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Then we pay more attention to the logic of proofs $LP$ (S.N. Artemov, 1995), which started an investigation in this area. We will prove that the modal analog of $LP$ is the modal logic $S4$ and describe translation of propositional intuitionistic logic in $LP$. Then we supply $LP$ with the arithmetical semantics in which propositional variables stand for arithmetical sentences and justification terms denote codes of arithmetical derivations. We will prove that $LP$ is sound and complete with respect to this semantics. Arithmetical completeness of $LP$ together with the realization results for $S4$ and intuitionistic logic enables us to describe arithmetical semantics for them. Another point of our interest is first order justification logic, for which we describe language and semantics and prove realization theorems for the appropriate first order modal logics. Also we will discuss possible ways of arithmetical reading for first order justification logic. RSS: Forthcoming seminars
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