What is meant by distribution of terms?

There are four types of propositions, namely (i) Universal affirmative (or A-proposition) (ii) Universal negative (or E-proposition), (iii) Particular affirmative (or I-proposition) and (iv) Particular negative (or O-proposition).

Only these types of propositions can occur in any logical argument. Of course, each proposition, in addition to quantifier and copula, must have a subject term and a predicate term. Now we wish to explain a crucial notion viz. ‘distribution of terms’ in a logical proposition, which plays an important role in developing rules for deductive arguments.

In categorical proposition, terms designate classes of objects. In other words, the subject and predicate terms in a categorical proposition designate classes of objects, and the categorical proposition may be regarded as about these classes.

For example, the proposition “All men are mortal” is about the class of men and the class of mortal beings. More over, it is about all men since we are talking about all the members of the class designated by the term, ‘men’. On the other hand, in asserting the proposition “All men are mortal”, we are not asserting or talking about all the members of the class designated by the predicate term ‘mortal’.

Hence, only a pan of the class of objects designated by the term ‘mortal’ is being considered. So, in any categorical proposition we may refer to either all or some members of a class designated by a term. To express this insight, Aristotle introduces a technical term called ‘distribution’.

A term is said to be distributed in a proposition if and only if it refers to the whole of the class designated by the term. Otherwise, the term is called undistributed. If a term refers to a part of the class designated by the term, it is called undistributed. Alternatively, we may say that if a term refers to the entire denotation of a term (or refers to all the members’ of the class designated by the term) either affirmatively or negatively, then the term in question is distributed. On the other hand, if only a part of a denotation is being referred to by the term then it is called undistributed.

Let us examine which term is distributed in which type of proposition. As we know, there are four types of categorical proposition namely A, E, I and O-propositions. For convenience, let us state the logical form or the structure along with a concrete example of each of the four types of proposition in a tabular form.

Proposition	Logical form	Concrete example
A	All S is P	All men are mortal
E	No S is P	No man is perfect
I	Some S is P	Some men are rich
0	Some S is not P	Some men are not rich

Here the variables S and P are called term variables (i.e. the terms are only to be substituted for S and P). More precisely, ‘S’ and ‘P’ stand for subject and predicate terms respectively.

A-Proposition (All S is P)

A-proposition is a universal affirmative proposition that has the logical structure of the form “All S is P”. It is clear that the subject term ‘S’ in an A-proposition is distributed. Because, here we are considering the entire denotation of the term S. Consider a concrete example, ‘All men are martial’. Here we are asserting that every member of the class designated the term ‘men’ is mortal. Hence this statement is about ‘all men’. So, according to the definition of distribution the term ‘men’ is distributed.

On the other hand, the term ‘mortal’ in are example “All men are mortal” is undistributed as we are not saying anything about the entire denotation of the term ‘mortal’ in the proposition, ‘All men are mortal’. Hence, in this ease at best a part of the denotation of the term ‘mortal’ is being considered. So the predicate term in an A-proposition is undistributed. In sum, in an A-Proposition the subject term is distributed and predicate term is undistributed.

E-Proposition (No S is P)

An E-proposition, otherwise known as universal negative proposition, has the logical structure ‘No S is P’, where ‘S’ and ‘P’ stand for subject and predicate term respectively. ‘No man is perfect’ is an instance of it. To assert ‘No S is P’, one is implying that S and P have no member in common, i.e., S and P are excluded from each other, In other words, no member of s is in P and no member of P is in S. Hence, here we are considering the whole of S as well as whole of P negatively. Hence, both S and P in an E-proposition are distributed. Thus, both the subject and the predicate terms of an E-proposition are distributed.

I-Proposition (Some S is P)

I-proposition is a particular affirmative proposition. It has the logical structure or form. “Some S is P”. Here both “S” and” P” are undistributed because in asserting ‘Some S is P’ (or “Some men are rich”), we are not talking of the entire denotation of term ‘S’ or ‘P’ (or ‘man’ and ‘rich’). Thus, we are considering a part of the denotation of S and P. Hence, the subject and predicate terms in an I-proposition are undistributed.

O-Proposition (Some S is not P)

An O-proposition is a particular negative proposition that has the logical structure or form, ‘Some S is not P’. Here clearly ‘S’ (the subject term) is undistributed as we are considering some S i.e. a part of the class designated by the term S.

The predicate term P is denied of “some S”. When we deny a term we deny the entire class designated by the term, otherwise the denial or negation has no meaning. Hence, the predicate term in an O- proposition is distributed as the class designated by ‘P’ as a whole is being denied of “some S”. Thus, in an O-proposition, the subject term is undistributed and predicate term is distributed.

Now we may summaries our above discussion on distribution in a tabular from as given below.

Distribution of Terms Proposition Logical form Subject Predicate A All S is P Distributed Undistributed E No S is P Distributed Distributes I Some S is P Undistributed Undistributed 0 Some S is not P Undistributed Distributed

From this table we may note the following facts with respect to distribution of terms in standard categorical proposition.

(i) In universal proposition, the subject term is distributed whereas in particular proposition subject term is undistributed.

(ii) In negative proposition the predicate term is distributed while in an affirmative proposition the predicate term remains undistributed.