Fallacy of four terms
Encyclopedia
The fallacy of four terms is the logical fallacy that occurs when a syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

has four (or more) terms rather than the requisite three. This form of argument is thus invalid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

.

## 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
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

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

The premises don't connect "humans" with "fins", so the reasoning is invalid. Notice that there are four terms: "fish", "fins", "goldfish" and "humans". Two premises aren't 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
Equivocation
Equivocation is classified as both a formal and informal logical fallacy. It is the misleading use of a term with more than one meaning or sense...

: 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" means the thing being described has only marginal 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 you use "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.

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

## 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. 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
Immediate inference
An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green." we can make the immediate inference that "No toads are not green." This new statement is known as the contrapositive of the original statement...

known as obversion
Obversion
In traditional logic, obversion is a "type of immediate inference in which from a given proposition another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original...

(that is, "No humans are immortal." is equivalent to "All humans are mortal.").

## Classification

The fallacy of four terms is a syllogistic fallacy
Syllogistic fallacy
Syllogistic fallacies are logical fallacies that occur in syllogisms. They include:Any syllogism type :*fallacy of four termsOccurring in categorical syllogisms:*related to affirmative or negative premises:...

. Types of syllogism to which it applies include statistical syllogism
Statistical syllogism
A statistical syllogism is a non-deductive syllogism. It argues from a generalization true for the most part to a particular case .-Introduction:Statistical syllogisms may use qualifying words like "most", "frequently", "almost never", "rarely",...

, hypothetical syllogism
Hypothetical syllogism
In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.-Propositional logic:...

, 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
Formal fallacy
In philosophy, a formal fallacy is a pattern of reasoning that is always wrong. This is due to a flaw in the logical structure of the argument which renders the argument invalid...

.

Equivocation
Equivocation
Equivocation is classified as both a formal and informal logical fallacy. It is the misleading use of a term with more than one meaning or sense...

of the middle term
Middle term
The middle term must distributed in at least one premises but not in the conclusion of a categorical syllogism. The major term and the minor terms, also called the end terms, do appear in the conclusion.Example:...

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. An argument that commits the ambiguous middle fallacy blurs the line between formal and informal (material)
Informal fallacy
An informal fallacy is an argument whose stated premises fail to support their proposed conclusion. The deviation in an informal fallacy often stems from a flaw in the path of reasoning that links the premises to the conclusion...

fallacies, however it is usually considered an informal fallacy because the argument's form appears valid.