Squares of Opposition

2.2 Squares of Opposition

A square of opposition shows the logical relations among categorical statements. There are two squares of opposition:

the Modern Square of Opposition
the Traditional Square of Opposition

2.2.1 The Modern Square of Opposition

The only logical relation in the Modern Square of Opposition is the contradictory relation.

The Contradictory Relation

To say that two statements are contradictory to each other means that they necessarily have opposite truth value. That is, if one of them is true, the other must be false, and if one of them is false, the other must be true.

For example, the A statement “All bats are mammals” and the O statement “Some bats are not mammals” contradict each other.

The contradictory relation also exists between the E statement “No swans are black birds” and the I statements “Some swans are black birds.”

We can use Venn Diagrams to illustrate why the contradictory relation holds between the A and O statements. When the A statement is true, the area α is empty. But if the area α is empty, then no member of S can be in the area α. This contradicts the O statement, which says that there is at least one member of S in the area α, i.e., the area α is not empty.

The contradictory relation exists between the A and O statements, and between the E and I statements. The relation can also be explicated in terms of the elimination of certain cases in the truth table.

A truth table lists all possible distributions of truth values. A single statement p has two possible truth values: truth (T) and falsehood (F). Given two statements p and q, there are four possible combinations of truth values, ranging from both p and q being true (TT) to both of them being false (FF). Accordingly, there are four rows (cases) in the truth table. In general, given n statements, there are 2ⁿ rows in the truth table.

Given A and O statements, there are four possible truth value combinations. We can view the contradictory relation as ruling out the logical possibility that A and O are both true and the logical possibility that they are both false. The same holds for E and I.

There is one advantage of using the truth table to understand the logical relations among the categorical statements. The tables can help us figure out whether the truth value of a statement can be determined when the truth value of another statement is known.

2.2.2 The Traditional Square of Opposition

If we assume that the set denoted by the subject term cannot be an empty set, then there are four logical relations among the A, E, I, O statements. They are shown in the Traditional Square of Opposition. The four relations are:

Contradictory

Contrary

Subcontrary

Implication

In the diagram below, we can actually see the complete square that shows the logical relations among the A, E, I, O statements.

The Contrary Relation

The contradictory relation has been explicated above. We now look at the contrary relation. Two statements are contrary to each other if they cannot both be true. The contrary relation exists between the A and E statements.

We can use Venn Diagrams to illustrate why the A and E statements cannot both be true. Suppose that both the A and E statements were true. In terms of their Venn Diagrams, this would mean that the area α and the area β were empty. But if both the area α and the area β were empty, then the set S would be empty. This would contradict the assumption that the set S cannot be an empty set.

The contrary relation can also be made clear in terms of the truth table. Notice that the contrary relation rules out the top case in the table.

The Subcontrary Relation

Two statements are subcontrary to each other if they cannot both be false. The subcontrary relation exists between the I and O statements.

Again, we can show why the I and O statements cannot both be false by using the Venn Diagrams. Suppose that both the I and O statements were false. According to the contradictory relation, this would amount to both the E and A statements being true. But then both the area β and the area α would be empty. But if both the area β and the area α were empty, then the set S would be empty. This again would contradict the assumption that the set S cannot be an empty set.

The subcontrary relation does not allow the logical possibility of both I and O being false in the truth table.

The Implication Relation

Implication is an important logic concept. If a statement p implies another statement q, then it cannot be the case that p is true, but q is false. Therefore, the implication relation rules out the second case in the truth table.

In the Traditional Square the A statement implies the I statement, and the E statement implies the O statement.

If p implies q, it also means that if q is false, then p must be false.

Exercise 2.2

Use the interactive program to determine the truth values. Write “T” for True, “F” for False and “?” for Undetermined.

Click on Check Answers to see if your answers are correct.
You can change the answers marked as incorrect and click on Check Answers again.
Reload the page to get rid of answers and correction marks.

Given that the A statement is false, use the Modern Square to determine the truth value of the other three statements.

A: All S are P. E: No S are P.