{{Short description|Proof strategy for showing a collection of statements are equivalent}}
In mathematics, a '''Ringschluss''' ({{Langx|de|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''',<ref name=hbk>{{cite book|title=Handbook of Philosophical Logic|volume=12|editor1-first=D. M.|editor1-last=Gabbay|editor2-first=Franz|editor2-last=Guenthner|edition=2nd|publisher=Springer|year=2005|isbn=9781402030925|page=261|url=https://books.google.com/books?id=Ikc5FKo7g4cC&pg=PA261}}</ref> '''closed chain inference''', or '''circular implication'''; however, it should be distinguished from circular reasoning, a logical fallacy.
In order to prove that the statements <math>\varphi_1,\ldots,\varphi_n</math> are each pairwise equivalent, proofs are given for the implications <math>\varphi_1\Rightarrow\varphi_2</math>, <math>\varphi_2\Rightarrow\varphi_3</math>, <math>\dots</math>, <math>\varphi_{n-1}\Rightarrow\varphi_n</math> and <math>\varphi_{n}\Rightarrow\varphi_1</math>.<ref>{{Cite book |last1=Plaue |first1=Matthias |url=https://books.google.com/books?id=-WCHDwAAQBAJ |title=Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis |last2=Scherfner |first2=Mike |date=2019-02-11 |publisher=Springer-Verlag |isbn=978-3-662-58352-4 |pages=26 |language=de |trans-title=Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis}}</ref><ref>{{Cite book |last1=Struckmann |first1=Werner |url=https://books.google.com/books?id=1epNDQAAQBAJ |title=Mathematik für Informatiker: Grundlagen und Anwendungen |last2=Wätjen |first2=Dietmar |date=2016-10-20 |publisher=Springer-Verlag |isbn=978-3-662-49870-5 |pages=28 |language=de |trans-title=Mathematics for Computer Scientists: Fundamentals and Applications}}</ref>
The pairwise equivalence of the statements then results from the transitivity of the material conditional.
== Example == For <math>n=4</math> the proofs are given for <math>\varphi_1\Rightarrow\varphi_2</math>, <math>\varphi_2\Rightarrow\varphi_3</math>, <math>\varphi_3\Rightarrow\varphi_4</math> and <math>\varphi_4\Rightarrow\varphi_1</math>. The equivalence of <math>\varphi_2</math> and <math>\varphi_4</math> results from the chain of conclusions that are no longer explicitly given:
:<math>\varphi_2 \Rightarrow \varphi_3 </math>. <math>\varphi_3 \Rightarrow \varphi_4</math>. This leads to: <math>\varphi_2 \Rightarrow \varphi_4</math> :<math>\varphi_4 \Rightarrow \varphi_1</math>. <math>\varphi_1 \Rightarrow \varphi_2</math>. This leads to: <math>\varphi_4 \Rightarrow \varphi_2</math>
That is <math>\varphi_2\Leftrightarrow \varphi_4</math>.
== Motivation == The technique saves writing effort above all. In proving the equivalence of <math>n</math> statements, it requires the direct proof of only <math>n</math> out of the <math>n(n-1)/2</math> implications between these statements. In contrast, for instance, choosing one of the statements as being central and proving that the remaining <math>n-1</math> statements are each equivalent to the central one would require <math>2(n-1)</math> implications, a larger number.<ref name=hbk/> The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.
== References == {{reflist}}
Category:Mathematical logic Category:Proof techniques