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 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 well-ordering principle, each $S\in C$ is well-ordered. [For each $S\in C$, let $<_{S}$ denote the well-ordering 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.

Title circular reasoning CircularReasoning 2013-03-22 16:06:32 2013-03-22 16:06:32 Wkbj79 (1863) Wkbj79 (1863) 15 Wkbj79 (1863) Definition msc 03F07 circular argument