Why does contrapositive imply conditional?

Question

Lets say P implies Q.

Therefore I understand that ~Q implies ~P, because if Q is not true then P can never be true.

However, I don’t get why it’s true the other way round. For example: If not B then not A, how do you get that A implies B from that statement.

If you are in a math class, you can always ask your teacher. Although the topic is short, you can still try to lengthen it for research on the topic. See Question guide. — Тyma Gaidash, Aug 29 '21 at 11:49
Welcome to MSE! <> Note that the relationship between a direct implication and its contrapositive is abstractly symmetric, since the negation of the negation of $P$ is $P$. In more detail, if we take $B = \neg P$ and $A = \neg Q$, then "$\neg B$ implies $\neg A$" is equivalent to "$P$ implies $Q$". — Andrew D. Hwang, Aug 29 '21 at 11:57

score 0 · Accepted Answer · answered Aug 29 '21 at 12:49

0

Ok let’s start with $$\lnot Q\implies \lnot P.$$

Letting $A=\lnot Q$ and $B=\lnot P,$ we can resymbolise this sentence as $$A\implies B.$$

This, as you agree, is equivalent to $$\lnot B \implies \lnot A. $$

Reversing the substitution: $$\lnot\lnot P\implies \lnot\lnot Q, $$ i.e., $$P\implies Q. $$

answered Aug 29 '21 at 12:49

ryang

1 Answers1