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.