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Common knowledge and the knowledge account of assertion. (English) Zbl 1429.03072
Yang, Syraya Chin-Mu (ed.) et al., Structural analysis of non-classical logics. The proceedings of the second Taiwan philosophical logic colloquium, TPLC 2014, Taipei, Taiwan, October 24–25, 2014. Berlin: Springer. Log. Asia: Stud. Log. Libr., 253-278 (2016).
Summary: In this chapter, I present the assertion account of common knowledge in the framework of a multi-agent system for the epistemic logic of knowledge and assertion: the propositional content of a formula $$\varphi$$ is common knowledge to a group of agents G iff everyone in G knows that $$\varphi$$ is true and that $$\varphi$$ is asserted. Three current accounts of common knowledge, including the iterated account, the fixed-point account, and shared environment approach, will be examined. I argue that common knowledge arises from communication which results from overtly observable interactions among agents in a group. I then propose that assertion plays a substantial role in communication, and a fortiori, in the acquisition of common knowledge, given the knowledge account of assertion – one must assert $$\varphi$$ only if one knows $$\varphi$$. I point out some semantic implications of the knowledge account of assertion in multi-agent systems, specifically, the transmission of individual knowledge to others, the transition of individual knowledge to common knowledge, and the luminosity of common knowledge. The assertion account of common knowledge is then proposed and justified by a class of Kripke models (referred to as TWC-models) appropriate for a multi-agent system of epistemic logic of common knowledge and assertion. The construction of TWC-models will be specified, and the related semantic rules will be given.
For the entire collection see [Zbl 1331.03012].
##### MSC:
 03B42 Logics of knowledge and belief (including belief change) 03A05 Philosophical and critical aspects of logic and foundations
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##### References:
 [1] Aumann, R.J.: Agreeing to disagree. Annuals Stat. 4, 1236-1239 (1976) · Zbl 0379.62003 [2] Baltag, A., Moss, L.S., Solecki, S.: The logic of public announcements, common knowledge, and private suspicions. In: TARK’98: Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 43-56. Morgan Kaufmann Publishers, San Francisco (1998) · Zbl 1386.03019 [3] Barwise, J.: Three views of common knowledge. In: TARK’88: Proceedings of the 2nd Conference on Theoretical Aspects of Reasoning about Knowledge, pp. 365-379. Morgan Kaufmann Publishers, San Francisco (1988) · Zbl 0704.03008 [4] Clark, H.H., Marshall, C.R.: Definite reference and mutual knowledge. In: Joshi, A.K., Webber, B.L., Sag, I.A. (Eds.) Elements of Discourse Understanding, pp. 10-63. Cambridge University Press, Cambridge (1981) [5] Davidson, D.: First person authority. In: Subjective, Intersubjective, Objective: Philosophical Essays Vol. 3, pp. 3-14. Clarendon Press, Oxford (1984/2001) [6] Davidson, D.: Knowing one’s own mind. In: Subjective, Intersubjective, Objective: Philosophical Essays Vol. 3, pp. 15-38. Clarendon Press, Oxford (1987/2001) [7] Davidson, D.: Three varieties of knowledge. In: Subjective, Intersubjective, Objective: Philosophical Essays Vol. 3, pp. 205-220. Clarendon Press, Oxford (1988/2001) [8] Davidson, D.: Subjective, intersubjective, objective—Philosophical essays, vol. 3. Clarendon Press, Oxford (2001) [9] Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about knowledge. MIT Press, Cambridge, Mass. (1995) · Zbl 0839.68095 [10] Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Common knowledge revisited. In: Hendricks, V.F., Jørgensen, K.F., Pedersen, S.A. (Eds.) Knowledge Contributors, pp. 87-104. Kluwer, Dordrecht (2003) · Zbl 1060.03008 [11] Gullvåg, I.: The logic of assertion. Theoria 44, 75-116 (1978) · Zbl 0418.03006 [12] Gupta, A., Belnap, N.D.: The revision theory of truth. MIT Press, Cambridge, Mass. (1993) · Zbl 0858.03010 [13] Halpern, J.Y., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. Assoc. Comput. Mach. 37, 549-587 (1990) · Zbl 0699.68115 [14] Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54, 319-379 (1992) · Zbl 0762.68029 [15] Hendricks, V.F., Jørgensen, K.F., Pedersen, S.A. (Eds.): Knowledge Contributors. Kluwer, Dordrecht (2003) [16] Hendricks, V.F., Symons, J.: Where’s the bridge? Epistemology and epistemic logic. Philos. Stud. 128, 137-167 (2006) [17] Hintikka, J.: Knowledge and belief: An introduction to the logic of the two notions. Cornell University Press, Ithaca (1962) [18] Lewis, D.: Convention: A philosophical study. Blackwell, Oxford (1969) [19] Lismont, L., Mongin, P.: On the logic of common belief and common knowledge. Theor. Decis. 37, 75-106 (1994) · Zbl 0831.03014 [20] Lismont, L., Mongin, P.: Belief closure: A semantics of common knowledge for modal propositional logic. Math. Soc. Sci. 30, 127-153 (1995) · Zbl 0886.90063 [21] Mertens, J.-F., Zamir, S.: Formulation of Bayesian analysis for games with incomplete information. Int. J. Game Theor. 14, 1-29 (1985) · Zbl 0567.90103 [22] Milgrom, P.: An axiomatic characterization of common knowledge. Econometrica 49, 219-222 (1981) · Zbl 0453.90028 [23] Monderer, D., Samet, D.: Approximating common knowledge with common beliefs. Game Econ. Behav. 1, 170-190 (1989) · Zbl 0755.90110 [24] Rescher, N.: Topics in philosophical logic. Kluwer D. Reidel, Dordrecht (1968) · Zbl 0175.26402 [25] Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5, 285-309 (1955) · Zbl 0064.26004 [26] van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic epistemic logic. Springer, Amsterdam (2008) · Zbl 1156.03015 [27] Williamson, T.: Knowing and asserting. Philos. Rev. 105, 489-523 (1996) [28] Williamson, T.: Knowledge and its limits. Oxford University Press, Oxford (2000) [29] Yang, S.C.-M.: TW-models for logic of knowledge-cum-belief. In: Downey, R., Brendle, J., Goldblatt, R., Kim, B. (Eds.) The Proceedings of 12th Asian Logic Conference, pp. 314-337. World Scientific Publishing Co., Singapore (2013)
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