Proof of Proposition 2.5: A Supplement to Common Knowledge

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Supplement to Common Knowledge

Proof of Proposition 2.5

Proposition 2.5.
omega

K^m_N(A) iff

(1) For all agents i₁, i₂, … , i_m N, K_i₁K_i₂ … K_{i_m}(A)

Hence,

K^*_N(A) iff (1) is the case for each m greater than or equal to

Proof.
Note first that

( 2 )

^{i₁ N}

K_i₁ (

^{i₂ N}

K_i₂ ( . . . (

^{i_{m - 1} N}

K_{i_{m - 1}} (

^{i_m N}

K_{i_m}( A ) ) ) ) )

^{i₁ N}

K_i₁ (

^{i₂ N}

K_i₂ ( . . . (

^{i_{m - 1} N}

K_{i_{m - 1}} (K¹_N ( A ) ) ) ) )

^{i₁ N}

K_i₁ (

^{i₂ N}

K_i₂ . . . (

^{i_{m - 2} N}

K_{i_{m - 2}} ( K²_N ( A ) ) ) )

= . . .

^{i₁ N}

K_i₁( K^{m - 1}_N ( A ) )

K^m_N ( A )

By ( 2 ), K^m_N ( A ) subset K_i₁K_i₂ . . . K_{i_m}( A ) for i₁, i₂, . . ., i_m N so if omega K^m_N ( A ) then condition ( 1 ) is satisfied. Condition ( 1 ) is equivalent to

^{i₁ N} K_i₁ (
^{i₂ N} K_i₂ ( . . . (
^{i_{m - 1} N} K_{i_{m - 1}} (
^{i_m N} K_{i_m} ( A ) ) ) ) )

so by ( 2 ), if ( 1 ) is satisfied then omega K^m_N ( A ). QED

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Peter Vanderschraaf
peterv@MAIL1.ANDREW.CMU.EDU

Stanford Encyclopedia of Philosophy Supplement to Common Knowledge

Proof of Proposition 2.5

Supplement to Common Knowledge Stanford Encyclopedia of Philosophy

Stanford Encyclopedia of Philosophy
Supplement to Common Knowledge

Supplement to Common Knowledge
Stanford Encyclopedia of Philosophy