Proof by example

Proof by example (also known as inappropriate generalization) is a logical fallacy whereby one or more examples are claimed as "proof" for a more general statement.

This fallacy has the following structure, and argument form:

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 have the property P.

The following example demonstrates why this is a logical fallacy:
 * I've seen a person shoot someone dead.
 * Therefore, all people are murderers.

The flaw in this argument is very evident, but arguments of the same form can sometimes seem somewhat convincing, as in the following example:


 * I've seen Gypsies steal. So, Gypsies must be thieves.

When valid
However, argument by example is valid when it leads from a singular premise to an existential conclusion. For example:


 * Socrates is wise.
 * Therefore, someone is wise.

(or)
 * I've seen a person steal.
 * Therefore, people can steal.

This is an informal version of the logical rule known as existential introduction (also known as particularisation or existential generalization).

Formally


 * Existential Introduction:
 * $$\underline{\varphi(\beta / \alpha)}\,\!$$
 * $$\exists \alpha\, \varphi\,\!$$