I am trying to decipher page 68 of Hermes' book on computability theory. One paragraph I am having trouble with is
Let $Q$ be an n-ary predicate ($n \geq 2$). Let $1 \leq i < k \leq n$. Then $P$ [a predicate] is called the $(i, k)$ identification of $Q$, if $P$ is $(n-1)$-ary and if for all $x_1, \dots, x_{k-1}, x_{k+1}, \dots, x_n$, $P x_1 \dots x_{k-1} x_{k+1} \dots x_n \iff Q x_1 \dots x_{k-1} x_i x_{k+1} \dots x_n.$
What does "identification" mean intuitively and what does it correspond to in more concrete terms?
There's also another paragraph I don't understand,
Let $Q$ be an n-ary predicate. Let $1 \leq i \leq n$. The (n-1)-ary predicate $P$ is called the ith generalization of $Q$, if for all $x_1 \dots x_{i-1}, x_{i+1}, \dots, x_n$, $P x_1 \dots x_{i-1} x_{i+1} \dots x_n \iff \wedge_{x_i} Q x_1 \dots x_n.$
On the right hand side, what is different between different terms of the conjunction? Isn't $Q x_1 \dots x_n$ the same no matter what the value of $x_i$ is? Also, what does "generalization" mean in intuitive terms?
Thanks :)