circular reasoning

Circular reasoning is an attempted proof of a statement that uses at least one of the following two things:

  • the statement that is to be proven

  • a fact that relies on the statement that is to be proven

Such proofs are not valid.

As an example, below is a faulty proof that the well-ordering principle implies the axiom of choiceMathworldPlanetmath ( The step where circular reasoning is used is surrounded by brackets [ ].

Let C be a collectionMathworldPlanetmath of nonempty sets. By the well-ordering principle, each SC is well-ordered. [For each SC, let <S denote the well-ordering of S.] Let mS denote the least member of each SC with respect to <S. Then a choice function f:CSCS can be defined by f(S)=mS.

The step surrounded by brackets is faulty because it actually uses the axiom of choice, which is what is to be proven. In the step, for each SC, an orderingMathworldPlanetmath is chosen. This cannot be done in general without appealing to the axiom of choice.

Title circular reasoning
Canonical name CircularReasoning
Date of creation 2013-03-22 16:06:32
Last modified on 2013-03-22 16:06:32
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 15
Author Wkbj79 (1863)
Entry type Definition
Classification msc 03F07
Synonym circular argument