# Categorical Propositions

### From Logic

An understanding of Classical logic begins with an understanding of Categorical Propositions.

## Contents |

### Definition

The Aristotelian or classical study of deduction focuses on arguments containing propositions of a special kind: called "categorical propositions". Categorical propositions are propositions that make declarations about entities belonging to, or not belong to, categories or classes. All categorical statements relate two classes or categories in some manner: we call these two terms the subject "S" and the predicate "P".

To understand all the parts of a categorical syllogism, first consider the following arguments:

No soldiers are cowards

All marines are soldiers

Therefore, no marines are cowards

and

Some Men are professors

Some professors are not tenured

Some men are not tenured.

All three of the propositions in this argument are categorical propositions. These propositions either affirm or deny that some class (S) is included in some other class (P), either in whole, or in part. There are 4 ways in which two classes or categories may be related to each other:

1) **Every** member of the first class is also a member of the second class - "All S are P". (All marines are soldiers)

2) **Some** but not all members of one class are also a member of the second class "Some S are P" (Some men are professors)

3) **Some** but not all members of one class are **not** a member of the second class "Some S are not P" (Some professors are not tenured)

4) The two classes have **no** members in common "No S are P" (No soldiers are cowards)

Here are the 4 types of propositions in brief form:

**All** S are P

**No** S are P

**Some** S are P

**Some** S are **not** P

Every standard form categorical proposition also has a quality, a quantity, and a specific manner of distributing the Subject and Predicate categories within the statement. Does this appear daunting at this point? While this may seem confusing right now, read through the following sections, and all will become clear.

### Quantity

There are two types of quantitative sentences in classical logic: **Universal and particular**. "All logicians are philosophers" is an example of the Universal quality. "Some logicians are philosophers" exemplifies the particular quality.

A universal categorical proposition, naturally enough, refers to all entities that belong to a particular class. A particular categorical proposition refers to at least one member of a class or excludes at least one member of a class.

One can also make statements regarding JUST a singular entity in classical logic: according to the traditional manner of exposition, singular sentences are treated as universal ones. The subject terms of singular sentences were seen as terms with an extension of only one element, e.g. "Socrates is a philosopher" was seen as equivalent to "All people identical to Socrates are philosophers", with the proviso that there is only one "Socrates".)

### Quality

There are two types of qualities: affirmative or negative. Statements that begin with the terms "All" or "some" are affirmative statements, whereas statements that declare "None" or "Some are not" are negative in quality. For example: "All S are P" is a universal affirmative sentence. "No S is P" is universal and negative. "Some S is P" is particular and affirmative and, finally, "Some S is not P" is particular and negative.

Now, let's put together what we have learned so far, here are the 4 types of propositions in brief form:

(A) All S are P (E) No S are P (The first two are Universal propositions) (I) Some S are P (O) Some S are not P : (The last two are Particular propositions)

**S** stands for Subject, **P** for Predicate, and words, "all", "no" and "some" are quantifiers. The quality of a statement is either affirmative (all, some) or negative (no, are not)
Thus, the first statement is a **Universal Affirmative Categorical Proposition**.

Notice the letters in parentheses before each of the four types propositions: **A, E, I, O**. These sentence forms were given short names, in a manner typical of the mnemonic strategies of the Middle Ages. The universal affirmative was called "A" (from the first vowel in the Latin "Affirmo", meaning "I affirm"). The universal negative was called "E" (from the first vowel in the Latin "Nego", meaning "I deny"), whereas the particular affirmative and particular negative were called "I" and "O" respectively (from the second vowel in "Affirmo" and "Nego" respectively).

### Distribution

The term "distribution" refers to the ways in which terms can occur in categorical propositions. A proposition distributes a term if it refers to all members of the class designated by the term. Let's take a look at the four types of statement (A, E, I, O) to see which terms they distribute:

(A) All dogs (S) are animals(P): the entire class "dogs" is *included* in the class "animals", so we can say that an A proposition distributes its subject. However, the predicate "animals' is not necessarily distributed, (i.e. all animals are not necessarily dogs) so an A proposition does not distribute its predicate.

(E) No dogs (S) are cats (P): the whole of the class "dogs" is said to be *excluded* from the class "cats", since all members of the class "dogs" (S) are referred to, we can say that the Subject has been distributed. At the same time, in asserting that the whole class of dogs is excluded from the class of cats, it is also asserted that the entire class of cats is excluded from the class of dogs. An E proposition therefore distributes both their subject and predicate terms

(I) Some dogs(S) are greyhounds(P): neither class is wholly included in the other, so *neither* the subject nor the predicate are distributed.

(O) Some dogs (S) are not greyhounds (P): again, the subject is not wholly included in the predicate, it says nothing about all dogs, but refers to some member of the class designated by the term. It says of this part of the class of all dogs that it is excluded from the class of greyhounds, that is from the whole of the latter class. Given the particular dogs referred to, it says that each and every member of the class of greyhounds is not one of those particular dogs. When something is said to be excluded from a class, the whole of the class is referred to, just as, when a person is excluded from a country, all parts of that country are forbidden to that person. Thus the particular negative categorical proposition *does distribute its predicate term*.

### An Easy Shortcut

Only Universal propositions, either affirmative and negative, distribute their subject terms. Particular propositions, both affirmative and negative, do NOT distribute their subject terms

Thus the quantity of any categorical proposition determines whether its subject is distributed!

Only Negative propositions, either universal or particular, distribute their predicate terms Affirmative propositions, either universal and particular, do NOT distribute their predicate terms

Thus the quality of any categorical proposition determines whether its predicate is distributed!

So, here is the key to determining the how categorical propositions distribute their Subjects and Predicate terms:

Universal Quantity distributes the Subject term

Negative Quality distributes the Predicate term!

### Review

You should now know that a categorical proposition deals with relationships between categories or classes, that we refer to the two classes in the proposition as the subject and the predicate, and that these statements have both a quantity and a quality. You should also understand how each of the four basic types: A, E, I, and O include or exclude (i.e. distribute) the subject and predicate in relation to each other, based on their quantity or quality. This is a very big part of your understanding of syllogistic logic, so make sure to grasp these points.

As a reminder, here are the 4 types of propositions in brief form:

(A) All S are P (E) No S are P (The first two are Universal propositions) (I) Some S are P (O) Some S are not P : (The last two are Particular propositions)

S stands for Subject, P for Predicate, and words, "all", "no" and "some" are quantifiers. The quality of a statement is either affirmative (all, some) or negative (no, are not) Thus, the first statement is a Universal Affirmative Categorical Proposition. It distributes it's Subject, but not its predicate.

Those interested in continuing to learn about classical logic can proceed to the next section, Traditional Square of Opposition.

## References

- Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
- Hurely, P. J. (2000) A Concise Introduction to Logic - 7th Edition