 Definition 5.2  Let a set \(\Omega\) of possible worlds together with a set of agents \(N\) be given.   (1) The proposition that \(A\) is (first level or first order) mutual p-belief for the agents of \(N, \mathbf{B}^p_{N^1}(A),\) is the set defined by   \[ \mathbf{B}^p_{N^1}(A) \equiv \bigcap_{i\in N} \mathbf{B}^p_i(A). \]   (2) The proposition that \(A\) is \(m\)th level (or \(m\)th order) mutual\(p\)-belief among the agents of \(N, \mathbf{B}^p_{N^m}(A),\) is defined recursively as the set   \[ \mathbf{B}^p_{N^m}(A) \equiv \bigcap_{i\in N} \mathbf{B}^p_i (\mathbf{B}^p_{N^{m-1}}(A)) \]   (3) The proposition that \(A\) is common p-belief among the agents of \(N, \mathbf{B}^p_{N^*}(A),\) is defined as the set  \[ \mathbf{B}^p_{N^*}(A) \equiv \bigcap_{m=1}^{\infty} \mathbf{B}^p_{N^m}(A). \]  