Form:

Any form of categorical syllogism in which the middle term is not distributed at least once.

Example:

[O]ne patient…concluded that there are money-trees because "money is green [in the United States] and so are trees, so money must grow on trees."

Counter-Example:

The 29 students in Mr Strang's classroom gravely considered the two sentences scrawled across the freshly washed blackboard:

All A's are C's.
All B's are C's.

"The apparent conclusion—that all A's are B's—does have a certain allure, a kind of appealing logic."

Mr Strang blinked myopically, his wrinkled face resembling that of a good-natured troll. Then he whirled, and his chalk drew a large screeching X through both sentences.

"Of course," he snapped, "it's also dead wrong. Its error can easily be verified by substituting 'teenager' for A, 'ostrich' for B, and 'two-legged' for C in the original premises. Thus, all teenagers are two-legged, all ostriches are two-legged, and therefore all teenagers are ostriches. I doubt you'd accept that conclusion."

Source: William Brittain, "Mr Strang Accepts a Challenge", from The Mammoth Book of Locked-Room Mysteries and Impossible Crimes, pp. 349-50.

Venn Diagram:

Both the Example and Counter-Example are represented by this diagram, where "S" represents the subject term, "P" is the predicate term, and "M" the middle term. Notice that the diagram does not show that the conclusion, "All S is P", is true.

Syllogistic Rule Violated:

In a valid categorical syllogism, the middle term is distributed in at least one of its occurrences.

Exposure:

Undistributed Middle is one of the most famous fallacies, and is sometimes used as a synonym for "fallacious argument", that is, some people know that an undistributed middle is somehow logically bad, and mistakenly think that any bad argument must have one. In other words, they seem to reason as follows:

All arguments with undistributed middle terms are bad arguments.
This is a bad argument.
Therefore, this argument has an undistributed middle.

This argument is a bad argument because it has an undistributed middle term, namely, "bad arguments"!

Resource:

Irving Copi & Carl Cohen, Introduction to Logic (Tenth Edition) (Prentice Hall, 1998), pp. 275-276.

[A] major component of reasoning takes place in a frontal module of the brain, a module being a specialized, relatively independent group of inter-connected nerve cells performing a particular category of tasks. Patients who have an injury confined to this module…have a…specific syndrome in which reason is disturbed without a generalized disorder of cognition. Appropriately enough, this isolated impairment of the module for reason is called "the dysexecutive syndrome," a condition analogous to selective disorders of language (the aphasias) or memory (the amnesias).

Patients with the dysexecutive syndrome…make mistakes in assembling rational sequences of thought. For example, one patient with the dysexecutive syndrome concluded that there are money-trees because "money is green [in the United States] and so are trees, so money must grow on trees."

Source: Donald B. Calne, Within Reason: Rationality and Human Behavior, Pantheon, 1999, pp. 15-16.