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

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

Example

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

${\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}$ . ${\displaystyle \varphi _{3}\Rightarrow \varphi _{4}}$ . This leads to: ${\displaystyle \varphi _{2}\Rightarrow \varphi _{4}}$ 

${\displaystyle \varphi _{4}\Rightarrow \varphi _{1}}$ . ${\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}$ . This leads to: ${\displaystyle \varphi _{4}\Rightarrow \varphi _{2}}$ 

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

Motivation

The technique saves writing effort above all. In proving the equivalence of ${\displaystyle n}$  statements, it requires the direct proof of only ${\displaystyle n}$  out of the ${\displaystyle n(n-1)/2}$  implications between these statements. In contrast, for instance, choosing one of the statements as being central and proving that the remaining ${\displaystyle n-1}$  statements are each equivalent to the central one would require ${\displaystyle 2(n-1)}$  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

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.