Fallacy of four terms

The fallacy of four terms (Latin: quaternio terminorum) is the formal fallacy that occurs when a syllogism has four (or more) terms rather than the requisite three. This form of argument is thus invalid.

Categorical syllogisms always have three terms:

Here, the three terms are: "goldfish", "fish", and "fins".

Using four terms invalidates the syllogism:

The premises do not connect "humans" with "fins", so the reasoning is invalid. Notice that there are four terms: "fish", "fins", "goldfish" and "humans". Two premises are not enough to connect four different terms, since in order to establish connection, there must be one term common to both premises.

In everyday reasoning, the fallacy of four terms occurs most frequently by equivocation: using the same word or phrase but with a different meaning each time, creating a fourth term even though only three distinct words are used:

The word "nothing" in the example above has two meanings, as presented: "nothing is better" means the thing being named has the highest value possible; "better than nothing" only means that the thing being described has some value. Therefore, "nothing" acts as two different words in this example, thus creating the fallacy of four terms.

Another example of equivocation, a more tricky one:

This is more clear if one uses "is touching" instead of "touches". It then becomes clear that "touching the pen" is not the same as "the pen", thus creating four terms: "the hand", "touching the pen", "the pen", "touching the paper". A correct form of this statement would be:

Now the term "the pen" has been eliminated, leaving three terms. [note: this argument is now valid but unsound because the major premise is untrue]

The fallacy of four terms also applies to syllogisms that contain five or six terms.

Sometimes a syllogism that is apparently fallacious because it is stated with more than three terms can be translated into an equivalent, valid three term syllogism. For example:

This EAE-1 syllogism apparently has five terms: "humans", "people", "immortal", "mortal", and "Greeks". However it can be rewritten as a standard form AAA-1 syllogism by first substituting the synonymous term "humans" for "people" and then by reducing the complementary term "immortal" in the first premise using the immediate inference known as obversion (that is, "No humans are immortal." is equivalent to "All humans are mortal.").

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