Fallacy of the Undistributed Middle Term

The fallacy of the undistributed middle term is a syllogistic fallacy in which the middle term is not distributed in either premise. In a valid syllogism, the middle term must be distributed at least once. (See my previous post for an explanation of distribution and other syllogistic logic terminology.)

In simpler terms, a syllogism with an undistributed middle term is one that attempts to equate two different subjects with a common predicate. This formal fallacy is similar to the informal fallacy of guilt by association, which also attempts to connect two things with a shared characteristic.

Examples

If the right terms are chosen, this kind of argument can be persuasive, even though it is poor. Consider this example:

Communists believe in socialized medicine.
Senator Jones believes in socialized medicine.
Therefore, Senator Jones is a Communist.

While it may very well be true that Senator Jones is a Communist, it would be in spite of this argument, not because of it. The middle term ("socialized medicine") is undistributed in both premises. At least one distributed middle term is necessary to link Senator Jones to the Communists. Otherwise, they may be two completely separate parties, who happen to believe the same thing concerning public health care.

By substituting more absurd terms, it's easy to see why this fallacy is poor reasoning:

All dogs are four-legged.
All tables are four-legged.
Therefore, all dogs are tables.

Several years ago, a KJV-onlyist calling himself "1611king" started thread at the Bible Versions Discussion Board about Virginia Mollenkott's involvement with the NIV. Several posters answered with a tu quoques (as I discussed in an earlier post) - the rationale being that if the presence of an immoral person on the NIV translation committee caused 1611king so much discomfort, he ought to feel the same about immoral persons translating the KJV as well. Instead, he dismissed the history as "tainted and spurious." But when it was stated that the history books cited came with documentation of their facts, 1611king replied:

Have you ever heard of the Satanic Bible? It's a REAL book with REAL references. Does this make it credible? No, and neither is your history credible.

[Source]

Syllogistically, this "rebuttal" goes something like this:

Premise 1. Some non-credible books are documented.
Premise 2. These secular history books are documented.
Conclusion. Therefore, these secular history books are non-credible books.

Since both premises are particular affirmatives, the middle term, documented, is undistributed in both. Therefore, the syllogism is invalid. It may still be that the history books are in error, but if so, it is in spite of 1611king's faulty logic, not because of it. His argument doesn't wash.

Remedy

Inform your opponent that he is committing guilt by association. Just because two things share a common characteristic, it doesn't necessarily mean they're connected.