Boolean wff is satisfiable?

Question

Definition:

A Boolean wff is satisfiable if and only if it is true for at least one assignment of truth values to the variables it contains.

Question:

The wff P v $\neg$Q is satisfiable. It is True if either P is True or Q is False.

The operator is OR. Why is this satisfiable? I understand P implies Q, but not the other way around. If P is true then we satisfy the statement. But if it's false, and Q is true, then the negative of Q would also be false?

Answer 1

When checking for satisfiability, you choose the values of $P$ and $Q$ to be true or false, so as to make the final proposition true. If the proposition is insastifiable, it means no combination of assignment of $P$ and $Q$ will make the proposition true.

So letting $P = \top$ and $Q = \top$ satisfies $P \vee \neg Q$ that is $P \vee \neg Q$ is satisfiable.

Now be careful $P \vee\neg Q$ is a rewriting of the implication operator in the other direction : $Q \Rightarrow P$. Indeed if $Q$ is true, then so must $P$. But if $Q$ is false, there are no requirements on $P$. Thus you can write the implication $( Q \wedge P) \vee \neg Q$, which can be distributed to \begin{equation} (Q \vee \neg Q) \wedge (P \vee \neg Q) = \top \wedge (P \vee \neg Q) = P \vee \neg Q \end{equation}