The fallacy of the undistributed middle (Latin: non distributio medii) is a formal error occurring in a categorical syllogism. This fallacy is committed when the middle term, which serves to connect the major and minor terms, is not distributed in either the minor premise or the major premise. In essence, it results in a syllogistic fallacy where the logical structure fails due to the inadequate distribution of the middle term across the premises.

Classical formulation

In classical syllogisms, statements follow the forms of "A" (all), "E" (none), "I" (some), or "O" (some not), each consisting of two terms. The first term is distributed in A statements, the second in O statements, both in E statements, and none in I statements.

The fallacy of the undistributed middle arises when the term connecting the two premises is never distributed. In this context, distribution is emphasized as follows:

In this invalid syllogism, the common term B (the middle term) is never distributed. Introducing a premise such as All B is Z or No B is Z would distribute B and validate the argument. - A related rule dictates that anything distributed in the conclusion must be distributed in at least one premise. For instance:

- Here, the middle term Z is distributed, but Y is distributed in the conclusion without being distributed in any premise, rendering this syllogism invalid.

Pattern

The fallacy of the undistributed middle takes the following form:

In this structure, the middle term "B" is not distributed across the premises, leading to an invalid conclusion where Y is asserted to be Z without proper logical support.

Example 1

Premises:

Conclusion:

Explanation: This is an example of the Fallacy of the Illicit Major. The Middle Term is actually distributed here in both premises in which it occurs. "All smartphones" is distributed as the subject of a universal proposition. "aren't smartphones" is distributed as the premise of a negative proposition. "touchscreens" however is distributed in the conclusion as the premise of a negative proposition but undistributed as the predicate of an affirmative proposition in the first premise.

All cats have four legs.
My dog has four legs.
Therefore, my dog is a cat.
In this example, "four legs" is the Middle Term. In both premises it is undistributed because in both cases it is the predicate of a Universal Affirmative proposition. In a Universal Affirmative proposition, the subject is distributed but the predicate is not. So in the first, "cats" is distributed because the statement refers to all cats. "Dogs" is distributed in the second example because the statement refers to all dogs.

Example 2

Premises:

Conclusion:

Explanation: In this case, the middle term "smartphones" is not distributed in the first premise because the premise is not making a statement about all smartphones. The first premise is a Particular Affirmative proposition and hence, neither the subject nor the predicate are distributed. If it had stated that 'All smartphones are waterproof' it would be a Universal Affirmative proposition and as such would (in this instance) be making a statement about ALL smartphones and therefore, the middle term would in fact be distributed. However, as the predicate of a negative proposition in the second premise, "smartphones" actually IS distributed therefore, this syllogism has the middle term distributed at least once and hence does not qualify as an example of the fallacy of the undistributed middle. It is a fallacy though. In this case, the Fallacy of the Illicit Major. This occurs when the major term (ie, the predicate of the conclusion) is not distributed in the premise in which it occurs. Stating, as the first premise does, that 'some smartphones are not waterproof' tells us nothing meaningful about either smartphones as an entire class, nor about the state of being waterproof. Neither the subject nor the predicate are distributed in this style of premise, (a Particular Affirmative). So, whilst "waterproof" is distributed in the conclusion (being the predicate of a negative proposition), it is not distributed in the premise in which it occurs. If we believe that the reason laptops are not waterproof is because they aren't smartphones, would we conclude that functional, well-maintained submarines are also not waterproof because they too are not smartphones?

Example 3

Premises:

Conclusion:

Explanation: Again, this is an example NOT of the Undistributed Middle but of the Illicit Major. See the above examples for comparison.

In popular culture

The fallacy of the undistributed middle has made its mark in popular culture, often depicted in various forms of media and entertainment. One notable example can be found in the TV series "Logic Labyrinth," where a character erroneously connects unrelated categories, leading to comedic misunderstandings.

It highlights instances where the fallacy has been humorously portrayed, providing a cultural perspective on the misunderstanding of logical connections in the broader context of entertainment.