Ringschluss

In mathematics, a Ringschluss (German: Beweis durch Ringschluss, lit. 'Proof by ring-inference') is a mathematical proof technique where the equivalence of several statements can be proven without having to prove all pairwise equivalences directly. In English it is also sometimes called a cycle of implications,^[1] closed chain inference, or circular implication; however, it should be distinguished from circular reasoning, a logical fallacy.

In order to prove that the statements $\varphi _{1},\ldots ,\varphi _{n}$ are each pairwise equivalent, proofs are given for the implications $\varphi _{1}\Rightarrow \varphi _{2}$ , $\varphi _{2}\Rightarrow \varphi _{3}$ , $\dots$ , $\varphi _{n-1}\Rightarrow \varphi _{n}$ and $\varphi _{n}\Rightarrow \varphi _{1}$ .^[2]^[3]

The pairwise equivalence of the statements then results from the transitivity of the material conditional.

Example

For $n=4$ the proofs are given for $\varphi _{1}\Rightarrow \varphi _{2}$ , $\varphi _{2}\Rightarrow \varphi _{3}$ , $\varphi _{3}\Rightarrow \varphi _{4}$ and $\varphi _{4}\Rightarrow \varphi _{1}$ . The equivalence of $\varphi _{2}$ and $\varphi _{4}$ results from the chain of conclusions that are no longer explicitly given:

\varphi _{2}\Rightarrow \varphi _{3}

.

\varphi _{3}\Rightarrow \varphi _{4}

. This leads to:

\varphi _{2}\Rightarrow \varphi _{4}

\varphi _{4}\Rightarrow \varphi _{1}

.

\varphi _{1}\Rightarrow \varphi _{2}

. This leads to:

\varphi _{4}\Rightarrow \varphi _{2}

That is $\varphi _{2}\Leftrightarrow \varphi _{4}$ .

Motivation

The technique saves writing effort above all. In proving the equivalence of $n$ statements, it requires the direct proof of only $n$ out of the $n(n-1)/2$ implications between these statements. In contrast, for instance, choosing one of the statements as being central and proving that the remaining $n-1$ statements are each equivalent to the central one would require $2(n-1)$ implications, a larger number.^[1] The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.

References

^ ^a ^b Gabbay, D. M.; Guenthner, Franz, eds. (2005). Handbook of Philosophical Logic. Vol. 12 (2nd ed.). Springer. p. 261. ISBN 9781402030925.
^ Plaue, Matthias; Scherfner, Mike (2019-02-11). Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis [Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis] (in German). Springer-Verlag. p. 26. ISBN 978-3-662-58352-4.
^ Struckmann, Werner; Wätjen, Dietmar (2016-10-20). Mathematik für Informatiker: Grundlagen und Anwendungen [Mathematics for Computer Scientists: Fundamentals and Applications] (in German). Springer-Verlag. p. 28. ISBN 978-3-662-49870-5.

[hbk-1] Gabbay, D. M.; Guenthner, Franz, eds. (2005). Handbook of Philosophical Logic. Vol. 12 (2nd ed.). Springer. p. 261. ISBN 9781402030925.

[2] Plaue, Matthias; Scherfner, Mike (2019-02-11). Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis [Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis] (in German). Springer-Verlag. p. 26. ISBN 978-3-662-58352-4.

[3] Struckmann, Werner; Wätjen, Dietmar (2016-10-20). Mathematik für Informatiker: Grundlagen und Anwendungen [Mathematics for Computer Scientists: Fundamentals and Applications] (in German). Springer-Verlag. p. 28. ISBN 978-3-662-49870-5.

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