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
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