Abstract
This paper investigates the relationships between three different views of common knowledge: the iterate approach, the fixed point approach, and the shared environment approach. We show that no two of these approaches are equivalent, contrary to accepted wisdom. We agree that the fixed point is the best conceptual analysis of the pretheoretic notion, but that the shared environment approach has its own role to play in understanding how common knowledge is used.
We also discuss the assumption under which various versions of the iterate approach are equivalent to the fixed point approach. We find that, for common knowledge, these assumptions are false, but that for simply having information, the assumptions are not so implausible, at least in the case of finite situations.
Jon Barwise was deceased at the time of publication.
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Notes
- 1.
F. Dretske, Knowledge and the Flow of Information (Cambridge, Mass.: Bradford Books/MIT Press, 1981).
- 2.
Obviously I have not read all, or even most, of the papers on common knowledge, so it could be that some or all of the points made in this paper are made elsewhere. If so, I would appreciate learning about it. But even if this is so, I am reasonably sure that the particular model I develop below is original, depending as it does on recent work in set theory by Peter Aczel.
- 3.
David Lewis, Convention, A Philosophical Study (Cambridge, Mass.: Harvard University Press, 1969).
- 4.
See, for example, the paper by Halpern and Moses, “Knowledge and common knowledge in distributed environments,” Proc. 3rd ACM Symp. on Principles of Distributed Computing (1984), 50–61, and the paper by Fagin, Halpern and Vardi, “A model-theoretic analysis of knowledge: preliminary report,” Proc. 25th IEEE Symposium on Foundations of C.S., 268–278.
- 5.
See Gilbert Harman’s review of Linguistic Behavior by Jonathan Bennett, Language 53 (1977): 417–24.
- 6.
R. J. Aumann, “Agreeing to disagree,” Annals of Statistics, 4 (1976), 1236–1239, and the working paper “On Aumann’s Notion of Common Knowledge – An alternative approach,” Tan and Ribeiro da Costa Werlang. University of Chicago Graduate School of Business, 1986.
- 7.
H. Clark and C. Marshall, “Definite reference and mutual knowledge,” in Elements of Discourse Understanding, ed. A. Joshi, B. Webber, and I. Sag (Cambridge: Cambridge University Press, 1981), 10–63.
- 8.
Op. Cit.
- 9.
See ch. 2 of Fred Dretske, Seeing and Knowing (Chicago: University of Chicago Press, 1969); or J. Barwise, “Scenes and other Situations”, Journal of Philosophical Logic 78 (1981): 369–97; or ch. 8 of J. Barwise and J. Perry, Situations and Attitudes (Cambridge, Mass.: Bradford Books/MIT Press, 1983).
- 10.
See ch. 3 of Seeing and Knowing or ch. 9 of Situations and Attitudes.
- 11.
J. Barwise and J. Etchemendy, The Liar: An Essay on Truth and Circularity (New York: Oxford University Press, 1987).
- 12.
P. Aczel, Non-well-founded Sets (CSLI Lecture Notes (Chicago: University of Chicago Press, 1987 (to appear)).
- 13.
For the reader unfamiliar with two card stud poker, here is all you need to know to follow the example. First each player is dealt one card which only he is allowed to see, and there is a round of betting. Then each player is dealt one card face up on the table and there is another round of betting. Hands are ranked and players bet if they think their hand is best. But they can also drop out of the round at any point. After both rounds of betting are over, the hands are displayed, so that all players can see who won. As far as the ranking, all that matters is that a hand with a matching pair is better than a hand with no pairs. But among hands with no pairs, a hand with an ace is better than a hand with no ace.
- 14.
In order to keep this paper within bounds, I am restricting attention only to positive, nondisjunctive facts.
- 15.
In more recent joint work with Aczel, a generalization of this relation takes center stage.
- 16.
Prashant Parikh, “Language and strategic inference,” Ph.D. Dissertation, Stanford University, 1987.
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Barwise, J. (2016). Three Views of Common Knowledge. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_37
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