Why is that: $P \Rightarrow T$, truth value(P) = ?, but $(P\Rightarrow F) \Rightarrow$ Truth value (P) = F

Question

Why is that:

If:

P :proposition.

T: true statement

F: false statement

$$P \Rightarrow T $$ In this statement, we can not have for sure the Truth value of P (if P is T or F) , but, in this statement $$P\Rightarrow F$$ P must be false.

Answer 1

Look at the truth-table of $P\to Q$: $$\begin{array}{cc|c}P&Q&P\to Q\\\hline T&T&T\\T&F&F\\F&T&T\\F&F&T\end{array}$$ As you can see, if $\text{TruthValue}\left(P\to Q\right)=T$, you cannot conclude on $\text{TruthValue}(P)$. But the only way to have $\text{TruthValue}(P\to Q)=T$ when $Q$ is false is to have $P$ false too.

Now, you can think of implication (the $\to$ connective) as saying: "it is never the case that when $P$ is true, $Q$ is false". This is precisely what is meant by the second line of the truth-table. All other cases are allowed.