Categorical proposition

Properties of categorical propositions

Categorical propositions can be categorized into four types on the basis of their "quality" and "quantity", or their "distribution of terms". These four types have long been named A, E, I, and O. This is based on the Latin affirmo (I affirm), referring to the affirmative propositions A and I, and nego (I deny), referring to the negative propositions E and O.

Quantity and quality

Quantity refers to the number of members of the subject class (A class is a collection or group of things designated by a term that is either subject or predicate in a categorical proposition.) that are used in the proposition. If the proposition refers to all members of the subject class, it is universal. If the proposition does not employ all members of the subject class, it is particular. For instance, an I-proposition ("Some S are P") is particular since it only refers to some of the members of the subject class.

Quality It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. The two possible qualities are called affirmative and negative. For instance, an A-proposition ("All S are P") is affirmative since it states that the subject is contained within the predicate. On the other hand, an O-proposition ("Some S are not P") is negative since it excludes the subject from the predicate.

An important consideration is the definition of the word some. In logic, some refers to "one or more", which is consistent with "all". Therefore, the statement "Some S are P" does not guarantee that the statement "Some S are not P" is also true.

Distributivity

The two terms (subject and predicate) in a categorical proposition may each be classified as distributed or undistributed. If all members of the term's class are affected by the proposition, that class is distributed; otherwise it is undistributed. Every proposition therefore has one of four possible distribution of terms.

Each of the four canonical forms will be examined in turn regarding its distribution of terms. Although not developed here, Venn diagrams are sometimes helpful when trying to understand the distribution of terms for the four forms.

A form (otherwise known as universal affirmative)

An A-proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals". All dogs are indeed mammals, but it would be false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is undistributed to "dogs".

E form (otherwise known as universal negative)

An E-proposition distributes bidirectionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.

The empty set is a particular case of subject and predicate class distribution.

I form (otherwise known as particular affirmative)

Both terms in an I-proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition, it is not possible to say that all Americans are conservatives or that all conservatives are Americans. Note the ambiguity in the statement: It could either mean that "Some Americans (or other) are conservatives" (de dicto), or it could mean that "Some Americans (in particular, Albert and Bob) are conservatives" (de re).

O form (otherwise known as particular negative)

In an O-proposition, only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed. The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the corrupt people group, namely, "All corrupt people are not some politicians", the predicate is distributed.

The distribution of the predicate in an O-proposition is often confusing due to its ambiguity. When a statement such as "Some politicians are not corrupt" is said to distribute the "corrupt people" group to "some politicians", the information seems of little value, since the group "some politicians" is not defined; This is the de dicto interpretation of the intensional statement (${\displaystyle ◻\mathrm{\exists }x[P{l}_x\wedge \mathrm{\neg }{C}_x]}$), or "Some politicians (or other) are not corrupt". But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes clearer; This is the de re interpretation of the intensional statement (${\displaystyle \mathrm{\exists }x◻[P{l}_x\wedge \mathrm{\neg }{C}_x]}$), or "Some politicians (in particular) are not corrupt". The statement would then mean that, of every entry listed in the corrupt people group, not one of them will be Albert: "All corrupt people are not Albert". This is a definition that applies to every member of the "corrupt people" group, and is, therefore, distributed.

Summary

In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not").

Criticism

Peter Geach and others have criticized the use of distribution to determine the validity of an argument.

It has been suggested that statements of the form "Some S are not P" would be less problematic if stated as "Not every S is P," which is perhaps a closer translation to Aristotle's original form for this type of statement.

Another criticism is that there is a little step from "All corrupt people are not some politicians" to "All corrupt people are not politicians" (whether meaning "No corrupt people are politicians" or "Not all corrupt people are politicians", which are different from the original "Some politicians are not corrupt"), or to "Every corrupt person is not some politician" (also different).

Operations on categorical statements

There are several operations (e.g., conversion, obversion, and contraposition) that can be performed on a categorical statement to change it into another. The new statement may or may not be equivalent to the original. [In the following tables that illustrate such operations, at each row, boxes are green if statements in one green box are equivalent to statements in another green box, boxes are red if statements in one red box are inequivalent to statements in another red box. Statements in a yellow box means that these are implied or valid by the statement in the left-most box when the condition stated in the same yellow box is satisfied.]

Some operations require the notion of the class complement. This refers to every element under consideration which is not an element of the class. Class complements are very similar to set complements. The class complement of a set P will be called "non-P".

Conversion

The simplest operation is conversion where the subject and predicate terms are interchanged. Note that this is not the same as the implicational converse in the modern logic where a material implication statement ${\displaystyle P\to Q}$ is converted (conversion) to another material implication statement ${\displaystyle Q\to P}$. Both conversions are equivalent only for A type categorical statements.

From a statement in E or I form, it is valid to conclude its converse (as they are equivalent). This is not the case for the A and O forms.

Obversion

Obversion changes the quality (that is the affirmativity or negativity) of the statement and the predicate term. For example, by obversion, a universal affirmative statement become a universal negative statement with the predicate term that is the class complement of the predicate term of the original universal affirmative statement. In the modern forms of the four categorical statements, the negation of the statement corresponding to a predicate term P, ${\displaystyle \mathrm{\neg }Px}$, is interpreted as a predicate term 'non-P' in each categorical statement in obversion. The equality of ${\displaystyle Px=\mathrm{\neg }(\mathrm{\neg }Px)}$ can be used to obvert affirmative categorical statements.

Categorical statements are logically equivalent to their obverse. As such, a Venn diagram illustrating any one of the forms would be identical to the Venn diagram illustrating its obverse.

Contraposition

Contraposition is the process of simultaneous interchange and negation of the subject and predicate of a categorical statement. It is also equivalent to converting (applying conversion) the obvert (the outcome of obversion) of a categorical statement. Note that this contraposition in the traditional logic is not the same as the contraposition (also called transposition) in the modern logic where the material implication statements ${\displaystyle P\to Q}$ and ${\displaystyle \mathrm{\neg }Q\to \mathrm{\neg }P}$ are logically equivalent. Both contrapositions are equivalent only for A type categorical statements.

Treatment in first-order logic

First-order logic is a much more expressive logic than that given by categorical propositions. In first order logic, the four forms can be expressed as:

• A form: ${\displaystyle \mathrm{\forall }x[S_x\to P_x]\equiv \mathrm{\forall }x[\mathrm{\neg }{S}_x\vee P_x]}$
• E form: ${\displaystyle \mathrm{\forall }x[S_x\to \mathrm{\neg }{P}_x]\equiv \mathrm{\forall }x[\mathrm{\neg }{S}_x\vee \mathrm{\neg }{P}_x]}$
• I form: ${\displaystyle \mathrm{\exists }x[S_x\wedge P_x]}$
• O form: ${\displaystyle \mathrm{\exists }x[S_x\wedge \mathrm{\neg }{P}_x]}$