Erroneous method of proof

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:

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

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]}

##
Valid cases of proof by example

In certain circumstances, examples can suffice as logically valid proof.

###
Proofs of existential statements

In some scenarios, an argument by example may be valid if it leads from a singular premise to an *existential* conclusion (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 as existential introduction, also known as *particularisation* or *existential generalization*:

- Existential Introduction
- ${\underline {\varphi (\beta /\alpha )))\,\!$
- $\exists \alpha \,\varphi \,\!$

(where $\varphi (\beta /\alpha )$ denotes the formula formed by substituting all free occurrences of the variable $\alpha$ in $\varphi$ by $\beta$.)

Likewise, finding a counterexample disproves (proves the negation of) a universal conclusion. This is used in a proof by contradiction.

###
Exhaustive proofs

Examples also constitute valid, if inelegant, proof, when it has *also* 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]}