Inlogic andmathematics,proof by example (sometimes known asinappropriate generalization) is alogical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledgedproof.[1][2]
The structure,argument form and formal form of a proof by example generally proceeds as follows:
Structure:
The following example demonstrates why this line of reasoning is a logical fallacy:
In the common discourse, a proof by example can also be used to describe an attempt to establish a claim usingstatistically insignificant examples. In which case, the merit of each argument might have to be assessed on an individual basis.[3]
In certain circumstances, examples can suffice aslogically valid proof.
In some scenarios, an argument by example may be valid if it leads from a singular premise to anexistential conclusion (i.e. proving that a claim is true for at least one case, instead of for all cases). For example:
(or)
These examples outline the informal version of the logical rule known asexistential introduction, also known asparticularisation orexistential generalization:
(where denotes theformula formed by substituting allfree occurrences of the variable in by.)
Likewise, finding acounterexample disproves (proves thenegation of) a universal conclusion. This is used in aproof by contradiction.
Examples also constitute valid, ifinelegant, proof, when it hasalso been 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.[4]
