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In my other notes on terms and propositions used in categorical logic, we learned that there are four (4) types of categorical propositions, namely:
- Universal affirmative (A),
- Universal negative (E),
- Particular affirmative (I), and
- 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 possible for the contrary to be either true or false. Again, since we cannot have an absolute truth-value to the contrary of the proposition “No pastors are corrupt”, then its truth-value is doubtful.
Subcontrary
Subcontrary is the relationship between particular affirmative (I) and particular negative (O) propositions. Hence, there is only one pair in subcontrary (that is, I-O). And as we can see in the image of a square of opposition above, subcontraries differ only in quality.
Example 1:
Some politicians are women. (I)
Some politicians are not women. (O)
Example 2:
Some mangoes in the basket are not ripe. (O)
Some mangoes in the basket are ripe. (I)
Rules in subcontrary: If one of the subcontraries is false, then the other is true; and if one is true, the other is doubtful. This is because subcontraries cannot be both false, but can be both true. Let us consider the examples above and assign truth-value to them.
If we assume that the proposition “Some politicians are women” is false, then its subcontrary “Some politicians are not women” is absolutely true. Of course, if it is not true (therefore false) that some of the politicians are women, then it is absolutely true that some of the politicians are not women.
However, if we assume that the proposition “Some mangoes in the basket are not ripe” is true, then its subcontrary “Some mangoes in the basket are ripe” is doubtful or cannot be absolutely determined; in other words, it can be true or it can be false. Consider this: Imagine we are facing a basket of mangoes. Now, suppose we see that all the mangoes in the side of the basket facing us are not ripe, then the proposition “Some mangoes in the basket are not ripe” is true. However, we cannot be certain about the truth-value (therefore doubtful) of the proposition “Some mangoes in the basket are ripe”. This is because we see only one side of the basket that is full of mangoes. For sure, it is possible that the rest of the mangoes in the basket are not ripe (therefore, “Some mangoes in the basket are ripe” is false) or there is at least one mango on the other side or in the middle of the basket (that we do not see because we are just facing one side of the basket that contains not ripe mangoes) that is ripe (hence, “Some mangoes in the basket are ripe” is true).
Subalternation
Subalternation is the relation between universal and particular propositions having the same quality. Hence, there are two pairs of subalternation, namely, universal affirmative (A) to particular affirmative (I) propositions, and universal negative (E) and particular negative (O) propositions.
Example 1:
All jasmine flowers are white. (A)
Some jasmine flowers are white. (I)
Example 2:
Some students are brilliant. (I)
All students are brilliant. (A)
Example 3:
No teachers are lazy. (E)
Some teachers are not lazy. (O)
Example 4:
Some fruits are not delicious. (O)
No fruits are delicious. (E)
Rules in subalternation: If the universal is true, then the particular is true. If the universal is false, then the particular is doubtful. If the particular is true, then the universal is doubtful. And if the particular is false, then the universal is false. Let us consider the examples above and assign truth-value to them.
If we assume that the proposition “All jasmine flowers are white” is true, then its subaltern “Some jasmine flowers are white” is absolutely true. As we can see, the truth of the universal affects the truth of the particular. Thus, if it is true that all jasmine flowers are white, then it is impossible for at least one of the jasmine flowers to be not white; hence, “Some jasmine flowers are white” is absolutely true.
If we assume that the proposition “No teachers are lazy” is false, then its subaltern “Some teachers are not lazy” is doubtful in the sense that it can either be true or false.
If we assume that the proposition “Some fruits are not delicious” is true, then its superaltern “No fruits are delicious” is doubtful because it can either be true or false, that is, it is possible that all fruits are not delicious is true and it is also possible that all fruits are not delicious is false.
If we assume that the proposition “Some fruits are not delicious” is false, then its superaltern “No fruits are delicious” must be false. As we can see, the falsity of the particular affects the falsity of the universal. Indeed, if the particular is false, then it is impossible for the universal to be true; it should be false.
Contradiction
A contradiction is a relation between universal and particular propositions having different quality. Hence, there are two pairs of contradiction, namely, universal affirmative (A) and particular negative (O) propositions, and universal negative (E) and particular affirmative (I) propositions.
Example 1:
All men are mortal. (A)
Some men are not mortal. (O)
Example 2:
No men are mortal. (E)
Some men are mortal. (I)
Example 3:
Some drivers are sweet lovers. (I)
No drivers are sweet lovers. (E)
Example 4:
Some students are not intelligent. (O)
All students are intelligent. (A)
Rule in contradiction: One member of each pair is a denial of the other. In other words, if the universal is true, then particular is false; and if the particular is false, then the universal is true. Let us consider some of the examples above and assign truth-value to them.
If we assume that the proposition “All men are mortal” is true, then its contradictory “Some men are not mortal” is absolutely false. Of course, obviously, if it is assumed that all men are mortal, then it is impossible for at least one man to be immortal.
If we assume that the proposition “Some students are not intelligent” is false, then its contradictory “All students are intelligent” is absolutely true. Of course, the first proposition says that there is not at least one student that is not intelligent; hence, we can logically conclude that all students are intelligent.