## Four Categorical Propositions and The Laws of Thought

*in*Aristotelian Logic

## FOUR STANDARD FORM CATEGORICAL PROPOSITIONS

Categorical propositions tell us about the relationship between classes. They tell us whether a particular class is included in another class or it is excluded from it, wholly or partially. A class is a collection of objects with the same features(s) or characteristic(s). So, there is hardly an object that does not belong to a class. For instance we have the class of chairs, men, dogs, lions, eagles, lizards, etc.

In addition, we should note that while a given class excludes members of another class, it can also be a part of a larger class. We have, for example, the following instances.

– The class of chairs excludes the class of cars, but it is part if the larger class of furniture.

– The class of men excludes the class of lizards, but is part of the larger class of humans.

– The class of dogs excludes that of ants, but it is part if the larger class of domestic animals.

– The class of lions excludes the class of fishes, but it is part of the larger class of wild animals.

– The class of eagles excludes the class of hawks, but they (eagles and hawks) are both part of the larger class of birds.

– The class of lizards excludes the class of monkeys, but it is part of a larger class of reptiles.

Having discussed what a class is let us now look at the Four Standard Form Categorical Propositions. They are as follows:

A. Universal Affirmative Proposition (also known as A proposition)

B. Universal Negative Proposition (also know as E proposition)

C. Particular Affirmative Proposition (also known as I proposition)

D. Particular Negative Proposition (also known as O proposition)

**A. UNIVERSAL AFFIRMATIVE PROPOSITIONS**

Universal Affirmative Propositions affirm or claim that all the members of a given class are included or contained in another class. Below are some examples:

All students are intelligent.

Nigerians are corrupt.

Every lecturer is hardworking.

The above propositions affirm that all (no exception) students (Nigerians, lecturers) are included or contained in the Class of intelligent (corrupt, hardworking) people.

**B. UNIVERSAL NEGATIVE PROPOSITIONS**

Universal Negative Propositions deny or reject the inclusion of a class in another class. In other words, a class is excluded completely from another class. Below are some examples:

No judge is wicked

Arabs are not peaceful

Every politician is not truthful

The above propositions exclude Judges (Arabs, Politicians) from the Class of wicked (peaceful, truthful) people.

**C. PARTICULAR AFFIRMATIVE PROPOSITIONS**

Particular Affirmative Propositions affirm or claim that some (not all) members of a class are included or contained in another class. Below are some examples:

Some Americans are black.

There are people who are good.

A good number of women are hardworking.

Few policemen are honest.

The above propositions affirm that some (not all) persons in the class of Americans (people, women, policemen) are also in the class of black (good, hardworking, honest) people.

**D. PARTICULAR NEGATIVE PROPOSITIONS**

Particular Negative Propositions exclude some (not all) members of a Class from another Class. We have some examples below:

Some workers are not punctual.

There are bosses who are not kind.

Few leaders are not wicked.

The above proposition exclude some (not all) members of the class of workers (Bosses, Leaders) from the class of punctual (kind, wicked) people.

**E. QUALITY AND QUANTITY**

The four standard form categorical propositions have both quality and quantity. If the proposition affirms some class inclusion, whether complete or partial, its quality is affirmative. In other words, propositions that affirm completely the inclusion of a Class, and that affirm partially the inclusion of a Class, are Affirmative. A and I propositions have this quality.

On the other hand, if the proposition denies Class inclusion, whether complete or partial, its quality is negative. So, all E propositions that deny completely the inclusion of a Class, and all O propositions that deny partially the inclusion of a Class are Negative.

Copi and Cohen are of the view that A and I propositions are thought to have been derived from the Latin word “Afflrmo” meaning “I affirm”, and E and O proposition are thought to have been derived from the Latin word ” nEgO” meaning “I deny”.

Quantity has to do with the number of members of the class under reference. If the proposition refers to all members of the class designated by its subject term, its quantity is universal. A and E propositions refer to all members of their class and so are Universal in terms of quantity. If the proposition refers only to some members of the Class designated by its subject term, its quantity is particular. I and O propositions refer to some of the members of their class and so are particular in terms of quantity.

Summarizing the four standard form of categorical propositions, we have

(a) All students are intelligent (A) Affirmative (Quality) Universal (Quantity)

(b) No students are intelligent (E) Negative (Quality) Universal (Quantity)

(c) Some students are intelligent (I) Affirmative (Quality) Particular (Quantity)

(d) Some students are not intelligent (O) Negative (Quality) Particular (Quantity)

It should be noted that a standard form categorical proposition has a structure: quantity, subject, copula, predicate. The quantifiers are All, No, Some, which are always implied even when not explicitly stated as in “Men are good”. The full proposition is ” All men are good”. In this example, All is the quantifier, men the subject, are the copula, and good is the predicate. The quantifier refer to the number of members in a given subject class. The copula links the subject men to the predicate good, and tells us about the relationship between them. The copula is often in the form of the verb “to be”, as in is, was, are, is not, was not, are not.

**F. DISTRIBUTION**

When we take a look at a standard form categorical proposition, we want to know whether the proposition is referring to the whole or part of the subject and predicate terms. Bello says “a term is distributed if it is intended to refer to every member of the class designated by the term.”Efimini puts it in another way thus: “a term is distributed if the proposition in which it occurs covers everything within the universe of discourse referred to by that term.” A term is undistributed if it refers to some (not all) members of the class designated by the term.

(a) Universal Affirmative Proposition (A Propositions).

An example of A proposition is: “All men are higher animals”.

(i) Quantifier ‘All’

(ii) Subject term ‘men’

(iii) Copula ‘are’

(iv) Predicate term ‘higher animals’

The quantifier and subject term say “All men” without any exception. So the subject term is distributed. But the proposition does not tell us anything about the other animals in the class of “higher animals”. So the predicate is undistributed. In summary, A propositions distribute their subject terms while their predicate terms are undistributed.

(b) Universal Negative Proposition (E Propositions)

An example of E proposition is: “No senator is poor”

(i) Quantifier “No”

(ii) Subject term “senator”

(iii) Copula “is”

(iv) Predicate term “poor”

The quantifier and the subject term say “No senator”. The class of senators without exception is being referred to. So the subject term is distributed. The proposition removes completely the class of senators from the class of “poor” people. However, the class of ‘poor’ people is, at the same time, completely removed from the class of senators. To demonstrate this we can rewrite the proposition to read: “No poor person is a senator” without distorting the meaning of the proposition. This being the case, we are right in saying that the predicate term, “poor”, is also distributed. In summary, E propositions distribute both their subject and predicate terms.

(c) Particular Affirmative Propositions (I Propositions)

An example of I proposition is: “Some students are hardworking”.

(i) Quantifier “Some”

(ii) Subject term “students”

(iii) Copula “are”

(iv) Predicate term “hardworking”

The quantifier and the subject term say “Some students”. Since it says “some students” and not “all students”, them the subject term is undistributed. The proposition adds some members of the class of students to the class of “hardworking” (predicate term) people, without telling us any more about the class of hardworking people as a whole. So we can say that the predicate term (hardworking) is undistributed. In summary, the subject and predicate terms of I propositions are undistributed.

(d) Particular Negative Proposition (O proposition)

An example of O proposition is: “Some footballers are not professional”

(i) Quantifier “Some”

(ii) Subject term “footballer”

(iii) Copula “are not”

(iv) Predicate term “professionals”

The quantifier and the subject term say “some footballer”. Since it says “some footballers” and not “All footballers”, then it is undistributed. The proposition excludes some members of the Class of footballers from the class of “professionals” as can be inferred from the copula, “are not”. At the same time, all the members of the class of professionals are also excluded from “some footballers”, thereby making the predicate term, “professionals”, distributed. In summary, while the subject term of O proposition are undistributed, the predicate term is distributed.

We may summarize distribution as follows:

Universal propositions, both affirmative and negative, distribute their subject terms, whereas particular propositions, whether affirmative or negative, do not distribute their subject terms… Affirmative propositions, whether universal or particular, do not distribute their predicate terms, whereas negative propositions, both universal and particular, do distribute their predicate terms.

**LAWS OF THOUGHT**

There are basically three Laws of Thought held to be fundamental principles behind all reasoning. In other words, it is traditionally held that “there are exactly three basic Laws of Thought, laws so fundamental that obedience to them is both the necessary and the sufficient condition of correct thinking.”

These laws are:.

A. The law of Identity

B. The law of Noncontradiction

C. The law of Excluded Middle

A. THE LAW OF IDENTITY

. This law or principle states that an object is what it is and nothing else. If you are an object man, then you are a man and nothing else. If I am not sure what the identity of an object is, I should not categorically say anything about it. But if I know what it is, then I should be able to distinguish it from every other thing that does not share the same characteristic(s) with it. So, if I am a man, then I am a man. Put it in a different way, “if any statement is true, then it is true.” Propositions like this are also said to be a tautology.

B. THE LAW OF NONCONTRADICTION

This law states that an object cannot be and not be at the same time. If am object is a car, it cannot be a non-car, or a chair, or a house at the same time. So, the statement, “This is a car and this is not a car” in reference to the same object is self-contradictory and, therefore, false.

C. THE LAW OF EXCLUDED MIDDLE

This law states that, of two contradictory propositions, the truth of one excludes the other and vice versa. The statement, “The lecturer is in class or the lecturer is not in class”, is always true. The lecturer is actually in class or he is not in class. If it is the case that the lecturer is in class, the statement is true, and if it is the case that the lecturer is not in class, the statement is also true.

So that does it, guys. This marks the end of this topic on Logic. We will meet in the next post. Stay Philosophical!