Question
Answered step-by-step
MajorIronMouse30
4. The Traditional Square of Opposition – The Subalternation…
4. The Traditional Square of Opposition – The Subalternation Relation
On the following traditional square of opposition, categorical propositions that differ only in their quantity are related by the subalternation relation. Thus, A statements and I statements are related by the subalternation relation, as are E statements and O statements.
The Traditional Square of Opposition
Because the Aristotelian (traditional) standpoint assumes existential import for universal propositions, it follows that the truth of a universal statement implies the truth of its corresponding particular statement. Thus, the truth of a universal A statement (“All S are P”) implies the truth of its corresponding particular I statement (“Some S are P”). Likewise, the truth of a universal E statement (“No S are P”) implies the truth of its corresponding particular O statement (“Some S are not P”). Therefore, if a given A statement is true, then its corresponding I statement must also be true. Likewise, if a given E statement is true, then its corresponding O statement must be true also. On the traditional square of opposition, the downward arrows (with a “T” to indicate a true A or E statement) illustrate that if an A or E proposition is true, then its corresponding I or O statement must also be true. But you can draw no conclusion about the truth value of a particular (I or O) statement from a given universal (A or E) statement that is false.
Because the truth of a universal statement implies the truth of its corresponding particular statement, it also follows that if a given particular (I or O) statement is false, then its corresponding universal (A or E) statement must also be false. For example, if you know that a given I statement (“Some S are P”) is false, then you know that its corresponding A statement (“All S are P”) must also be false. Likewise, if you know that a given O statement (“Some S are not P”) is false, then you know that its corresponding E statement (“No S are P”) must also be false. On the traditional square of opposition, the upward arrows (with an “F” to indicate a false I or O proposition) illustrate that if an I or O proposition is false, then its corresponding universal (A or E) statement must also be false. But knowing that a particular (I or O) statement is true does not tell you whether its corresponding universal (A or E) statement is true or false. The truth value of the universal proposition in this case is logically undetermined by the truth of the particular statement.
You should note that the subalternation relation holds only on the traditional square of opposition (which assumes the Aristotelian standpoint) because of the assumption of existential import. The modern square of opposition (which assumes the Boolean standpoint) does not allow for any inferences to be drawn between an A statement and an I statement, or between an E statement and an O statement.
Following are several categorical propositions with assumed truth values. Use the provided selection lists to indicate the subalternate for each given proposition. Then indicate whether each subalternate is true, false, or logically undetermined by the truth value of the given proposition.
Given Statement: All non-H are non-Q.
Truth Value of Given Statement: False
Subalternate:
Truth Value of Subalternate:
Given Statement: All H are non-Q.
Truth Value of Given Statement: True
Subalternate:
Truth Value of Subalternate:
Given Statement: Some non-H are Q.
Truth Value of Given Statement: True
Subalternate:
Truth Value of Subalternate:
Given Statement: No H are Q.
Truth Value of Given Statement: False
Subalternate:
Truth Value of Subalternate:
Given Statement: No H are Q.
Truth Value of Given Statement: True
Subalternate:
Truth Value of Subalternate:
