As we may already know, our main goal in logic is to determine the validity of arguments.
And in categorical logic, we will employ the Eight (8) Rules of Syllogisms for us to be able to determine the validity of an argument. But since the 8 rules of syllogisms talk about the quantity and quality of terms and propositions, then it is but logical enough to discuss the nature of terms and propositions before we delve into the discussion on the 8 rules of syllogisms. In what follows, I will discuss the nature of terms and propositions.
First of all, logicians define a term as an idea expressed in words either spoken or written. Of course, an idea is understood as the mental representation of something. Hence, when one says, for example, “a table”, then we have at term, that is, a “table”.
Classification of Terms
There are four (4) classifications of terms in terms of quantity, namely: singular, collective, particular, and universal.
A singular term is one that stands for only one definite object.
Examples:
1) Table
2) Peter
3) Tree
A collective term is one that is applicable to each and every member of a class taken as a whole but not to an individual taken singly.
Examples:
1) Orchestra
2) Platoon
3) Choir
A particular term is one that refers to an indefinite number of individuals or groups. Some signifiers of a particular term are: some, a number of, several, almost all, a few of, practically all, at least one, not all, and the like. Hence, if a term is signified by at least one of these signifiers, then we conclude that that term is a particular one.
Examples:
1) Some Asians
2) Almost all students
3) Several politicians
A universal term is one that is applicable to each and every member of a class. Some of the signifiers of a universal term are: no, all, each, every, and the like.
Examples:
1) All Asians
2) Every politician
3) No student
A proposition, on the other hand, is a judgment expressed in words either spoken or written. When we say a judgment, it refers to the mental act of affirming or denying something.
Example:
1) President Trump is a good president.
2) President Trump is not a good president.
The first example above is an act of affirmation because the copula (or linking verb) is does not contain a negation sign “not”. The second example is an act of negation because the copula (or linking verb) is contains a negation sign “not”.
Kinds of Propositions used in Logic
There are two types of propositions used in logic, namely, categorical and hypothetical propositions. On the one hand, a categorical proposition is one that expresses an unconditional judgment. For example, we may say “The Japanese people are hard-working.” According to logicians, this proposition is a categorical one because it does not pose any condition. On the other hand, a hypothetical proposition is one that expresses a conditional judgment. For example, we may say “If it rains today, then the road is wet.” Please note that in categorical logic we always use categorical propositions.
Elements of a Categorical Proposition
A categorical proposition has three elements, namely: Subject (S), Copula (C), and Predicate (P).
Example:
Quantity of a Categorical Proposition
In terms of quantity, a categorical proposition can be classified into two, namely: 1) particular and 2) universal.
A particular proposition is one that contains a particular subject term.
Example:
- Some Asians are excellent basketball players.
A universal proposition is one that contains a universal subject term.
Example:
1) All men are mortal.
As we can see, it is the quantity of the subject that determines the quantity of the proposition. Thus, if the subject is particular, then the proposition is particular, and if the subject is universal, then the proposition is universal.
Now if the subject of the proposition does not contain a signifier, then the quantity of the proposition must be based on what the proposition denotes. Consider the example below:
Nuns are girls.
As we can see, the subject of the proposition does not contain a signifier. But if we analyze it, it would become clear that the proposition is universal. This is because there is not at least 1 nun that is not a girl. In other words, all nuns are girls. Let us consider another example:
Americans are rich.
Obviously, the example above denotes particularity because it’s not sound to assume that all Americans are rich. Of course, many Americans are rich, but reason tells us that not all of the Americans are rich. Hence, the above proposition can be translated as follows: “Some Americans are rich”.
Quality of a Categorical Proposition
Categorical propositions can be either affirmative or negative.
A proposition is affirmative if the copula of the proposition does not contain a negation sign “not”.
Example: 1) Some students are brilliant.
A proposition is negative if the copula of the proposition contains a negation sign “not”.
Example: 1) Some students are not brilliant.
Four Basic Types of Categorical Propositions
If we combine the quantity and quality of propositions, the result is the four (4) types of categorical propositions, namely: 1) Universal Affirmative, 2) Universal Negative, 3) Particular Affirmative, and 4) Particular Negative. Logicians use the letter “A” to represent a universal affirmative proposition, “E” for universal negative, “I” for particular affirmative, and “O” for particular negative. Consider the examples below:
Universal Affirmative (A) : All men are mortal.
Universal Negative (E) : No men are mortal.
Particular Affirmative (I) : Some men are mortal.
Particular Negative (O) : Some men are not mortal.
Distribution of Terms
In a universal proposition, the subject term is distributed, while in a particular proposition subject term is undistributed. And in a negative proposition, the predicate term is distributed while in an affirmative proposition the predicate term remains undistributed. In other words, the subject terms of all universal propositions are always universal, while the subject terms of all particular propositions are always particular. And the predicate terms of all affirmative propositions are always particular, while the predicate terms of all negative propositions are always universal.
Translating Categorical Propositions into their Standard Form:
To avoid confusion when we analyze the 8 rules of syllogisms, it is helpful to translate categorical propositions into their standard form. Below are the standard forms of an A, E, I, and O propositions.
A proposition : All + subject + copula + predicate
E proposition : No + subject + copula + predicate
I proposition : Some + subject + copula + predicate
O proposition : Some + subject + copula + not + predicate
Examples:
A: Every priest is religious.
Standard form: All priests are
E: Every priest is not religious.
Standard form: No priest is religious.
I: Almost all politicians are corrupt.
Standard form: Some politicians are corrupt.
O: Several politicians are not corrupt.
Standard form: Some politicians are not corrupt.