
## 3 The Generalized Pcr Rules

In the equation (8), the PCR5 is given for two experts only. Two extensions for three experts and two classes are given in [17], and the equation for M **experts,**
for X ∈ DΘ, X 6≡ ∅ **is given in [4] by:**

$$m_{\text{PCRS}}(X)=m_{\text{c}}(X)+$$ $$\sum_{i=1}^{M}m_{i}(X)\sum_{\begin{subarray}{c}Y_{\sigma_{i}(j)}\in\{X\}\\ M\cap Y_{\sigma_{i}(M-1)}\in\{D^{\Theta}\}\end{subarray}}\frac{\left(\prod_{j=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})\mathds{1}_{j>i}\right)\prod_{Y_{\sigma_{i}(j)}=X}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})}{\sum_{\begin{subarray}{c}Y_{\sigma_{i}(j)}=X\\ M\cap Y_{\sigma_{i}(M)}\cap X=\emptyset\end{subarray}}\left(\prod_{j=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)}).T(x_{\sim\mathbb{Z},m_{i}(X)})\right)},\tag{15}$$

where σi counts from 1 to M **avoiding** i:

$$\left\{\begin{array}{l l}{\sigma_{i}(j)=j}&{{\mathrm{if~}}j<i,}\\ {\sigma_{i}(j)=j+1}&{{\mathrm{if~}}j\geq i,}\end{array}\right.$$

and:

$\left\{\begin{array}{l l}T(B,x)=x&\text{if}B\text{is true,}\\ T(B,x)=1&\text{if}B\text{is false.}\end{array}\right.$
We propose another generalization of the equation (8) for M **experts, for**
X ∈ DΘ, X 6= ∅:

$$\begin{array}{r l}{{}}&{{=}}\quad m_{\mathrm{c}}(X)+}\\ {{}}&{{}}\\ {{\sum_{i=1}^{M}m_{i}(X)^{2}\sum_{\stackrel{M-1}{\prod_{k=1}^{M-1}Y_{\sigma_{i}(k)}\cap X\equiv\emptyset}}}}\\ {{}}&{{}}\\ {{\quad(Y_{\sigma_{i}(1)},\ldots,Y_{\sigma_{i}(M-1)})\in(D^{\circledcirc})^{M-1}}}\end{array}\quad\quad\left({\frac{\prod_{j=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})}{\prod_{m_{i}(X)+\sum_{j=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})}}}\right),$$  and all $m_{i}(M)$ are $(M,\alpha)$. 
$$\left(16\right)$$

where σ **is defined like in (16).**
mi(X) +
M
X−1 j=1 mσi(j)(Yσi (j)) 6= 0, mc **is the conjunctive consensus rule given**
by the equation (5).

We can propose two more generalized rules given by:

$$\begin{array}{r c l}{{m_{\mathrm{PCR6f}}(X)}}&{{=}}&{{m_{\mathrm{c}}(X)+}}\\ {{}}&{{}}&{{}}\\ {{\sum_{i=1}^{M}m_{i}(X)f(m_{i}(X))\sum_{\stackrel{M-1}{k=1}Y_{\sigma_{i}(k)}\cap X\equiv\emptyset}}}&{{\left(\frac{\prod_{j=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})}{f(m_{i}(X))+\sum_{j=1}^{M-1}f(m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)}))}\right)}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{(Y_{\sigma_{i}(1)},\ldots,Y_{\sigma_{i}(M-1)})\in(D^{\alpha})^{M-1}}}\end{array}\right),}}\end{array}$$
