Knowledge Means ‘All’, Belief Means ‘Most’
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Abstract
We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalized ‘majority’ quantifier, interpreted as truth in many (a ‘majority’ of the) epistemically alternative states. This doxastic interpretation is of interest in KR applications and it also has an independent philosophical and technical interest. The logic KBM comprises an S4 epistemic modal operator, a doxastic modal operator of consistent and complete belief and ‘bridge’ axioms which relate knowledge to belief. To capture the notion of a ‘majority’ we use the ‘large sets’ introduced independently by K. Schlechta and V. Jauregui, augmented with a requirement of completeness, which furnishes a ‘weak ultrafilter’ concept. We provide semantics in the form of possible-worlds frames, properly blending relational semantics with a version of general Scott-Montague (neighborhood) frames and we obtain soundness and completeness results. We examine the validity of certain epistemic principles discussed in the literature, in particular some of the ‘bridge’ axioms discussed by W. Lenzen and R. Stalnaker, as well as the ‘paradox of the perfect believer’, which is not a theorem of KBM.
Keywords
modal epistemic logic majorities large setsPreview
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References
- 1.Askounis, D., Koutras, C.D., Zikos, Y.: Knowledge means ‘all’, belief means most. Technical Report, draft version (May 2012), http://www.uop.gr/~ckoutras, http://users.uop.gr/%7Eckoutras/AKZ-KBM-Full.pdf
- 2.Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press (2001)Google Scholar
- 3.Bratman, M.: Practical reasoning and acceptance in a context. Mind 101(401), 1–16 (1992)CrossRefGoogle Scholar
- 4.Chagrov, A., Zakharyashev, M.: Modal Logic. Oxford Logic Guides, vol. 35. Oxford University Press (1997)Google Scholar
- 5.Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. Studies in Logic and the Foundations of Mathematics, vol. 73. North-Holland, Amsterdam (1990)zbMATHGoogle Scholar
- 6.Chellas, B.F.: Modal Logic, an Introduction. Cambridge University Press (1980)Google Scholar
- 7.Cohen, L.: An essay on Belief and Acceptance. Oxford University Press (1995)Google Scholar
- 8.Dubois, D., Welty, C.A., Williams, M.-A. (eds.): Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR 2004), Whistler, Canada, June 2-5. AAAI Press (2004)Google Scholar
- 9.Dǒsen, K.: Duality between modal algebras and neighbourhood frames. Studia Logica. 48(2), 219–234 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Engel, P.: Believing, holding true, and accepting. Philosophical Explorations: An International Journal for the Philosophy of Mind and Action 1(2), 140–151 (1998)MathSciNetCrossRefGoogle Scholar
- 11.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)Google Scholar
- 12.Gilbert, M.P.: Modeling collective belief. Synthese 73, 185–204 (1987)CrossRefGoogle Scholar
- 13.Gochet, P., Gribomont, P.: Epistemic logic. In: Gabbay and Woods [11], vol. 7, pp. 99–195 (2006)Google Scholar
- 14.Goldblatt, R.: Mathematical Modal Logic: A View of its Evolution. In: Gabbay and Woods [11], vol. 7, pp. 1–98 (2006)Google Scholar
- 15.Hakli, R.: Group beliefs and the distinction between belief and acceptance. Cognitive Systems Research 7(2-3), 286–297 (2006)CrossRefGoogle Scholar
- 16.Halpern, J.: Should knowledge entail belief? Journal of Philosophical Logic 25(5), 483–494 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Hintikka, J.: Knowledge and Belief: an Introduction to the Logic of the two notions. Cornell University Press, Ithaca (1962)Google Scholar
- 18.Jauregui, V.: Modalities, Conditionals and Nonmonotonic Reasoning. PhD thesis, Department of Computer Science and Engineering, University of New South Wales (2008)Google Scholar
- 19.Kracht, M., Wolter, F.: Normal modal logics can simulate all others. Journal of Symbolic Logic 64(1), 99–138 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
- 20.Lenzen, W.: Recent Work in Epistemic Logic. North-Holland (1978)Google Scholar
- 21.Lenzen, W.: Epistemologische Betrachtungen zu [S4,S5]. Erkenntnis 14, 33–56 (1979)CrossRefGoogle Scholar
- 22.Pacuit, E.: Neighborhood semantics for modal logic: an introduction. Course Notes for ESSLLI 2007 (2007)Google Scholar
- 23.Pacuit, E., Salame, S.: Majority logic. In: Dubois, et al. [8], pp. 598–605Google Scholar
- 24.Salame, S.: Majority Logic and Majority Spaces in contrast with Ultrafilters. PhD thesis, Graduate Center, City University of New York (2006)Google Scholar
- 25.Schlechta, K.: Defaults as generalized quantifiers. Journal of Logic and Computation 5(4), 473–494 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Schlechta, K.: Filters and partial orders. Logic Journal of the IGPL 5(5), 753–772 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Segerberg, K.: An essay in Clasical Modal Logic. Filosofiska Studies, Uppsala (1971)Google Scholar
- 28.Stalnaker, R.: On logics of knowledge and belief. Philosophical Studies 128(1), 169–199 (2006)MathSciNetCrossRefGoogle Scholar
- 29.Voorbraak, F.: As Far as I Know - Epistemic Logic and Uncertainty. PhD thesis, Department of Philosophy, Utrecht University (1993)Google Scholar