Show that X is logically equivalent to Y if and only if the arguments "X. therefore Y and "Y. Therefore X" are both valid.

Take the argument "X. Therefore Y." as an example, with X as the premise and Y as the conclusion. There is a counterpoint, an interpretation wherein X is true and Y is false if this argument is invalid. As a result, if "X. Therefore Y." is incorrect, X and Y are not logically equivalent, and a counterexample to the argument is likewise a counterexample demonstrating that X and Y are not logically equivalent. The argument "Y. Therefore X." follows the similar pattern, with the second statement, Y, serving as the premise and the first sentence, X, serving as the conclusion.

If this argument is invalid, there is indeed a counterexample, ie an interpretation in which Y is true and X is false, and so a counterexample to X and Y being logically equivalent once more.

if both "X. Therefore Y." and "Y. Therefore X." are valid arguments, every interpretation of X that is true is also an interpretation of Y that is true (the validity of "X. Therefore Y."), and every interpretation of Y that is true is also an interpretation of X that is true (the validity of "Y. Therefore X."). But it's just another way of saying that X and Y have the same truth value in each interpretation. We can't have a circumstance (an interpretation) where one is true and the other is false if whenever X is true, Y is true, and whenever Y is true, X is true. As a result, if both "X. Therefore Y." and "Y. Therefore X." are true, X and Y are logically equivalent