Abstract
Artemov and Protopopescu (2018) introduced a Brouwer-Heyting-Kolmogorov (BHK) interpretation of knowledge operator to define the intuitionistic epistemic logic IEL, where the axiom \(A\supset KA\) is accepted but the axiom \(KA\supset A\) is refused. This paper studies the notion of distributed knowledge on an expansion of the multi agent variant of IEL. We provide a BHK interpretation of distributed knowledge operator to define the intuitionistic epistemic logic with distributed knowledge DIEL. We construct Hilbert system and cut-free sequent calculus for \(\mathbf {DIEL}\) and show that they are sound and complete for the intended Kripke semantics.
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- 1.
To derive \( D_{\{a\}} A \supset \lnot \lnot A \) in \(\mathcal {H}\mathbf {(DIEL)}\), i.e., the extension of \(\mathcal {H}(\mathbf {DIEL}^-)\) by the axiom \(\lnot D_{\{a\}} \bot \), it is noted that the following are derivable in Hilbert system \(\mathcal {H}\mathbf {(DIEL)}\) : \((D_{\{a\}} A \wedge \lnot A )\supset (D_{\{a\}} A \wedge D_{\{a\}}\lnot A )\) and \((D_{\{a\}} A \wedge D_{\{a\}}\lnot A ) \supset D_{\{a\}}( A \wedge \lnot A )\). Thus, \(\mathcal {H}\mathbf {(DIEL)}\vdash (D_{\{a\}} A \wedge \lnot A )\supset D_{\{a\}} \bot \). Since \(\lnot D_{\{a\}} \bot \) holds in the extension \(\mathcal {H}\mathbf {(DIEL)}\), we have \(\mathcal {H}\mathbf {(DIEL)}\vdash (D_{\{a\}} A \wedge \lnot A ) \supset \bot \). This gives us \(\mathcal {H}\mathbf {(DIEL)}\vdash D_{\{a\}} A \supset \lnot \lnot A\), as desired. Conversely, we derive \(\lnot D_{\{a\}} \bot \) in the extension of \(\mathcal {H}(\mathbf {DIEL}^-)\) by the axiom \( D_{\{a\}} A \supset \lnot \lnot A\). This is easy by taking \(\bot \) as A in the axiom.
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Acknowledgment
We would like to thank the reviewers for their helpful comments. The work of the first author was supported by JSPS KAKENHI Grant Number JP 20J11427. The work of the second author was supported by JSPS KAKENHI Grant Number JP 21J10573. The work of the third author was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 17H02258 and (C) Grant Number 19K12113.
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Su, Y., Murai, R., Sano, K. (2021). On Artemov and Protopopescu’s Intuitionistic Epistemic Logic Expanded with Distributed Knowledge. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_18
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