Abstract
We introduce KBE, a modal epistemic logic for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent’s decision-making process. In our logic, Knowledge and Belief are captured by S4.2, a modal logic holding a distinguished position among the epistemic logics investigated in AI and Philosophy. The Estimation operator of KBE is a kind of generalized ‘many’ or ‘most’ quantifier, whose origins go back to the work of J. Burgess and A. Herzig, but its model-theoretic incarnation (‘weak filters’) has been introduced by K. Schlechta and V. Jauregui. We work with complete weak filters (‘weak ultrafilters’) as we are interested in situations where an estimation can be always reached. The axiomatization of KBE comprises ‘bridge’ axioms which reflect the intuitive relationship of ‘estimation’ to ‘knowledge’ and ‘belief’, several introspective properties are shown to hold and it comes out that believing ϕ can be equivalently defined in KBE as ‘estimating that ϕ is known’, an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The model theory of KBE comprises a class of frames combining relational Kripke frames with Scott-Montague semantics, in which neighborhoods are collections of ‘large’ sets of possible worlds. Soundness and completeness is mentioned and a tableaux proof procedure is sketched.
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References
Askounis, D., Koutras, C.D., Zikos, Y.: Knowledge means ‘all’, belief means ‘most’. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 41–53. Springer, Heidelberg (2012)
Aucher, G.: Principles of knowledge, belief and conditional belief. In: Rebuschi, M., Batt, M., Heinzmann, G., Lihoreau, F., Musiol, M., Trognon, A. (eds.) Dialogue, Rationality, and Formalism. Logic, Argumentation & Reasoning, vol. 3, Springer (2014)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. (53). Cambridge University Press (2001)
Burgess, J.P.: Probability logic. J. Symb. Log. 34(2), 264–274 (1969)
Carnielli, W.A., Sette, A.M.: Default operators. In: Workshop on Logic, Language, Information and Computation, WOLLIC 1994, UFPE, Recife (1994)
Carnielli, W.A., Veloso, P.A.S.: Ultrafilter logic and generic reasoning. In: Gottlob, et al. (eds.) [12], pp. 34–53
Chellas, B.F.: Modal Logic, an Introduction. Cambridge University Press (1980)
Fitting, M., Mendelsohn, R.L.: First-Order Modal Logic. Synthése Library, vol. 277. Kluwer Academic Publishers (1998)
Fitting, M.C.: Proof Methods for Modal and Intuitionistic Logics. D. Reidel Publishing Co., Dordrecht (1983)
Gabbay, D.M., Woods, J. (eds.): Logic and the Modalities in the Twentieth Century. Handbook of the History of Logic, vol. 7. North-Holland (2006)
Gochet, P., Gribomont, P.: Epistemic logic. Gabbay and Woods [10], vol. 7, pp. 99–195 (2006)
Gottlob, G., Leitsch, A., Mundici, D. (eds.): KGC 1997. LNCS, vol. 1289. Springer, Heidelberg (1997)
Halpern, J.: The relationship between knowledge, belief and certainty. Annals of Mathematics and Artificial Intelligence 4, 301–322 (1991)
Halpern, J., Samet, D., Segev, E.: Defining knowledge in terms of belief: The modal logic perspective. Review of Symbolic Logic (to appear)
Herzig, A.: Modal probability, belief, and actions. Fundamenta Informaticae 57(2-4), 323–344 (2003)
Hughes, G.E., Cresswell, M.J.: A New Introduction to Modal Logic. Routledge (1996)
Jauregui, V.: Modalities, Conditionals and Nonmonotonic Reasoning. PhD thesis, Department of Computer Science and Engineering, University of New South Wales (2008)
Kaminski, M., Tiomkin, M.L.: The modal logic of cluster-decomposable kripke interpretations. Notre Dame Journal of Formal Logic 48(4), 511–520 (2007)
Koutras, C.D., Moyzes, C., Nomikos, C., Zikos, Y.: On the ‘in many cases’ modality: tableaux, decidability, complexity, variants. In: Likas, A., Blekas, K., Kalles, D. (eds.) SETN 2014. LNCS, vol. 8445, pp. 207–220. Springer, Heidelberg (2014)
Koutras, C.D., Moyzes, C., Zikos, Y.: A modal logic of Knowledge, Belief and Estimation. Technical report, Graduate Programme in Algorithms and Computation (2014) (available through the authors’ webpages)
Koutras, C.D., Zikos, Y.: A note on the completeness of S4.2. Technical report, 2013, Graduate Programme in Logic, Algorithms and Computation (December 2013)
Lenzen, W.: Recent Work in Epistemic Logic. North-Holland (1978)
Lenzen, W.: Epistemologische Betrachtungen zu [S4,S5]. Erkenntnis 14, 33–56 (1979)
Pacuit, E.: Dynamic epistemic logic I: Modeling knowledge and belief. Philosophy Compass 8(9), 798–814 (2013)
Schlechta, K.: Defaults as generalized quantifiers. Journal of Logic and Computation 5(4), 473–494 (1995)
Stalnaker, R.: On logics of knowledge and belief. Philosophical Studies 128(1), 169–199 (2006)
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Koutras, C.D., Moyzes, C., Zikos, Y. (2014). A Modal Logic of Knowledge, Belief, and Estimation. In: Fermé, E., Leite, J. (eds) Logics in Artificial Intelligence. JELIA 2014. Lecture Notes in Computer Science(), vol 8761. Springer, Cham. https://doi.org/10.1007/978-3-319-11558-0_47
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DOI: https://doi.org/10.1007/978-3-319-11558-0_47
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