Ringschluss

In mathematics, a Ringschluss (Template:Langx) 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 ${\phi }_1,\dots ,{\phi }_n$ are each pairwise equivalent, proofs are given for the implications ${\phi }_1\Rightarrow {\phi }_2$, ${\phi }_2\Rightarrow {\phi }_3$, $\dots$, ${\phi }_{n-1}\Rightarrow {\phi }_n$ and ${\phi }_n\Rightarrow {\phi }_1$.

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

Example

For $n=4$ the proofs are given for ${\phi }_1\Rightarrow {\phi }_2$, ${\phi }_2\Rightarrow {\phi }_3$, ${\phi }_3\Rightarrow {\phi }_4$ and ${\phi }_4\Rightarrow {\phi }_1$. The equivalence of ${\phi }_2$ and ${\phi }_4$ results from the chain of conclusions that are no longer explicitly given:

${\phi }_2\Rightarrow {\phi }_3$. ${\phi }_3\Rightarrow {\phi }_4$. This leads to: ${\phi }_2\Rightarrow {\phi }_4$

${\phi }_4\Rightarrow {\phi }_1$. ${\phi }_1\Rightarrow {\phi }_2$. This leads to: ${\phi }_4\Rightarrow {\phi }_2$

That is ${\phi }_2\iff {\phi }_4$.

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 ${\phi }_i\Rightarrow {\phi }_j$ 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.