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,
Structure:
- I know that X is such.
- Therefore, anything related to X is also such.
- 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]
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
Valid cases of proof by example
In certain circumstances, examples can suffice as
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
(where denotes the
Likewise, finding a
Exhaustive proofs
Examples also constitute valid, if
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]
See also
- Affirming the consequent
- Anecdotal evidence
- Bayesian probability
- Counterexample
- Hand-waving
- Inductive reasoning
- Modus ponens
- Proof by construction
- Proof by intimidation
References
- ^ "Logical fallacies". www.auburn.edu. Archived from the original on 31 July 2002.
- ^ a b "Proof by example – Grammarist". Grammarist. Retrieved 2019-12-01.
- ^ "What is Proof By Example?". Simplicable. Retrieved 2019-12-01.
- ^ "Proofs". www.mit.edu. Retrieved 2019-12-01.