Question

By doing which of these things do we show that p, q, and r are equivalent statements?

1) Proving p implies q, q implies r, and r implies p
2) Proving p if and only if q, q if and only if r, and r if and only if p
3) Proving p is true, q is true, and r is true
4) Proving p is false, q is false, and r is false

Answer 1

To show equivalence between statements p, q, and r, proving the biconditional relationships between them is the correct method. This establishes a two-way logical relationship that proves the statements are equivalent.

Step-by-step explanation:

To show that statements p, q, and r are equivalent, one must prove that each statement implies the other in a circular fashion, meaning you must prove that p implies q, q implies r, and r implies p. However, the stronger condition for equivalence is to prove the biconditional relationships between the statements, which means to prove p if and only if q, q if and only if r, and r if and only if p, establishing a two-way relationship between each pair. Therefore, option 2), proving p if and only if q, q if and only if r, and r if and only if p, is the correct approach to demonstrate that the statements are equivalent. This kind of logical reasoning is essential to constructing solid arguments, understanding the logical meaning of statements, and is foundational in the study of propositional logic.