Proof by Examples
- Inappropriate Generalization
- Hasty Generalization
- No-Limits Fallacy
- Taking one or more non-exhaustive examples from a group that have a property, and making a generalization that everything in that group has that property.
3 is odd, and it is a prime number.
13 is odd, and it is a prime number.
97 is odd, and it is a prime number.
Therefore, all odd numbers are prime numbers.
In short, mistaking inductive reasoning for deductive reasoning
- A common version is to assume that anything can be extended off to infinity, or that since having a little of something is good, having more must be better. It's a line of thinking commonly used by those talking about future technology.
"If I told you fifty years ago that you'd have a phone smaller than a deck of cards, that computers would be small enough to put into a pocket, and that your car would be able to call for help if it was involved in a crash, you'd have laughed at me. So if you say that faster-than-light travel is impossible, you're just being small-minded, since technology continues to improve all the time."
- In mathematics proof by example is usually no proof at all. Many famous examples are used to illustrate this.
- A simple one is a conjecture by Christian Goldbach that "every odd composite number can be written as the sum of a prime and twice a square number" which certainly seems to be true if you try casually testing a few example. It wasn't until much later that a counter example (5777) was found.
- An even more dramatic example is Euler's sum of power conjecture for which the first counter-example is 61,917,364,224!
- There are problem in mathematics that have been tested for trillions upon trillions of examples without finding a counter-example but still lack proof. (The huge numbers of examples are both used to search for counter-examples and in hopes of discovering patterns that might lead to a proof.)
- Things like (literal) witch hunts and radical Islamic terrorism being used to imply that everything about a certain religion is wrong/dangerous.
- The idea that computers that can design replacements better than themselves could do so at a rate that accelerates infinitely, often citing a graph by Ray Kurzweil which appears to show an upward trend. This ignores that there are physical ceilings to such development; such a system might hit these faster, but would not gain the magical ability to simply ignore them.
- Note Kurzweil himself does not argue that any particular method of computation can be improved infinitely and supports his arguments about where technology is headed with current research and mathematical theories about the absolute limits of computational density.
Looks like this fallacy but is not
- If the list of examples is exhaustive, in which case it is known as "proof by exhaustion" or "proof by cases". Meaning that you prove, using groups as examples, both that the statement is true for all examples, and it is impossible for any relevant example to not be in one or more of the groups. For example:
The sum of the angles of any acute triangle on a Euclidean plane add up to 180°.
The sum of the angles of any right triangle on a Euclidean plane add up to 180°.
The sum of the angles of any obtuse triangle on a Euclidean plane add up to 180°.
All triangles on a Euclidean plane are either acute, right, or obtuse, therefore the sum of the angles of any triangle on a Euclidean plane add up to 180°.
- (While triangles on the surface of a sphere, for example, do weird things, the surface of a sphere is not a Euclidean plane; therefore, that case is not relevant to the proof.)
- When you are disproving by example — this is termed a "counter-example."
9 is not a prime number, and it is odd.
Therefore, not all odd numbers are prime numbers.
- Proving an existential statement (i.e. "There exists...") by example. One example is all that's needed. The prime (pardon the pun) example might well be this: "2 is an even number and is prime. Therefore, there exists at least one prime number that is even."
- An attempt at real induction. Inductive logic admits that its conclusions are not necessarily true, but rather that they are probably true, and it tends to attempt to be as exhaustive as possible and to eliminate as many alternative explanations as possible, to reduce the possibility that the conclusion is wrong to as close to zero as possible. However, an honest scientist (i.e. practitioner of inductive logic) would freely admit that there is the possibility, however slim, that the entirety of his/her science is entirely wrong.
- A (simple) example of this might be: "A new plant found seems to fit into a particular category with several others. All of the plants within this category seem to need three things to thrive - carbon dioxide, water and a light source. Therefore, it seems likely this plant will also need those three to survive. We should study it to confirm or deny this theory."