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.

单词	CircularReasoning
释义	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.
随便看	operations on multisets OperationsOnRelations operations on relations OperationsWithEvents operations with events Operator operator operator OperatorInducedByAMeasurePreservingMap operator induced by a measure preserving map OperatorMonotone operator monotone operatornamearcTanWithTwoArguments operatornamecfoperatornamecfalphaoperatornamecfalpha operatornamekerL0IfAndOnlyIfLIsInjective operatornamepvfrac1xIsADistributionOfFirstOrder OperatorNorm operator norm OperatorNormOfMultiplicationOperatorOnL2 operator norm of multiplication operator on L^2 OperatorTopologies operator topologies OpposingAnglesInACyclicQuadrilateralAreSupplementary opposing angles in a cyclic quadrilateral are supplementary Opposite