In this article we will explore a wide range of ideas and concepts related to Proof by example. From its origin to its current applications, Proof by example has been a topic of interest and debate in various areas. We will analyze its importance in contemporary society, as well as its influence on the development of different disciplines. Through research and interviews with experts, we will seek to shed light on the most relevant aspects related to Proof by example, providing our readers with a more complete and in-depth vision of this topic.
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.
The structure, argument form and formal form of a proof by example generally proceeds as follows:
Structure:
The following example demonstrates why this line of reasoning is a logical fallacy:
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.
In certain circumstances, examples can suffice as logically valid proof.
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:
(or)
These examples outline the informal version of the logical rule known as existential introduction, also known as particularisation or existential generalization:
(where denotes the formula formed by substituting all free occurrences of the variable in by .)
Likewise, finding a counterexample disproves (proves the negation of) a universal conclusion. This is used in a proof by contradiction.
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.