# Proof by example

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{{Short description|Erroneous method of proof}}
In [logic](/source/logic) and [mathematics](/source/mathematics), '''proof by example''' (sometimes known as '''inappropriate generalization''') is a [logical fallacy](/source/Informal_fallacy) whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged [proof](/source/mathematical_proof).<ref>{{cite web|archive-url=https://web.archive.org/web/20020731165349/http://www.auburn.edu/~marchjl/fallacies.htm|url=http://www.auburn.edu/~marchjl/fallacies.htm|title=Logical fallacies|archive-date=31 July 2002|publisher=www.auburn.edu}}</ref><ref name=":1">{{Cite web|url=https://grammarist.com/rhetoric/proof-by-example/|title=Proof by example – Grammarist|website=[Grammarist](/source/Grammarist)|access-date=2019-12-01}}</ref>

The structure, [argument form](/source/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](/source/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''.<ref name=":1" />

[Formal form](/source/Mathematical_logic):
:<math>\exists x:P(x)\;\;\vdash\;\;\forall x:P(x)</math>

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](/source/statistically_insignificant) examples. In which case, the merit of each argument might have to be assessed on an individual basis.<ref>{{Cite web|url=https://simplicable.com/new/proof-by-example|title=What is Proof By Example?|website=Simplicable|access-date=2019-12-01}}</ref>

==Valid cases of proof by example==
In certain circumstances, examples can suffice as [logically valid](/source/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](/source/List_of_rules_of_inference), also known as ''particularisation'' or ''existential generalization'':

;Existential Introduction
: <math>\frac{\varphi(\beta / \alpha)}{\exists \alpha\, \varphi}</math>

(where <math>\varphi(\beta / \alpha)</math> denotes the [formula](/source/formula_(logic)) formed by substituting all [free occurrences](/source/Free_occurrence) of the variable <math>\alpha</math> in <math>\varphi</math> by <math>\beta</math>.)

Likewise, finding a [counterexample](/source/counterexample) disproves (proves the [negation](/source/logical_negation) of) a universal conclusion. This is used in a [proof by contradiction](/source/proof_by_contradiction).

===Exhaustive proofs===
{{Main article|Proof by exhaustion}}

Examples also constitute valid, if [inelegant](/source/Mathematical_elegance), 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.<ref>{{Cite web|url=http://www.mit.edu/people/dmredish/wwwMLRF/links/Humor/Math_Proofs.html|title=Proofs|website=www.mit.edu|access-date=2019-12-01}}</ref>

==See also==
*[Affirming the consequent](/source/Affirming_the_consequent)
*[Anecdotal evidence](/source/Anecdotal_evidence)
*[Bayesian probability](/source/Bayesian_probability)
*[Counterexample](/source/Counterexample)
*[Hand-waving](/source/Hand-waving)
*[Inductive reasoning](/source/Inductive_reasoning)
**[Problem of induction](/source/Problem_of_induction)
*[Modus ponens](/source/Modus_ponens)
*[Proof by construction](/source/Proof_by_construction)
*[Proof by intimidation](/source/Proof_by_intimidation)

==References==
<references/>

== Further reading ==

* [Benjamin Matschke: Valid proofs by example in mathematics (arXiv)](/source/arxiv%3A1909.00480)
{{Formal Fallacy}}

{{DEFAULTSORT:Proof By Example}}
Category:Quantificational fallacies

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