Square of Opposition: Categorical Logic
In my other notes on terms and propositions used in categorical logic, we learned that
there are four (4) types of categorical propositions, namely:
1) Universal affirmative (A),
2) Universal negative (E),
3) Particular affirmative (I), and
4) Particular negative (O).
Now, the relationship between and among these four types of categorical propositions
is what logicians call the “square of opposition”.
There are four types of relations in the square of opposition, namely:
1) Contrary,
2) Subcontrary,
3) Subalternation, and
4) Contradiction.
Please see the two models of a square of opposition below.
, Contrary
Contrary is the relationship between universal affirmative (A) and universal negative (E)
propositions. Hence, there is only one pair in contrary (that is, A-E), and the pair differs
only in quality. As we can see, both are universal propositions, but one is affirmative and
the other negative.
Example 1:
All philosophers are deep thinkers. (A)
No philosophers are deep thinkers. (E)
Example 2:
No pastors are corrupt. (E)
All pastors are corrupt. (A)
Rules in Contrary: If one of the contraries is true, then the other is false. But if one is
false, then the other is doubtful, that is, its truth-value cannot be determined; this is
because contraries cannot be both true but can be both false. Let us consider the
examples above and assign truth-value to them.
If we assume that the proposition “All philosophers are deep thinkers” is true, then
obviously its contrary “No philosophers are deep thinkers” is absolutely false. Of course,
if it is already assumed that all philosophers are indeed deep thinkers, then it is
impossible for philosophers to be not deep thinkers.
However, if we assume that the proposition “No pastors are corrupt” is false, then we
cannot absolutely say that its contrary “All pastors are corrupt” is true. For sure, it’s
In my other notes on terms and propositions used in categorical logic, we learned that
there are four (4) types of categorical propositions, namely:
1) Universal affirmative (A),
2) Universal negative (E),
3) Particular affirmative (I), and
4) Particular negative (O).
Now, the relationship between and among these four types of categorical propositions
is what logicians call the “square of opposition”.
There are four types of relations in the square of opposition, namely:
1) Contrary,
2) Subcontrary,
3) Subalternation, and
4) Contradiction.
Please see the two models of a square of opposition below.
, Contrary
Contrary is the relationship between universal affirmative (A) and universal negative (E)
propositions. Hence, there is only one pair in contrary (that is, A-E), and the pair differs
only in quality. As we can see, both are universal propositions, but one is affirmative and
the other negative.
Example 1:
All philosophers are deep thinkers. (A)
No philosophers are deep thinkers. (E)
Example 2:
No pastors are corrupt. (E)
All pastors are corrupt. (A)
Rules in Contrary: If one of the contraries is true, then the other is false. But if one is
false, then the other is doubtful, that is, its truth-value cannot be determined; this is
because contraries cannot be both true but can be both false. Let us consider the
examples above and assign truth-value to them.
If we assume that the proposition “All philosophers are deep thinkers” is true, then
obviously its contrary “No philosophers are deep thinkers” is absolutely false. Of course,
if it is already assumed that all philosophers are indeed deep thinkers, then it is
impossible for philosophers to be not deep thinkers.
However, if we assume that the proposition “No pastors are corrupt” is false, then we
cannot absolutely say that its contrary “All pastors are corrupt” is true. For sure, it’s
