Abstract
Hilbert style, Gentzen style sequent and Kanger style sequent calculi for logic S5n(ED) are considered in this paper. Gentzen style sequent calculus is constructed and its equivalence with Hilbert style system is proved, getting soundness and completeness of Gentzen style system. Kanger style indexed sequent calculus is defined for cut elimination.
Highlights
S5n(ED) is one of the epistemic logics
Suggestions of proof theory of epistemic logic are usually limited to Hilbert style axiomatizations [2]
Several works were done by Regimantas Pliuškevičius and Aida Pliuškevičienė in [7], and Raul Hakli and Sara Negri in [2], proposing cut free Gentzen style sequent calculi for S4n(D) and S5n(D)
Summary
S5n(ED) is one of the epistemic logics. Epistemic logic is the logic of knowledge and belief. Suggestions of proof theory of epistemic logic are usually limited to Hilbert style axiomatizations [2]. Aida Pliuškevičienė in [6] presented indexed sequent calculus, which is known as Kanger style, for logic S52. Our work is strongly related to [6] and describes Kanger style sequent calculus for logic S5n(ED). We define syntax of S5n(ED) in Section 1, prove equivalence between Hilbert style calculus HS-S5n(ED) and Gentzen style sequent calculus GS-S5n(ED) in Sections 2 and 3, to get soundness and completeness of GS-S5n(ED).
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