Categorical Propositions

Categorical Propositions

Categorical propositions are assertions about members of categories or classes. Every categorical proposition is a statement about the members of two classes and their relationship to one another. For example,

No bachelors are married.Some Volkswagons are not made in Germany.

These sort of subject-predicate statements are the kind found in a form of logic, known as Aristotelian, traditional or categorical syllogistic.Aristotle (384-322 BCE) was the first to study ways of arguing and formulate logic as a discipline. The form of argument that he identified and systematized used subject-predicate statements in a syllogism (two premises and a conclusion). Because this was the form of logic that, for all practical purposes, was used until the nineteenth century, it is known as traditional logic. Because it was first worked out by Aristotle, it is also known as Aristotelian logic. And, finally, because it deals with categorical statements in a syllogistic form, it is known as the logic of the categorical syllogism. Because we will study a modern form of this traditional or Aristotelian logic, we will refer to it as the categorical syllogistic.
Although modern logic has modified this traditional logic and indeed gone beyond it, the categorical syllogistic is worthy of study for two reasons. One, traditional logic has played a major role in the history of western thought. Indeed, it is the logic most people recognize as logic. Two, the categorical syllogism is a relatively accessible deductive system. It employs a limited number of propositional forms and its syllogisms can be tested for validity without too much technical difficulty. Moreover, one encounters categorical syllogisms in ordinary language. So, we will begin our study of deductive logic with an up-dated version of the traditional syllogism. But to do this, we need to study the categorical proposition more closely.

The Four Kinds of Categorical Propositions

As noted earlier, a categorical proposition is a statement that relates two classes, or categories. The two classes in any given categorical proposition are placed in a subject-predicate relationship. Something is predicated, or said about, some subject. What is said is that a class (indicated by the subject term) is either included in or excluded from the class indicated by the predicate term. Thus, to refer to one of the examples above, “No bachelor is married” states that the class indicated by the subject term (bachelors) is not found at all in the class indicated by the predicate term (married persons). Similarly, to say that all Catholic priests are male is to observe that everyone who is a Catholic priest (the subject term) is included in the male-class (the predicate term).
There are four kinds of categorical propositions. Using “S” and “P” as symbols (to stand for “subject” and “predicate”), they are

Universal affirmative:     All S are P.            

Universal negative:         No S are P.

Particular affirmative:     Some S are P.

Particular negative:         Some S are not P.

The words “all” and “some” are called “quantifiers” because they indicate the quantity of the subject. That is, they specify how much of the subject class is included in the predicate class. (“No” indicates that zero members are included.) The verb in a properly expressed categorical proposition is always some form of the verb “to be”, and is known as the “copula”. Thus we get the following schema:

quantifier:    all (every), no, somesubject:        the class which is included in or excluded from the predicate
copula:         is, are, was, were
predicate:     the class of which the subject is or is not a part

This analysis does not, however, clearly indicate whether a proposition is affirmative or negative in quality. An affirmative proposition is one that states that the subject is included in the predicate class; a negative one that the subject is excluded from the predicate. Thus, a
more complete scheme would add:

negative qualifier:     no, not

Since the four basic categorical propositions have a subject and predicate term and a copula, one way to distinguish them is by their quantity and quality. Each proposition will be universal or particular (distinguished by quantity) and affirmative or negative (distinguished according to quality). Thus, above, we were able to distinguish them as follows:

Universal affirmative:     All S are P.Universal negative:         No S are P.
Particular affirmative:    Some S are P.
Particular negative:         Some S are not P.

The Square of Opposition

There is another way to distinguish these four propositions. We can arrange them in a square of opposition. This indicates that the universal affirmative and particular negative are contradictories and that the universal negative and particular affirmative are contradictories. That is, if one contradictory is true, the other must be false. Here is a table sorting the four propositions according to quantity and quality:

Affirmative Negative
Universal All S are P No S are P
Particular Some S are P Some S are not P

Traditionally, the universal affirmative proposition was called “the A proposition and the particular afirmative was called “the I proposition.” (The letters “A” and “I” come from the first two vowels in the Latin word, affirmo, “I affirm.”) The universal negative proposition was called “the E proposition” and the particular negative was called “the O proposition” (so called, from the vowels in the Latin word, nego, “I deny”). We will follow this use, referring to the propositions as “A, E, I and O propostions.” So this gives us a chart that looks like this:

Affirmative Negative
Universal A E
Particular I O