conditional associated in propositional logic

Question

I know it is a basic logic question but I always wanted to get rid of the doubt and I can not find an explanation.

I understand that in the propositional logic I can transform a conditional in these three different ways

$P \Rightarrow Q :$

$Q \Rightarrow P$

$\neg P \Rightarrow \neg Q$

$\neg Q \Rightarrow \neg P$

I do not understand how they can be "replaced" if their truth tables are not equivalent. Can I transform $P \Rightarrow Q$ into any of them?

I can not find the explanation of how it can happen.

Thanks.

Answer 1

$P\to Q$ is equivalent to its contraposition: $\lnot Q\to \lnot P$.   If indeed $P$ implies $Q$, but $Q$ is false, then $P$ cannot be true.   So $P\to Q$ entails $\lnot Q\to \lnot P$.   Also vice versa (in classical logic).   Thus they are syntactically equivalent.   You may also show semantic equivalence by truth tables.

$Q\to P$ is the converse of $P\to Q$.   It is not logically equivalent to $P\to Q$ .

NB: $Q\to P$ is, of course, equivalent to its contraposition, $\lnot P\to \lnot Q$.

Answer 2

Relative to $P \to Q$:

$Q \to P$ is its converse

$\neg P \to \neg Q$ is its inverse

$\neg Q \to \neg P$ is its contrapositive