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Philosophy 103: Introduction to Logic
Fallacy of the Undistributed Middle Term

Mary Wollstonecraft, etching A.L. Merritt Abstract: The Fallacy of the Undistributed Middle Term is discussed and illustrated.

  1. Fallacy of the undistributed middle term is a formal fallacy committed in standard form syllogisms whenever the term appearing in both premises is undistributed. (I.e., each of those terms refers to some but not all of the individuals in the premise statements.)

    1. Example of a distributed and undistributed term:

      In the statement …
      All [persons]D are [primates].U
      …the subject term “persons” is said to be a distributed term since it refers to, or denotes, each and every person.

      The predicate term “primates” is said to be undistributed since it does not refer to each and every primate — “primates” in that sentence only makes reference to those individuals who are “persons” and not to other kinds of individuals such as as lemurs or gorillas.

      [If you want to review the topic of the distribution of subject and predicate terms in categorical statements, here's the link: Distribution of a Term.]

      The “undistributed middle” is the second fallacy in our list of syllogistic fallacies.

    2. Example: In the following argument, persons likely to approve of the reasoning are those who focus only on the truth of the premises:

      All [modern feminists]D are [social equality seekers].U
      All [protofeminists]D are [social equality seekers].U
      All [protofeminists]D are [modern feminists].U


      Venn diagram 
    of AAA-3 syllogism The Venn Diagram would be sketched like this:

    3. Intuitively, it is fairly evident that for the conclusion to follow validly, one would have to presuppose in the major premise “All social equality seekers are modern feminists,” not “All modern feminists are are social equality seekers.” So when the given major premise is converted in this manner, the subject class “social equality seekers” now becomes distributed since A statements always distribute their subject terms.

      So, with the major premise converted, the validity of the argument can be illustrated by the argument's nesting of terms:

      protofeminists > social equality seekers > modern feminists[1]

      However, since not all seekers of social equality are feminists, the revised argument is not a sound argument.

      But in the original argument, the class “social equality seekers” is undistributed since it appears in the predicate of the A statement. To repeat, in the original argument, since the middle term is undistributed in both premises, the argument commits the undistributed middle fallacy.

  2. The Fallacy of the Undistributed Middle Term occurs when the middle term is undistributed in both premises.

    1. Rule: In a valid standard form categorical syllogism, the middle term must be distributed in at least one premise.

    2. Diagram of Undistributed Middle Term
    3. Reason: for the two terms of the conclusion to be connected through the third, as in the mechanism shown here, at least one term must be related to the whole of the class designated by the middle term.

      Otherwise, the connection might be with different parts of the middle term, as illustrated below, and no connection can be made.

      John Buridan, a fourteenth century logician, explained the reason for this rule in this manner:
      Since the middle is not distributed in either [premise] it is possible that its conjunction with the major extreme is true for one thing and its conjunction with the minor is true for another; and from this no conjunction of the extremes with one another can be inferred …[2]
      The premises can related to each other only if at least one of the middle terms is distributed.

    4. In the history of logic, occasionally, an undistributed middle term has been considered an example of equivocation (fallacia æquivocationis) presumably because the denotation of the terms are different since the references of the terms are not necessarily identical in the context of their use.[3]

    5. Note:  Remember for the Fallacy of the Undistributed Middle Term to occur, the middle term must be undistributed in both premises, not just one premise.

  3. Optional Exercise: Try this challenge exercise. I think you'll find the mental mastery obtained well worthwhile.
    (1) Visualize the position of the middle term in the following syllogistic forms.
    (2) Then picture the two distributions statuses by the kind of statement.
    (3) State whether or not the fallacy of the undistributed middle term occurs.
    At first, picturing in your mind the distribution statuses to determine the fallacy occurs will be challenging, but working through a few problems will presently become rewarding.

    AAA-2
    III-1
    EAE-1
    AAA-1
    AAI-3
    IOO-3
    AII-4
    OIO-3
    OAO-4
    OOO-1

    Notes

    1. To say the revised argument is valid does not imply the revised argument is sound unless both of the premises presented are known to be true. This argument would be sound if all “social equality seekers” were understood to be narrowly classified as “feminists.”

    2. John Buridan, Treatise on Consequences trans. Stephen Read (New York: Fordham University Press, 2015), 121. doi: 0.5422/fordham/9780823257188.001.0001

    3. E.g., see Henry Joseph Turrell, A Manual of Logic, Or, A Statement and Explanation of the Laws of Formal Thought (London: Rivingtons, 1870), 105.)

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