Question

In logic, a proposition is a subaltern of another proposition if it must be true when its superaltern is true, and the superaltern must be false when the subaltern is false. Given the two derivations:

1. If proposition P is true, then proposition Q is true.
2. If proposition Q is false, then proposition P is false.

Do these two derivations imply each other? In other words, can the subalternation relationship between a superaltern and its subaltern be simplified to either of the following statements:

1. A proposition is a subaltern of another if it must be true when its superaltern is true.
2. A proposition is a subaltern of another if the superaltern must be false when the subaltern is false.

Is the subalternation relationship between a superaltern and its subaltern a derivational relationship between two propositions, or an implicational relationship between component propositions in a compound proposition?

Answer 1

In logic, the subalternation relationship is an implicational relationship where the truth of a superaltern proposition guarantees the truth of its subaltern, and vice versa for falsity.

Step-by-step explanation:

The subalternation relationship between a superaltern and its subaltern in logic is an implicational relationship between propositions, rather than a derivational relationship. This implicational relationship entails that the truth of the superaltern proposition guarantees the truth of the subaltern, and the falsity of the subaltern ensures the falsity of the superaltern. In the derivations provided, proposition P as the superaltern implies proposition Q as the subaltern (modus ponens) and the falsity of Q implies the falsity of P (modus tollens).

The ability to provide counterexamples is a critical tool in assessing whether such conditional or universal affirmative statements are false, which in turn implies whether the necessary and sufficient conditions they express hold true. In your example, the two derivations you provided do suggest a strong relationship between P and Q, such that you can reason about the truth of one based on the truth of the other. Nonetheless, it's essential to evaluate the logical structure and the truth of the premises independently when assessing arguments.