0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.