# 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.

## Explanation

Categorical syllogisms always have three terms:

Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All goldfish have fins.

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

Using four terms invalidates the syllogism:

Major premise: All fish have fins.
Minor premise: All goldfish are fish.
Conclusion: All humans have fins.

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:

Major premise: Nothing is better than eternal happiness.
Minor premise: A ham sandwich is better than nothing.
Conclusion: A ham sandwich is better than eternal happiness.

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:

Major premise: The pen touches the paper.
Minor premise: The hand touches the pen.
Conclusion: The hand touches the paper.

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:

Major premise: All that touches the pen, touches the paper.
Minor premise: The hand touches the pen.
Conclusion: The hand touches the paper.

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.[1]

## Reducing 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.[2] For example:

Major premise: No humans are immortal.
Minor premise: All Greeks are people.
Conclusion: All Greeks are mortal.

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.").[3]

## Classification

The fallacy of four terms is a syllogistic fallacy. Types of syllogism to which it applies include statistical syllogism, hypothetical syllogism, and categorical syllogism, all of which must have exactly three terms. Because it applies to the argument's form, as opposed to the argument's content, it is classified as a formal fallacy.

Equivocation of the middle term is a frequently cited source of a fourth term being added to a syllogism; both of the equivocation examples above affect the middle term of the syllogism. Consequently this common error itself has been given its own name: the fallacy of the ambiguous middle.[4] An argument that commits the ambiguous middle fallacy blurs the line between formal and informal (material) fallacies, however it is usually considered an informal fallacy because the argument's form appears valid.[5]

## References

Notes
1. Copi & Cohen 1990, pp. 206–207.
2. Copi & Cohen 1990, pp. 214–217.
3. Cogan 1998, pp. 95–96.
4. Copi & Cohen 1990, p. 206.
5. Coffey 1912, pp. 302–304.
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