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Access-based intuitionistic knowledge. (English) Zbl 07332115
Summary: 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 (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.
##### MSC:
 03-XX Mathematical logic and foundations 68-XX Computer science
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