# Categorical Propositions

Categorical propositions are simple sentences composed in a noun-verb-direct object structure with a quantifying prefix. The first noun is the subject (S), the direct object is the predicate (P). The verb is called the copula and is almost always a conjugation of the verb "to be." In most cases sentences using other verbs can be rewritten using the verb "to be." The predicate (also known as a direct object) is typically either a noun or an adjective.

The quantifier can be depicted visually using a diagram developed by Venn, hence their name: "Venn diagrams." Three quantifiers are used in categorical propositions: all, none, and some.

## Quality

Categorical propositions can have one of two qualities. The proposition is either affirmative or negative. Affirmative propositions are represented by the letters A and I from the Latin AffIrmo. Negative propositions are represented by the letters E and O from the Latin nEgO.

## Quantity

Categorical propositions may refer to all, some, or no members of a category. Propositions which refer to all or none are termed universal propositions. Propositions which refer to some are termed particular propositions.

### All

The diagram for "all" is an open, unfilled circle.

### None

The diagram for "none" is an open, filled, darkened, or cross-hatched circle. That "all" is an empty circle and "none" is a filled in circle can be confusing later on in the course. The filled-in circle is equivalent to an "erased" circle. That category no longer exists. In these documents the circle is a darkened circle, on a white board or sheet of paper the category is usually "cross-hatched" with diagonal lines.

### Some

The diagram for "some" is an open, unfilled, circle with a single "X" in the circle.

### Some not

The diagram for "some (are) not" a member is an open, unfilled, circle with a single "X" outside the circle.

## Categorical Propositions

The four quantifiers form the four types of categorical propositions.

### Universal Affirmative (A)

The Universal Affirmative (A) is in the form "All ______ is ______, sometimes referred to as "All S is P" where S is the Subject and P is the Predicate.

An example would be "All cats are animals." That categorical proposition happens to be true. A categorical proposition can be false, such as "All planets are flat." The diagram below depicts "All Sepes are Kosraen"

Note that the left circle consists of Sepes (or women named Sepe) and the right circle contains all Kosraens. Since there are no Sepes who are not Kosraen, the part of the Sepe circle outside the Kosraen circle is "erased" by "blacking" it out. The diagram depicts the proposition, whether the proposition is true or false is not a consideration.

### Universal Negative (E)

The Universal Negative (E) is in the form "No ______ is ______" or "No S is P."

"No fish are birds" would be a universal negative. "No sakau en Pohnpei is alcoholic" is diagrammed below. Put more formally, "No sakau en Pohnpei is a beverage that contains alcohol."

The diagram for the Universal Negative is

### Particular Affirmative (I)

The particular affirmative states that there are members of the subject in the predicate category. The proposition does not rule out the possibility that all members of the subject are members of the predicate category.

The particular affirmative takes the form "Some _____ is _______" or "Some S is P." Examples would include "Some fish are animals which swim" or "Some years are leap years." The diagram below depicts "Some stone money is from Yap." Remember, whether the proposition is factually true or false is not being tested or asserted.

### Particular negative (O)

The particular negative states that there are some members of the subject that are not in the predicate category. The proposition does not rule out the possibility that no members of the subject are members of the predicate category. The proposition is also not a guarantee that some members of the subject are in the predicate.

The particular negative takes the form "Some _____ is not _______" or "Some S is not P." Another way to think about the particular negative that restores some symmetry with the universal negative would be to say, "Not every S is P." Examples would include "Some birds are not flying birds" or "Some jugglers are not joggers." The diagram below depicts "Some Japanese navy ships in Chuuk state are not floating ships."

## Distribution

The statement that "All women named Sepe are Kosraen" clearly does not also mean that "All Kosraens are women named Sepe." Yet the statement that "No sakau en Pohnpei is a beverage containing alcohol" clearly also means that "No beverage containing alcohol is sakau en Pohnpei."

This difference in the reversibility of a proposition is referred to as "distribution." The term distribution refers to whether the category is distributed to all members of the category. In the A proposition the subject is distributed to the predicate, but not the reverse.

In an E proposition both the subject and predicates are distributed. "No sakau en Pohnpei is alcoholic" refers to all sakau en Pohnpei, the predicate refers to all alcoholic beverages. This inclusion of all in both means both the subject and predicate are distributed.

The I proposition "Some stone money is from Yap" does not mean that all stone money is from Yap. Nor does it mean all things from Yap are stone money.

The O proposition that "Some Japanese Navy ships in Chuuk are not floating ships" does imply that "Some not floating ships are Japanese Navy ships in Chuuk." The predicate is distributed. The subject, however, is not distributed. Japanese Navy ships in Chuuk could include floating Japanese Navy ships visiting Weno.

## Existential Import

A term, whether subject or predicate, is said to have existential import if the term implies the actual existence of members of that category. Aristotelian logic inferred affirmative propositions such as "All cats are felines" include that there are such entities as cats. Aristotelian logic also inferred the existence of members of I propositions. "Some diesel engines are engines powered by coconut oil," also implied the existence of such diesel engines.

There are affirmative categories, however, that one wants to be empty categories. Consider the A propositions "All students who are intoxicated on campus will be suspended." The hope of the college is that this subject will remain empty and will not be substantiated by the existence of entities (drunk students). There are other propositions, "Some sokolei are college educated," where the existential import of the subject (sokolei: forest elves of Pohnpei) is uncertain.

Modern Boolean logic does not presume that the universal propositions (A and E) contain members. These propositions are not said to assert the existence of members. There is no "existential import" implied. The particular propositions, however, are taken to have members. The assertion that "Some stone money is from Yap" is taken to imply the existence of something called stone money. Existential import is carried by, or implied by, I and O propositions. The statement "Some sokolei are college educated," establishes the existence of sokolei.

Aristotle and the Peripatetic school of philosophers he founded did not write the O proposition in Latin as "Some S is not P." The construction "Some S is not P" may be due to a translation by Boethius circa 500 CE. Aristotle formulated O as "Not every S is P." This restores a symmetry to the propositions (quantifier-subject-copula-predicate) by returning the quantifier to the front of the proposition. This also removes the construction "Some S" which implies that members of S must exist, and replaces it with "Not every S." Although the change is subtle, "not every" includes the possibility of "none." This allows the O proposition to lack existential import, O can be empty.

The result is that one has to specify whether one is using Aristotelian or modern logic for purposes of determining existential import.

## Summary table

The following table attempts to depict the above terminologies in a single table using Boolean existential import assumptions. Note that whether George Boole, for whom Boolean logic is named, subscribed to existential import for I and O propositions is not universally accepted.

Quality Predicate distribution Quantity Subject distribution Exist. Import Affirmative AffIrmo Negative nEgO Predicate undistributed Predicate distributed Universal Subject distributed AAll S is P ENo S is P None Particular Subject undistributed ISome S is P O Some S is not P Implied

## Conversion

Conversion is the switching of the subject S and the predicate P. If a categorical proposition is true, then only the universal negative E and the particular affirmative I are assured to be also true.

The converse of All cats are animals is All animals are cats. The first proposition is true, the converse is not true. The way to refer to this situation is to say that "the universal affirmative" does not convert. When we say this, we mean that as a general rule the universal affirmative does not remain true after conversion.

There are situation that one can contrive to make the universal affirmative convert. All rocks are stones converts to All stones are rocks. When the subject S and the predicate P refer to the same thing, then the universal affirmative converts. This conversion would be termed "trivial." A special case. Despite the existence of contrived, special cases, the universal affirmative is still said to not convert.

The universal negative, however, does convert. No cats are dogs converts to No dogs are cats. If the original universal negative is true, then the converted universal negative is always true. There are no special cases one can find where the original is true and the conversion is not.

## Practice

Identify the type of proposition below, the subject, copula, and predicate in the following four propositions. Draw a Venn diagram for each proposition.

1. Some cloudy days are not rainy days.
2. No rainy days are dry days.
3. Some cloudy days are rainy days.
4. All sunny days are hot days.
5. Some students have homework.
6. Not every pwisehn malek is chicken feces.
7. Yesleen, Noleen, and Maybesoleen are Pohnpeian women.
8. Cows are fish.

9. Create a true particular affirmative proposition I and then convert that proposition. Write down both and explain whether the particular affirmative converts.

10. Create a true particular negative proposition O and then convert that proposition. Write down both and explain whether the particular negative converts.

11. Write an A, E, I, and O proposition in your first language.