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.

| | | | | 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