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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. 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 $<_{S}$ denote the wellordering of $S$.] Let $m_{S}$ denote the least member of each $S\in C$ with respect to $<_{S}$. Then a choice function $\displaystyle f\colon C\to\bigcup_{{S\in C}}S$ 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.
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Comments
example
I thought it would be a good idea to put an example of circular reasoning in the article; however, I would greatly appreciate it if people have ``simpler'' examples. The one that I provide was the first one that I could come up with. I think it's pretty simple, but it may not be so to someone with little or no exposure to group theory. As always, feel free to attach examples to this entry if you would like, but, in this particular case, I highly encourage it. If someone attaches an example that I particularly like, I may remove the example that I provide from the entry.
Warren
Tautology
Known in Lat. as {\em petitio principii}. Tautologies use a circular reasoning within an argument or enunciate. For example, the entry ``circular reasoning'' by Wkbj79 expresses a logic tautology because we are using in the reasoning the selfevident $g*g^{1}=e_G$.
perucho