Fallacies of distribution

If the distributivity of the terms is not taken into account in a logical conclusion, this can lead to an invalid conclusion.

For example, in the following syllogism Show in Syllogism-Finder App:

All rectangles are polygons.
All hexagons are polygons.
All rectangles are hexagons.

In both premises, the term “polygons” is in an undistributed position, i.e. it is used in a way that refers only to a subset of all polygons (namely those that are rectangles or hexagons, respectively). For this reason, the terms refer to different extensions and thus cannot connect the two premises. A conclusion based on such premisses can not be valid.

The term distribution or distributivity describes if a term is used refers to a subset of the objects it designates or to all objects.

In the example above, a statement such as “all rectangles are polygons” describes all rectangles, but only a part of the polygons - in particular those that are also rectangles. Thus, the term “rectangles” is distributed, while “polygons” is undistributed.

The following table gives an overview of the distributivity of the terms in the four categorical statement types:

Specifically for syllogisms, the following rules apply:

• The middle term, which connects the two premises, must in at least one of the premises occur in a distributed position. A syllogism where the middle term is undistributed in both premises commits the fallacy of the undistributed middle.

• If a term appears in the conclusion statement in a distributed position, it must also be distributed in the premise in which it appears. Traditionally, there a distinction between fallacies of undistributed major and minor term is often made, but for simplicity, these both are treated here as illicit process.

• Syllogism
• Distributivity

• Undistributed middle
• Illicit process

• Affirmative conclusion from a negative premise
• Fallacy of exclusive premises
• Negative conclusion from affirmative premises