Four-term fallacy

A four-term fallacy (also quaternio terminorum, fallacy of four terms, and if using equivocation, ambiguous middle term) occurs when a syllogism uses four or more terms instead of the required three.

The fallacy is a formal fallacy, because the conclusion does not follow even if the premises are correct.

Form
A four term fallacy uses (at least) four terms, represented here as letters W, X, Y, and Z. The fallacy (when limited to 4 terms) takes two forms. First, where the premises contain 4 terms:

Second, where a new term is introduced in the conclusion:

In both cases, the conclusion does not follow, because all of the terms used in the conclusion are not used in the premises. Compare this to a proper syllogism:

Here, only three terms (X, Y, and Z) are used, and both terms used in the conclusion are used in the premises.

Often, however, the four term fallacy technically uses only three apparent terms, but the truth of either premise and the conclusion or both premises depends on the alternation of two meanings of one term:

Second, where the middle term alternates two meanings (ambiguous middle term proper):

Third, where the predicate alternates two meanings:

Examples
Using the first form:

Using the second form:

Many first-cause arguments are based on a four-term fallacy:

Wow, that God really came out of nowhere! *ba-dum-tss*