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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 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.
- 2.
References
Ågotnes, T., Wáng, Y.N.: Resolving distributed knowledge. Artif. Intell. 252, 1–21 (2017). https://doi.org/10.1016/j.artint.2017.07.002
Ågotnes, T., Wáng, Y.N.: Group belief. In: Dastani, M., Dong, H., van der Torre, L. (eds.) CLAR 2020. LNCS (LNAI), vol. 12061, pp. 3–21. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-44638-3_1
Artemov, S., Protopopescu, T.: Intuitionistic epistemic logic. Rev. Symb. Logic 9, 266–298 (2016). https://doi.org/10.1017/S1755020315000374
Blackburn, P., Rijke, M.d., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press (2001). https://doi.org/10.1017/CBO9781107050884
Dummett, M.A., Lemmon, E.J.: Modal logics between S4 and S5. Math. Log. Q. 5(14–24), 250–264 (1959)
Fagin, R., Halpern, J.Y., Vardi, M.Y.: What can machines know? On the properties of knowledge in distributed systems. J. ACM 39, 328–376 (1996)
Fagin, R., Moses, Y., Halpern, J.Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (2003)
Gabbay, D.M.: Semantic proof of the Craig interpolation theorem for intuitionistic logic and extensions. In: Gandy, R., Yates, C. (eds.) Logic Colloquium 1969, Studies in Logic and the Foundations of Mathematics, vol. 61, pp. 391–410. Elsevier (1971). https://doi.org/10.1016/S0049-237X(08)71239-4
Gerbrandy, J.: Bisimulations on planet Kripke. Ph.D. thesis, University of Amsterdam (1999)
Giedra, H.: Cut free sequent calculus for logic \({S5_n (ED)}\). Lietuvos matematikos rinkinys 51, 336–341 (2010)
Hakli, R., Negri, S.: Proof theory for distributed knowledge. In: Sadri, F., Satoh, K. (eds.) CLIMA 2007. LNCS (LNAI), vol. 5056, pp. 100–116. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-88833-8_6
Hermant, O.: Semantic cut elimination in the intuitionistic sequent calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 221–233. Springer, Heidelberg (2005). https://doi.org/10.1007/11417170_17
van der Hoek, W., van Linder, B., Meyer, J.J.: Group knowledge is not always distributed (neither is it always implicit). Math. Soc. Sci. 38(2), 215–240 (1999)
Jäger, G., Marti, M.: A canonical model construction for intuitionistic distributed knowledge. In: Advances in Modal Logic, vol. 11, pp. 420–434. College Publications (2016)
Krupski, V.N., Yatmanov, A.: Sequent calculus for intuitionistic epistemic logic IEL. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 187–201. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-27683-0_14
Murai, R., Sano, K.: Craig interpolation of epistemic logics with distributed knowledge. In: Herzig, A., Kontinen, J. (eds.) FoIKS 2020. LNCS, vol. 12012, pp. 211–221. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39951-1_13
Murai, R., Sano, K.: Intuitionistic epistemic logic with distributed knowledge. In: Procedia Computer Science. Elsevier (to appear)
Ono, H.: Craig’s interpolation theorem for the intuitionistic logic and its extensions: a semantical approach. Studia Logica, 19–33 (1986)
Pliuskevicius, R., Pliuskeviciene, A.: Termination of derivations in a fragment of transitive distributed knowledge logic. Informatica Lith. Acad. Sci. 19, 597–616 (2008)
Protopopescu, T.: Three essays in intuitionistic epistemology. Ph.D. thesis. The Graduate Center, City University of New York (2016)
Roelofsen, F.: Distributed knowledge. J. Appl. Non-Classical Logics 17(2), 255–273 (2007)
Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logic. In: Studies in Logic and the Foundations of Mathematics, vol. 82, pp. 110–143. Elsevier (1975)
Simpson, A.K.: The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis. University of Edinburgh (1994)
Su, Y., Sano, K.: First-order intuitionistic epistemic logic. In: Blackburn, P., Lorini, E., Guo, M. (eds.) LORI 2019. LNCS, vol. 11813, pp. 326–339. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-60292-8_24
Su, Y., Sano, K.: Cut-free and analytic sequent calculus of intuitionistic epistemic logic. In: The Logica Yearbook 2019, pp. 179–192. College Publications, London (2020)
Troelstra, A.S.: History of Constructivism in the 20th Century. Lecture Notes in Logic, pp. 150–179, Cambridge University Press (2011). https://doi.org/10.1017/CBO9780511910616.009
Wáng, Y.N., Ågotnes, T.: Simpler completeness proofs for modal logics with intersection. In: Martins, M.A., Sedlár, I. (eds.) Dynamic Logic. New Trends and Applications - Third International Workshop. LNCS, vol. 12569, pp. 259–276. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-65840-3_16
Wolter, F., Zakharyaschev, M.: Intuitionistic modal logics as fragments of classical bimodal logics. In: Logic At Work, pp. 168–186 (1997)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-88708-7_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-88707-0
Online ISBN: 978-3-030-88708-7
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)