Common knowledge (logic): Difference between revisions

Revision as of 22:15, 7 April 2013

Revision as of 09:14, 11 September 2013

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By abbreviating the expression <math>E_GE_G^{n-1} \varphi</math> with <math>E_G^n \varphi</math> and defining <math>E_G^0 \varphi = \varphi</math>, we could then define common knowledge with the axiom

:<math>C \varphi \Leftrightarrow \bigwedge_{i = 10}^n\infty E^ni \varphi</math> with <math>n = 1, 2,\dots</math>

There is however a complication. The languages of epistemic logic are usually ''finitary'', whereas the [[axiom]] above defines common knowledge as an infinite conjunction of formulas, hence not a [[well-formed formula]] of the language. To overcome this difficulty, a ''fixed-point'' definition of common knowledge can be given. Intuitively, common knowledge is thought of as the fixed point of the "equation" <math>E_G (\varphi \wedge C_G \varphi)</math>. In this way, it is possible to find a formula <math>\psi</math> implying <math>E_G (\psi \wedge C_G \varphi)</math> from which, in the limit, we can infer common knowledge of <math>\varphi</math>.