# 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 ◈:

All rectanglesare.polygonsAll hexagonsare.polygons∴ ~~All~~.rectanglesarehexagons

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.

## Rules of distributivity

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:

Statement | Subject | Predicate | |
---|---|---|---|

A | “All S are P” | distributed | not distributed |

E | “No S is P” | distributed | distributed |

I | “Some S are P” | not distributed | not distributed |

O | “Some S are not P” | not distributed | distributed |

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.