Four-term fallacy

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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:

P1: W is X.
P2: Y is Z.

C1: W is Z.

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

P1: X is Y.

P2: Y is Z.

C1: W is X.

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:

P1: X is Y.

P2: Y is Z.

C1: X is Z.

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

Examples

Using the first form:

P1 : All fish have fins.

P2 : All humans have hair.

C1 : All fish have hair.

Using the second form:

P1: All fish have fins.
P2: All goldfish are fish.
C1: All humans have fins.

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

P1: All events have a cause.
P2: Causal chains have to start somewhere.
C1: God is the cause!

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