Undistributed Middle (Fallacy)

The undistributed middle is a logical fallacy of deduction in which the middle term of a syllogism is not distributed in at least one of the premises.

According to the rules of logic, a term is "distributed" when a sentence says something about everything the term designates. A syllogism is invalid if both middle terms are undistributed.

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Definition and Examples of Sorites in Rhetoric

By Richard Nordquist

British educator Madsen Pirie illustrates the fallacy of the undistributed middle with this "schoolboy" argument: "because all horses have four legs and all dogs have four legs, so all horses are dogs."

"Both horses and dogs are indeed four-legged," Pirie points out, "but neither of them occupies the whole of the class of four-legged beings. This leaves convenient room for horses and dogs to be different from each other, and from other beings which might also without any overlap be in the four-legged class" (How to Win Every Argument: The Use and Abuse of Logic, 2007).

Examples and Observations

• "The 'middle' which carelessly omitted to get itself distributed is the term which appears in the first two lines of a three-line argument, but which disappears in the conclusion. The classic three-liner requires that this middle term must cover the whole of its class at least once. If not, it is undistributed. All men are mammals. Some mammals are rabbits, therefore some men are rabbits.
  (Even though the first two lines are correct, the middle term 'mammals' never once refers to all mammals. The middle term is thus undistributed and the deduction invalid.) . . . The standard three-liner (called a 'syllogism') works by relating one thing to another by means of a relationship they both have with a third. Only if at least one of those relationships applies to all the third thing, do we know that it is certain to include the other relationship."
  (Madsen Pirie, How to Win Every Argument: The Use and Abuse of Logic. Continuum, 2007)
• "It's Speaking English That Kills You"
  "[P]ersuaders use the undistributed middle principle to sway opinion and alter behavior in significant ways. For example, because someone serves on the school board, many critics assume that the person must favor all the board's decisions. This example appeared in a small-town newspaper recently: Consider these facts: The Japanese eat very little fat and suffer fewer heart attacks than the British or the Americans. On the other hand, the French eat a lot of fat and suffer fewer heart attacks than the British or the Americans. The Italians drink excessive amounts of red wine and also suffer fewer heart attacks than the British or the Americans. Therefore eat and drink what you like. It's speaking English that kills you (Consider the Facts, 2002, p. 10). This fallacy also underlies any appeal suggesting that using a certain popular brand will make us like others who use it."
  (Charles U. Larson, Persuasion: Reception and Responsibility, 12th ed. Wadsworth, 2010)
• "Some Humans Are Cows"
  "Consider [this] example: Some mammals are cows.
  All humans are mammals.
  So, some humans are cows. The middle term here is 'mammals,' which is undistributed in both major and minor premises. As a result, these premises only refer to some mammals. The major premise refers to cows, which are mammals, and the minor premise refers to humans, which are mammals. But, obviously, the conclusion is invalid because the middle term in each of its occurrences refers to distinct classes of mammals but never to all mammals. For example, the syllogism would indeed be valid (but needless to say not sound) if the major premise said that all mammals are cows."
  (Elliot D. Cohen, Critical Thinking Unleashed. Rowman & Littlefield, 2009)
• Long-Haired Radicals
  "The following invalid syllogism . . . illustrates what happens when the middle term is undistributed in both premises:
  All radicals are people with long hair.
  Ed is a person with long hair.
  Therefore, Ed is a radical.
  In this syllogism, the middle term, 'people with long hair,' is undistributed in both premises, since in both it is the predicate term of an A statement. Both the major and the minor terms are related to the middle term in the premises, but neither the major nor the minor class is related to the entire class referred to by the middle term, so their relationship to each other is not known. The first premise does not rule out the possibility that the class of people with long hair contains members who are not radicals, and the second premise would permit Ed to be such a person."
  (Robert Baum, Logic, 4th ed. Harcourt, 1996)
• Umberto Eco's Fallacy of the Undistributed Middle
  "Triumphantly, I completed the syllogism: " . . . Venantius and Berengar have blackened fingers, ergo they touched the substance!'
  "'Good, Adso,' William said, 'a pity your syllogism is not valid, because aut semel aut iterum medium generaliter esto, and in this syllogism the middle term never appears as general. A sign that we haven't chosen the major premise well. I shouldn't have said that all those who touch a certain substance have black fingers, because there could also be people with black fingers who have not touched the substance. I should have said that all those and only all those who have black fingers have certainly touched a given substance."
  (Umberto Eco, The Name of the Rose, 1980; trans. 1983)