Ringschluss

In mathematics, a Ringschluss (German: Beweis durch Ringschluss, lit. 'Proof by ring-closure') is a mathematical proof technique where the equivalence of several statements can be proven without having to prove all pairwise equivalences directly.

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 $\varphi _{n}\Rightarrow \varphi _{1}$.

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. By dispensing with the formally necessary chain of conclusions, only $n$ direct proofs need to be provided for $\varphi _{i}\Rightarrow \varphi _{j}$ instead of $n(n-1)$ direct proofs. The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.

See also

• The term should not be confused with the invalid circular reasoning.