[Math] Intuition: Why is the biconditional true if both statements are false

Question

I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand).

Source: p 333, A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley

The truth table shows that the biconditional is true when its two components have the
same truth value and that otherwise it is false. These results are required by the fact
that $P ≡ Q$ is simply a shorter way of writing $(P ⊃ Q) \wedge (Q ⊃ P)$. If P and Q are either both
true or both false, then $P ⊃ Q$ and $Q ⊃ P$ are both true, making their conjunction true. …

I already understand the above, but am seeking an even more intuitive explanation.

Answer 1

Consider the sentence:

I will jump off the cliff if and only if you do it as well

You did not jump off the cliff - so why should I?

The moral of the story is that a biconditional statements only states that $\alpha$ holds whenever $\beta$ is the case, they are, say, 'logically attached'. In the cliff analogy, they either jump 'together' or not.