◀ ▲ ▶Branches / Logic / Definition: Interpretation of Propositions - the Law of the Excluded Middle

As mentioned above, the German mathematician Gottfried Wilhelm von Leibniz (1646 - 1716) 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.

Table of Contents

Examples: 1

1. Definition: Paradox

Mentioned in:

Chapters: 1
Corollaries: 2 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14 15 16
Examples: 17 18
Explanations: 19
Lemmas: 20 21 22 23 24 25 26 27
Motivations: 28
Parts: 29
Proofs: 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Propositions: 45 46
Sections: 47

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References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
2. Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000