Undistributed Middle

7 Undistributed Middle

Charlene Elsby

Minds are computers, since minds calculate and computers calculate, too…

The typical freshman in college

Related errors include the exclusive premises fallacy (Chapter 4), the fallacy of four terms (Chapter 5), and the illicit major and minor terms fallacies (Chapter 6).

In order to understand the fallacy of undistributed middle, we need to know that the middle term is the term that appears in both of the premises but not in the conclusion. In addition, we need the concept of distribution. “Distribution” is meant to describe the extension of the term, that is, how many things it applies to. If it’s all of them, then the term is distributed, as is the term “cats” in “All cats are blue.” I have said something about all of the cats, and so that term is distributed in my premise. “Blue” is not distributed, since I haven’t included all of the blue things in my statement. (There could be other blue things besides cats, like the sky, or it could be that all and only cats are blue, but the statement does not tell us whether that is true.) With regard to its place in the history of logic, William Kneale and Martha Kneale (1985) credit the medieval logicians with applying the term “distribution” to this property of terms: “it came later to indicate the property which a general term is supposed to have when it is used to stand for all the individuals to which it is applicable” (272).

Stated succinctly, the fallacy of undistributed middle occurs when the middle term isn’t distributed in either premise. This fallacy was apparent to medieval scholars like John Buridan in his Treatise on Consequences in Summulae de Dialectica (1350/2001): “Sixth Conclusion: No syllogism is valid in which the middle is distributed in neither premise, unless the middle is used in the minor with a relative of identity” (121).

That’s not to say that the problem wasn’t already apparent in Aristotle. It is the reason many of the possible combinations of premises result in “no syllogism.” Many forms of argument fail due to their middle term’s being undistributed, and while Henry Aldrich (1862) uses the undistributed middle term to explain why an argument is invalid, it doesn’t seem to have been formalized as a “fallacy” until Richard Whately’s Elements of Logic (1827). Still, in Aristotle, we get the idea through his examples of situations where no syllogism is possible. One of these situations is where the middle term is undistributed. Another note on terminology – where the middle term is that which is repeated twice in the premises, the “extremes” are the other two terms of the syllogism. If, for instance, we have “substance” as a middle term and “animal” and “human” as extremes, then the start of a non‐syllogism with an undistributed middle would look like this:

1. All animals are substance.
2. All humans are substance.

These premises do not determine a categorical relation between animals and humans. Each class belongs to the larger class of substances, but the premises do not tell us whether the two classes overlap.

Aristotle formulated this rule in the Prior Analytics: “But if M is predicated of every N and O, there cannot be a syllogism. Terms to illustrate a positive relation between the extremes are substance, animal, man; a negative relation, substance, animal, number—substance being the middle term” (27a19).

We cannot infer the relation between animals and humans from those two premises alone, since neither addresses all substances. It is completely possible that the animals that are substances and the humans who are substances aren’t the same substances. Alexander of Aphrodisias (1991) fully comprehends the reasoning behind this, although he doesn’t use the term “distribution.”

The reason why there is no syllogistic combination from two universal affirmatives in the second figure is that the middle is predicated of both extremes, and the predicate is the major term. Thus, being major in relation to both, it may be predicated of one extreme term in virtue of one of its parts, and of the other in virtue of another. And in this way there is nothing which the extremes share with each other, if they each share with the middle term in virtue of different parts of it. For the extremes must share in one and the same thing if there is to be a syllogism. (151)

For a categorical syllogism to be valid, at least one premise must refer to the whole class denoted by the middle term. This does not guarantee overlap between the extremes: a valid conclusion may instead exclude it. If the middle term is undistributed in both premises, the syllogism is invalid.

There exist any number of ways this fallacy could be avoided; that is, the way to avoid the fallacy of undistributed middle is to create any valid syllogism. While not all syllogisms with distributed middles are valid, a syllogism must have a distributed middle in order to be valid. Distribution of the middle term is thus a necessary but not sufficient condition for a valid syllogism. If we imagine a world where no humans are substance, then we could reform the example above to look like this:

1. All animals are substance.
2. No humans are substance.

And from these premises we could conclude

1. No humans are animals.

In order to do so, we have had to distribute the middle term – the fact that the classes of “human” and “substance” are mutually exclusive means that, at some point in our syllogism, we have taken all of substance into account. That is, all of substance is excluded from being human.