As mentioned above, the German mathematician Gottfried Wilhelm von Leibniz postulated that Every judgment is either true or false. This became known as the Law of Excluded Middle. The law of excluded middle only holds in propositional logic $PL0$ because of its special semantics. We are now able to formally restate the Law of Excluded Middle and define the semantics of $PL0$.
Let \(L\) be a formal language with the syntax of propositional logic.
For every interpretation $I(U,L)$ in any domain of discourse $U$, the valuation function of any string $s\in L$ is a partial function
\[
[[s]]_I:=\begin{cases}
\in\mathbb B,&\text{ if }s\text{ is a proposition} \\
undefined,&\text{otherwise}
\end{cases}
\]
Remember that we denote by $\mathbb B=\{1,0\}$ the set of truth values.
If $s\in L$ is a proposition, then either $I\models s$ or $I\not{\models} s$ for all models $I$, shortly $$\models s\text{ or }\not {\models} s,$$ where “$\models$” is the satisfaction relation.
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| created: 2014-06-22 17:03:53 | modified: 2020-05-04 18:48:25 | by: bookofproofs | references: [656], [7878]
[7878] Beierle, C.; Kern-Isberner, G.: “Methoden wissensbasierter Systeme”, Vieweg, 2000
[656] Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011