I find myself completely lost proceeding to the following section of the text.

"Let $R$ be a relation, $A$ a mathematical object, and $x$ a letter (i.e., a "totally indeterminate" mathematical object). In the assembly of letters and fundamental signs which constitutes the relation $R$, replace the letter $x$ wherever it occurs by the assembly $A$. One of the criteria for forming relations is that the assembly so obtained is again a relation, which is denoted $(\ast )$ by the notation

$(\mathrm{A}\mid x)\mathrm{R}$

and is called the relation obtained by substituting $A$ for $x$ in $R$, or by giving $x$ the value $A$ in $R$. The mathematical object A is said to satisfy the relation $R$ if the relation $(\mathrm{A}\mid x)\mathrm{R}$ is true. It goes without saying that if the letter $x$ does not appear at all in the assembly $R$, then the relation $(\mathrm{A}\mid x)\mathrm{R}$ is just $R$, and in this case to say that $A$ satisfies $R$ means that $R$ is true."

However, I do appreciate the textbook that is self-contained and appreciate the author devoted to mathematical reasoning so rigorously at the beginning of the chapter. I tried to find some textbook about mathematical logic but they are either too abstract or not thorough enough that seems to start from the most fundamental (i.e. from axiom and the most basic rule).

I have read relevant posts on the subject I am asking but can't decide the material right for me. I am wondering if there are any materials or textbooks that introduce mathematical logic rigorously and serve as a supplementary text for me to understand the first chapter of the book? If there really isn't any textbook that is not too abstract but rigorous enough, I am wondering if there are any other textbooks on abstract algebra that start from mathematical logic and build the whole system from the scratch?