Proof by example

In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof.^[1][2]

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

Structure:

I know that X is such.

Therefore, anything related to X is also such.

Argument form:

I know that x, which is a member of group X, has the property P.

Therefore, all other elements of X must have the property P.^[2]

Formal form:

${\displaystyle \exists x:P(x)\;\;\vdash \;\;\forall x:P(x)}$

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 using statistically insignificant examples. In which case, the merit of each argument might have to be assessed on an individual basis.^[3]