Proof by example

Inlogicandmathematics,proof by example(sometimes known asinappropriate generalization) is alogical fallacywhereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledgedproof.

The structure,argument formand formal form of a proof by example generally proceeds as follows:

Structure:

I know thatXis such.

Therefore, anything related toXis also such.

Argument form:

I know thatx,which is a member of groupX,has the propertyP.

Therefore, all other elements ofXmust have the propertyP.

Formal form:

$\exists x:P(x)\;\;\vdash \;\;\forall x:P(x)$

The following example demonstrates why this line of reasoning is a logical fallacy:

I've seen a person shoot someone dead.

Therefore, all people are murderers.

In the common discourse, a proof by example can also be used to describe an attempt to establish a claim usingstatistically insignificantexamples. In which case, the merit of each argument might have to be assessed on an individual basis.

Valid cases of proof by example

In certain circumstances, examples can suffice aslogically validproof.

Proofs of existential statements

In some scenarios, an argument by example may be valid if it leads from a singular premise to anexistentialconclusion (i.e. proving that a claim is true for at least one case, instead of for all cases). For example:

Socrates is wise.

Therefore, someone is wise.

(or)

I've seen a person steal.

Therefore, (some) people can steal.

These examples outline the informal version of the logical rule known asexistential introduction,also known asparticularisationorexistential generalization:

Existential Introduction

${\underline {\varphi (\beta /\alpha )}}\,\!$

$\exists \alpha \,\varphi \,\!$

(where$\varphi (\beta /\alpha )$denotes theformulaformed by substituting allfree occurrencesof the variable$\alpha$in$\varphi$by$\beta$.)

Likewise, finding acounterexampledisproves (proves thenegationof) a universal conclusion. This is used in aproof by contradiction.

Exhaustive proofs

Main article:Proof by exhaustion

Examples also constitute valid, ifinelegant,proof, when it hasalsobeen demonstrated that the examples treated cover all possible cases.

In mathematics, proof by example can also be used to refer to attempts to illustrate a claim by proving cases of the claim, with the understanding that these cases contain key ideas which can be generalized into a full-fledged proof.

See also

• Affirming the consequent
• Anecdotal evidence
• Bayesian probability
• Counterexample
• Hand-waving
• Inductive reasoning
  • Problem of induction
• Modus ponens
• Proof by construction
• Proof by intimidation

Further reading

• Benjamin Matschke: Valid proofs by example in mathematics (arXiv)