Part I: Lesson

“And yet, if both subject and predicate are used in their fullest extension, the resulting proposition will be false. For, indeed, no affirmation at all could, in those circumstances, be true.”

(Aristotle, On Interpretation 7, trans. Harold P. Cook, 1962)

1. Distribution

In a previous lesson, we discussed the helpful notion of extension. Recall that the extension of a term is just the totality of the members of the category the term refers to. For example, the extension of the term “tree” is the totality of all trees (since those are all the members of the category “tree”). Using this notion, we have the following two definitions, allowing us to classify terms in categorical propositions:

• Distributed: a term is distributed if we are using it to refer to its entire extension;
• Not Distributed/Undistributed: a term is not distributed or undistributed if we are using it to refer only to a part of its extension.

In the following, we will look at the distribution of terms – both subjects, and predicates – in each type of categorical sentence.

a) Universal Affirmative (UA)

Recall that Universal Affirmative (UA) propositions express that “All S is P.” Let’s look at our earlier example – “All cats are animals.” In this case, we are talking about all cats; thus, in (UA) propositions, the subject term is always distributed; ‘all’ distributes over all the items of the subject term in (UA) propositions. The predicate, on the other hand, is not. When we express that all cats are animals, we are saying that every single cat qualifies as an animal and that only some animals qualify as cats. Through (UA) propositions, we are only referring to a part of the extension of the predicate.

b) Universal Negative (UN)

Recall that Universal Negative (UN) propositions express that “No S is P.” From our earlier lesson, consider the proposition “No snakes are birds.” Again, we are referring to the entire extension of our subject. To see this more clearly, think of the proposition as saying that, of all the snakes, none of them qualify as a bird. Then, we observe that the subject term is again distributed; ‘no’ distributes over all the items of the subject term. In (UN) propositions, the predicate is also distributed. When we say of all the snakes that they are not birds, we also mean to say of all the birds that they are not snakes either.

c) Particular Affirmative (PA)

Recall that Particular Affirmative (PA) propositions express that “Some S is P.” One previous example was “Some snakes are poisonous.” In this situation, neither the subject nor the predicate is distributed. In the case of the subject, this is fairly easy to see: we are using ‘Some’ to refer to the snakes, so we are only picking out a subset of all the snakes (quite simply, some of the snakes!). In the case of the predicate, we are also looking at a smaller subset of all the things that fall are ‘poisonous’, in particular those that also happen to be snakes. Clearly, not all poisonous things are snakes; only some of them are. Therefore, we are referring to some of the snakes and to some of the poisonous things, so both terms are not distributed.

c) Particular Negative (PN)

Recall that Particular Negative (PN) propositions express that “Some S is not P”. In the previous lesson, we looked at “Some shirts are not blue” as an example. As with (PA) propositions, ‘Some’ ensures that we are only talking about a subset of all the shirts, so we know that the subject is not distributed. In the case of the predicate, we are taking the relevant subset of shirts and denying it the entire category of blue things. In other words, the shirts we are left with after subtracting the blue ones are shirts about which no part of the extension of ‘blue’ can be predicated. So the predicate is undistributed. 

[Activity] Draw a table with a partner representing the distribution of terms for each sentence type. As you do this, recall why each term is distributed or undistributed.