Reasoning about proof and knowledge
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In previous work [Lewitzka, 2017], we presented a hierarchy of classical modal logics, along with algebraic semantics, for the reasoning about intuitionistic truth (i.e. proof), belief and knowledge. Interpreting as a proof predicate, the systems also express properties of intuitionistic belief and knowledge established in [Artemov and Protopopescu, 2016] where epistemic principles are in line with Brouwer-Heyting-Kolmogorov (BHK) semantics. In this article, we further develop our approach and show that the S5-style systems of our hierarchy are complete w.r.t. a relational semantics based on intuitionistic general frames. This result can be seen as a formal justification of our modal axioms as adequate principles for the reasoning about proof combined with belief and knowledge. In fact, the semantics turns out to be a uniform framework able to describe also the intuitionistic epistemic logics of [Artemov and Protopopescu, 2016]. The relationship between intuitionistic epistemic principles and their representation by modal laws of our classical logics becomes explicit.
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