Proof by Examples

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.

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."

Also called:

Inappropriate Generalization
Hasty Generalisation
No-Limits Fallacy

Looks like this Logical 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.

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 plenty.
- 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.
If you are trying to prove that something is possible, as opposed to trying to prove that something is true. Demonstrating that X is always Y is a substantial burden of proof. Demonstrating that X is sometimes Y requires a minimum of one positive example.
The speaker has a definitive authority to name the characteristics of something. If Anne McCaffrey says that a dragon is a genetically-engineered beast ridden by an order of warriors on a planet called "Pern" one can assume that to be an accurate description of dragons, in the writings of Anne McCaffrey.
- On the other hand that would only refer to Pern dragons and has no bearing on dragons in Ring Cycle or Middle Earth, nor is it intended to.

Examples of Proof by Examples include:

Usually, in mathematics, no matter how many examples you might have, proof by example is not a good idea.
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.