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, closed chain inference, or circular implication; however, it should be distinguished from circular reasoning, a logical fallacy.

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. 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. The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.