access-based intuitionistic knowledge, formalizing how agents know through proof access, extending classical modal logics, and unifying various approaches within a Heyting algebra framework.

Contribution

It develops a new formalization of intuitionistic knowledge based on proof access, extending existing modal logics and unifying different concepts in a Heyting algebra setting.

Findings

Formalizes access-based intuitionistic knowledge with proof access.

Extends classical modal logics with intuitionistic principles.

Provides a unifying semantic framework using Heyting algebra expansions.

Abstract

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.