Access-based Intuitionistic Knowledge
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We introduce the concept of access-based intuitionistic knowledge which relies on the intuition that agent i knows φ if i has found access to a proof of φ. Basic principles are distribution and factivity of knowledge as well as □φ→ K_iφ and K_i(φ∨ψ) → (K_iφ∨ K_iψ), where □φ reads `φ 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.
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