# conditional associated in propositional logic

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.

## 2 Answers

$$P\to Q$$ 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, $$\lnot P\to \lnot Q$$.

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