Part I: Lesson

The Universal Affirmative and Universal Negative are two of the four types of categorical propositions (see next lesson for the other two types). They are called “categorical”  because they relate two categories of things. Because this notion is so crucial, we must once again briefly review what we mean by categories.

By category, we mean a collection of things that exclusively share some common property/properties. Terms refer to categories. So, we mean the same thing when we express these these five sentences:

1. Apples are red.
2. Things which have the property of being apples are red.
3. Apples are things which have the property of being red.
4. Things which have the property of being apples are also things which have the property of being red.
5. Members of the category “apples” are also members of the category “red”.

Categories can have many members, like the category “dogs,” but they can also have only one member, like the category “Aristotle.” “Aristotle” has only one member, namely Aristotle himself! Since he is the only thing that is Aristotle, he is the only member of such a category. We can represent categories like “countries” using circles:

As one can see, all of the smaller circles, representing the categories of specific countries, are members of the larger circle, representing the category “countries.” This is because Japan, France, Brazil, and the United States are all countries. Note that if we had space, we would put all countries inside the circle. It is worth noting that there is no category “all dogs” or “all countries”. There are the categories “dogs” (or “dog”) and “country” (or “countries”). However, “all” is what’s called a quantifier. A quantifier is an expression that indicates how many of the members of a certain category are being talked about in a proposition. “No”, “Some”, and “Every” are some other quantifiers. Now that we understand what categories and quantifiers are, we can define some types of propositions:

• Universal Affirmative (All A’s are B’s): a proposition stating that all members of a given category (A) are also members of another category (B). These propositions are of the form “All S is P,” where S is the Subject and P the Predicate. These are also sometimes referred to as A Sentences.
• Universal Negative (No A’s are B’s): a proposition stating that no members of a given category (A) are also members of another category (B). These propositions are of the form “No S is P,” where S is the Subject and P the Predicate. These are also sometimes referred to as E Sentences.

Generally, to affirm is to say “yes” to something, or to take it as true ; to negate is to say “no”, or to take it as false. Universal Affirmative and Universal Negative propositions get their names by affirming or denying that some predicate is true of a subject. For instance, in the Universal Affirmative proposition, “All cats are mammals,” we affirm that the predicate, ‘is an animal’, is true of every cat. In other words, we affirm that everything that has the property of being a cat also has the property of being an animal. Put one more way, every member of the category “cats” is also a member of the category “animals.” In the Universal Negative proposition, “No cats are reptiles”, we deny that the predicate, ‘is a reptile’, is true of any cat. In other words, we are saying that no members of the category “cats” are also members of the category “reptiles” (and vice-versa).  

It is important to observe, however, that in calling a proposition “affirmative”, we are not saying that we affirm it, in the sense that we think the proposition as a whole is true. Affirmative and negative propositions get their names for what they express about their subjects: affirmative propositions say that some predicate is true of their subject, negative propositions deny that some predicate is true of their subject. Any one of these propositions can turn out to be true or false, as the examples below will show. 

Lastly, we should note that these propositions are universal because their subject talks about all members of the category referred to by the subject, not just merely some of them. They say that all members of a category either satisfy or do not satisfy some predicate. Compare, for example,  “All cats are mammals” with “Some cats are mammals”; only the first is a universal proposition, speaking about all cats, rather than just some particular cats. These latter propositions are called particular propositions; we’ll deal with those in the next lesson.