Distributivity (logic)

Describes the property of an identifier in a logical statement that it can also refer to any subset of its extension.

Description and examples

The concept of distribution of a term can be best explained with some examples. Consider the following statement:

All dogs are mammals.

Herein, the term “dogs” is distributed, i.e. it can be replaced with an expression describing any subset of the term “dogs” without changing the truth value of the statement: You don’t need additional information to deduce that likewise poodles, dachshunds, hunting dogs, long-haired dogs, etc. are also all mammals.

Take for comparison the following statement:

Some dogs are dachshunds.

Here, the term “dogs” is not distributed, i.e. the term cannot simply be replaced by its subset: Statements like “some poodles are dachshunds” is obviously nonsensical, while others, such as “some hunting dogs are dachshunds” or “neighbour’s dog is a dachshund” could not be evaluated without additional (external) information.

Rules of distribution

The following rules for the distribution of terms apply to the four types of statement that can appear in a syllogism:

Type Statement Distributivity
A All S are P Subject only
E No S is P Both
I Some S are P Neither
O Some S are not P Predicate only

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.

Significance

The concept of distributivity is important for understanding the limitations of certain types of statements, in particular, it determines which conclusions can be derived from a statement. Failure to observe these limitations can lead to the formal fallacies of distribution.