Reasoning about proof and knowledge |
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Authors: | Steffen Lewitzka |
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Institution: | Instituto de Matemática e Estatística, Departamento de Ciência da Computação, Universidade Federal da Bahia UFBA, 40170-110 Salvador, BA, Brazil |
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Abstract: | In previous work 15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in 13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL 4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems. |
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Keywords: | 03B45 03B42 03F45 03G10 Modal logic Epistemic logic Intuitionistic logic Proof predicate BHK interpretation Relational semantics |
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