Determine $H_2$ according to the interpretation of $P$

Question

$$ H_2 = ((P\to Q)\to(((P\land Q)\leftrightarrow P)\land((P\lor Q)\leftrightarrow Q)))\to P $$

The answer is $I[H_2]=0$, when $I[P] = 0$

I tried to do, but in my work I conclude that it depends on the interpretation of $Q$, $P$ is not sufficient to determine if $H_2$ is false

Answer 1

Replace $P$ with $0$ into the formula to get:

$((0 → Q) → (((0 ∧ Q) ↔ 0) ∧ ((0 ∨ Q) ↔ Q))) → 0$.

Now simplify: $0 → Q$ is $1$. $0 ∧ Q$ is $0$ and $0 ↔ 0$ is $1$. Finally: $0 ∨ Q$ is $Q$ and $Q ↔ Q$ is $1$.

Thus, what we get is:

$(1 → (1 ∧ 1)) → 0$.