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The generation of formulas held in common knowledge

Author

Listed:
  • Robert Samuel Simon

    (Institut fuer Mathematische Stochastik, Lotze str. 13, 37083 Goettingen, GERMANY)

Abstract

Common knowledge of a finite set of formulas implies a special relationship between syntactic and semantic common knowledge. If S, a set of formulas held in common knowledge, is implied by the common knowledge of some finite subset of S, and A is a non-redundant semantic model where exactly S is held in common knowledge, then the following are equivalent: (a) S is maximal among the sets of formulas that can be held in common knowledge, (b) A is finite, and (c) the set S determines A uniquely; otherwise there are uncountably many such A. Even if the knowledge of the agents are defined by their knowledge of formulas, 1) there is a continuum of distinct semantic models where only the tautologies are held in common knowledge and, 2) not assuming that S is finitely generated (a) does not imply (c), (c) does not imply (a), and (a) and (c) together do not imply (b).

Suggested Citation

  • Robert Samuel Simon, 2001. "The generation of formulas held in common knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(1), pages 1-18.
    • Handle: RePEc:spr:jogath:v:30:y:2001:i:1:p:1-18
    Note: Received November 1999/Revised version January 2000
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