Is the goal of natural deduction to prove logical equivalence, or an implies statement from the proposition to the conclusion?

Question

When we are using natural deduction as a method of proving a statement, does it prove logical equivalence or rather that the final statement is true if the proposition is true? For example, one could use V-intro with the following example. P (prop) would allow us to deduce ∴P V Q using V-into, but P and ∴P V Q are not logically equivalent. From my understanding, natural deduction allows us to say if our proposition is true, then the conclusion we come to using the laws of natural deduction is also true.

Answer 1

Your understanding is correct: The existence of a natural deduction derivation $A_1, ... A_n \vdash B$ proves a logical implication: If all premises $A_i$ are true, then the conclusion $B$ is true as well.
The reverse direction does in general not hold. To prove that two propositions $A$ and $B$ are logically equivalent, one has to show a derivation from $A$ to $B$ and a derivation from $B$ to $A$.