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 wellordering principle implies the axiom of choice^{} (http://planetmath.org/WellOrderingPrincipleImpliesAxiomOfChoice). The step where circular reasoning is used is surrounded by brackets [ ].
Let $C$ be a collection^{} of nonempty sets. By the wellordering principle, each $S\in C$ is wellordered. [For each $S\in C$, let $$ denote the wellordering of $S$.] Let ${m}_{S}$ denote the least member of each $S\in C$ with respect to $$. Then a choice function $f:C\to {\displaystyle \bigcup _{S\in C}}S$ can be defined by $f(S)={m}_{S}$.
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 $S\in C$, an ordering^{} is chosen. This cannot be done in general without appealing to the axiom of choice.
Title  circular reasoning 

Canonical name  CircularReasoning 
Date of creation  20130322 16:06:32 
Last modified on  20130322 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 