Let $U$ be the domain of discourse, whose elements or (any kind of mathematical objects) we want to study, and let $L\subset (\Sigma^*,\cdot)$ be a formal language over an alphabet $\Sigma$ with a syntax appropriate to generate strings that are interpretable under the interpretation $I$, and let the corresponding truth function $[[]]_I:L\to \mathbb B$ be given.
The rules of inference is a non-empty, finite set of rules of the form
$$\begin{array}{rcl}Rule_{(1)}&:=&\frac{f^{(1)}_1,\ldots,f^{(1)}_{m_{(1)}}}{f_1},\\&\vdots\\ Rule_{(r)}&:=&\frac{f^{(r)}_1,\ldots,f^{(r)}_{m_{(r)}}}{f_r}\end{array}$$
meaning for the $j$-th rule, $j=1,\ldots,r$ that if $f^{(j)}_1,\ldots,f^{(j)}_{m_{(j)}}\in L$ are strings interpreted as true, then also the string $f_j\in L$ can be taken for granted, in other words, from $[[f^{(j)}_1]]_I=1,\ldots,[[f^{(j)}_{m_{(j)}}]]_I=1$ it follows that $[[f_j]]_I=1$ for $j=1,\ldots,r$.
We also say that $f_j$ are (logically) derivable from the $f^{(j)}_1,\ldots,f^{(j)}_{m_{(j)}},$ $j=1,\ldots,r$ and write this using the symbol "$\vdash$", i.e.
$$\begin{array}{rcl}f^{(1)}_1,\ldots,f^{(1)}_{m_{(1)}},&\vdash& f_1,\\ &\vdots\\ f^{(r)}_1,\ldots,f^{(r)}_{m_{(r)}},&\vdash& f_r.\end{array}$$
Chapters: 1 2
Definitions: 3 4
Examples: 5
