TITLE: What's the difference between biconditional iff and logical equivalence?
QUESTION [11 upvotes]: I am confused about the difference between ↔ (biconditional iff) and ≡ (logical equivalence). For instance, p→q can be rewritten as ∼p∨q. Would it be correct to say p→q↔∼p∨q or p→q≡∼p∨q?
Secondly, is ⇔ another symbol for ≡?
Finally, what's the difference between → and ⇒?

REPLY [6 votes]: Consider this analogy. You wouldn't say the following:

Prove that 3 + 5.

You might instead say this:

Prove that 3 = 5.


Similarly, following doesn't make sense:

Prove that $p \leftrightarrow q$.

Instead one of these would be correct:

Prove that $p \leftrightarrow q$ is always true.
  or
  Prove that $p \equiv q$.