Undistributed middle

The textbook case

All cats are mammals. All dogs are mammals. So all cats are dogs.

The premises are true and the conclusion is plainly false. “Mammals” appears in both premises, but both times the statement is about every cat or every dog, not every mammal. Being in the same large group doesn’t put two things in the same place within it.

A folder full of paperwork

All the documents that need a signature are in the blue folder. The lease renewal is in the blue folder. So the lease renewal needs a signature.

This sounds like a reasonable shortcut, and the conclusion may even be true. But the first statement is about every document that needs signing, not about everything in the folder. The folder could also hold receipts, copies and cover letters. The same move turns up in Affirming the consequent‘s “Hidden inside ‘all’” example: sharing a trait with a group doesn’t show you belong to it.

Overlapping “some” groups

Some of the people in the book club are in the running group. Some of the people in the running group take the pottery class. So some of the book club members take pottery.

The middle group here is the running group, and “some” statements say nothing about all its members. The book club’s runners and the pottery class’s runners could be entirely different people. This version is easy to miss because the conclusion is modest and the overlap feels likely. It may well be true, but the premises don’t show it.

A syllogism is an argument with two premises and a conclusion that uses three terms (groups). The minor term (P, “cats”) is the subject of the conclusion, and the major term (R, “dogs”) is its predicate. The middle term (Q, “mammals”) appears in both premises but not in the conclusion. Its job is to connect the other two. The major premise is the one containing the major term; it is listed first here, as textbooks do.

Distribution, in plain terms. A term is distributed in a statement when the statement says something about every member of that group. “All cats are mammals” says something about every cat (each one is a mammal), but nothing about every mammal. So “cats” is distributed and “mammals” is not. The four standard statement types work out like this:

“No P are Q” distributes both because it rules out every P from every Q. “Some P are not Q” counts as distributing Q because it places that P outside the whole Q group. Hurley’s textbook gives a memory aid: “Unprepared Students Never Pass” (Universal statements distribute Subjects, Negative statements distribute Predicates).

The rule this fallacy breaks: the middle term must be distributed at least once. Hurley explains why. If at least one premise is about the whole middle group, the other premise’s connection to that group has to meet it. If neither is, the minor and major terms “may be related to different parts” of the middle group, and nothing links them.

The middle column shows the same arrangement made valid by a single word. “No P are Q” is about every mammal, so it does distribute the middle term, and the conclusion follows.

A note on history. Aristotle named the major, minor and middle terms, but the rules based on distribution came much later: Hamblin traces them to the later Middle Ages, and they have been a textbook fixture since the seventeenth century. Hamblin also notes that the idea of distribution has had critics, and that the usual explanation fits “Some P are not Q” poorly. Whatever its philosophical footing, the full set of textbook rules does sort valid syllogisms from invalid ones.

• When one premise really is about the whole middle group. A “no” statement about it, an “all” statement with it as the subject, or an “only” statement (“Only documents that need a signature go in the blue folder”) covers every member, and the argument can be valid.
• As a clue rather than a proof. Sharing a trait with a group can be evidence of belonging to it, when that trait is rare outside the group. “It’s in the blue folder, so it probably needs a signature” is reasonable if the folder is almost never used for anything else. The error is claiming the conclusion follows.
• When the speaker plainly means “all and only”. If the office rule is that the blue folder is reserved for documents needing signatures, stating that as a premise makes the argument valid.

The test: does either premise say something about every member of the middle group?

The same shape with a “no”

All cats are mammals. No fish are mammals. So no fish are cats.

Like the textbook case, “mammals” is the predicate in both premises, which is the pattern people learn to flag. But “No fish are mammals” says something about every mammal: none of them is a fish. That distributes the middle term, so the argument is valid. It’s the first condition under “When it isn’t an error”.

“Only” instead of “all”

Only documents that need a signature go in the blue folder. The lease renewal is in the blue folder. So the lease renewal needs a signature.

This is the folder example with one word changed. “Only documents that need a signature go in the blue folder” means everything in the blue folder needs a signature, a statement about every item in the folder. Now the renewal’s place in the folder does carry the conclusion. Whether the office actually follows that rule is a question about the premise’s truth, not the argument’s form.

A shared category feels like a link. Cats and dogs really do have something in common, and “both are mammals” sounds like a reason to group them. The argument trades on that feeling of connection while the premises supply only two separate memberships.

It also often rides on quietly reversing an “all” statement. “All the documents that need a signature are in the blue folder” is easy to hear as “everything in the blue folder needs a signature”. Hurley names that reversal, “All A are B, so all B are A”, as its own formal fallacy, illicit conversion (“All cats are animals” does not give “all animals are cats”).

Ask whether anything else belongs to the middle group. If the blue folder could hold anything other than documents needing signatures, or mammals include things other than cats, the shared membership doesn’t settle the question. That shows the argument doesn’t prove its conclusion, not that the conclusion is false (see Valid vs. true).