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The difference between common knowledge of formulas and sets

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  • Robert Samuel Simon

    (Institute for Mathematical Stochastics, Lotzestr. 13, D-37083 GÃttingen, Germany)

Abstract

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.

Suggested Citation

  • Robert Samuel Simon, 1999. "The difference between common knowledge of formulas and sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 367-384.
    • Handle: RePEc:spr:jogath:v:28:y:1999:i:3:p:367-384
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    Cited by:

    1. Robert Samuel Simon, 2001. "On the Unique Extensibility and Surjectivity of Knowledge Structures," Discussion Paper Series dp274, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

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