# Four-term fallacy

Invalid syllogism containing more than three terms (typically exactly four). In its simplest form, this may look like this:

All squares are rectangles.
All circles are ellipses.
Therefore: ???

No conclusion can be derived from the two statements, which are not linked by a common middle term.

## Other names

• Quaternio terminorum
• Fallacy of four terms

## Description

A “four-term fallacy” describes a syllogism that contains four (or more) terms instead of the required three. This breaks the formal conditions for syllogisms, in other words: it is invalid.

That these terms are easily recognizable as four different words – as shown in the example above – is likely to be the exception. More often, this fallacy is caused by an ambiguity in the terms themselves (equi­vocation), as in the following example:

Nothing light can ever be dark.
All feathers are light.
Therefore, no feather can be dark.

Although the syllogism has the form of a Modus Celarent Open in Syllogism-Finder App, it only appears at first glance to consist of the required three terms; instead, the term “light” is used in two different, incompatible meanings.

In many cases (but not always!), such ambiguity concerns the middle term, i.e. the term which connects the major and minor premises and which therefore does not appear in the conclusion.

### Distinction from other fallacies

The “four-term fallacy” is a rather general formal fallacy in the formulation of a syllogism. To apply this concept to other forms (e.g. constructive dilemma) at least the number of terms has to be adapted.

A specific variant is the fallacy of the ambiguous middle term, where the middle term occurs in two different meanings. This is probably the form in which the “four-term fallacy” occurs most often.

However, if the individual premises are inherently ambiguous, specifically through an amphiboly (an ambiguity in the grammatical structure of a statement), we speak of a syntactic ambiguity.

## Valid use

### Enthymeme

It is however not fallacious (but still often a weak form of an argument) if a premise has only been implied, for example by omitting intermediate steps of a multipart argument (Ent­hymeme).

Example:

All humans are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

The connection from the major to the minor statement could in this case be established by an implied syllogism, which may have the following form:

All humans are mortal.
All men are humans.
Therefore, all men are mortal.

This then gives rise to the following (valid) syllogism:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

However, such forms are also frequently used to distract from weaknesses in the argument.