In epistemic logic, with knowledge operators ${\text{K}}_i$ corresponding to a set of agents, any formula that is known by all agents, that all agents know to be known by all agents, such that all agents know that all agents know to be known by all agents, etc. Formally, let the operators ${\text{K}}_i$ represent the knowledge of $n$ agents; then a common knowledge operator corresponds to the infinitary conjunction:

• • $\text{C}\phi =\phi \wedge {\bigwedge }_{i_0,\dots ,i_{m<n}}\left({\text{K}}_{i_0}...{\text{K}}_{i_m}\phi \right)$

That is, $\text{C}$ is the weakest modal operator such that $\text{C}\phi$ entails that for any sequence of knowledge operators ${\text{K}}_{i_0},...,{\text{K}}_{i_m}$, the formula ${\text{K}}_{i_0}...{\text{K}}_{i_m}\phi$ is true. Alternatively, if one defines an intermediate operator corresponding to ‘everyone knows’:

• • $\text{E}\phi ={\text{K}}_0\phi \wedge ...\wedge {\text{K}}_{n-1}\phi$

Then common knowledge captures the infinitary conjunction:

• • $\text{C}\phi =\phi \wedge \text{E}\phi \wedge ...\wedge \underset{n\text{times}}{\underbrace{\text{E}...\text{E}}}\phi \wedge ...$

Common knowledge of $\phi$ is stronger than the related notion of mutual knowledge of $\phi$, that is, that $\phi$ is known by all agents.