Definition: Rules of Inference

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

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011