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 $PL0$.

Definition: Interpretation of Propositions - the Law of the Excluded Middle 

Let \(L\) be a formal language with the syntax of propositional logic.

Law of Excluded Middle (modern formulation)

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.

An equivalent formulation

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]

1.Example: Examples of Propositions With a More Complex Syntax 

2.Definition: Paradox