Closed chain inference explained

A closed chain inference is a mathematical proof technique with which the pairwise 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 ⇒ \varphi} _{2} , \varphi _{2 ⇒ \varphi} _{3} , ... , \varphi _{n-1} ⇒ \varphi _{n} and \varphi _{n} ⇒ \varphi _{1} .^{[1] [2]}

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 ⇒ \varphi} _{2} , \varphi _{2 ⇒ \varphi} _{3} , \varphi _{3 ⇒ \varphi} _{4} and \varphi _{4 ⇒ \varphi} _{1} . The equivalence of \varphi _{2} and \varphi _{4} results from the chain of conclusions that are no longer explicitly given:

\varphi _{2} ⇒ \varphi _{3}
	\varphi _{3} ⇒ \varphi _{4}
	This leads to:
It follows	\varphi _{2} ⇒ \varphi _{4}
	And
	\varphi _{4} ⇒ \varphi _{1}
	\varphi _{1} ⇒ \varphi _{2}
	This leads to:
It follows	\varphi _{4} ⇒ \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 ⇒ \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.

Notes and References

1. Book: Plaue . Matthias . Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis . Scherfner . Mike . 2019-02-11 . Springer-Verlag . 978-3-662-58352-4 . 26 . de . Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis.
2. Book: Struckmann . Werner . Mathematik für Informatiker: Grundlagen und Anwendungen . Wätjen . Dietmar . 2016-10-20 . Springer-Verlag . 978-3-662-49870-5 . 28 . de . Mathematics for Computer Scientists: Fundamentals and Applications.