I am having a bit of trouble understanding the pasted excerpt. I think I might be missing something basic. As I understand it, the contrapositive of a conditional statement is where we take a conditional statement and both 1) flip the hypothesis and conclusion and 2) negate the q and p so we have
¬q -> ¬p
Looking at the truth table of the original
p -> q I can convert each possibility to the contrapositive
¬q -> ¬p . So, for example, when p is True and q is False, the
p -> q is false. I can now turn this case into the contrapositive by taking the
q and negating it which is True and then take the
p and negating it which is False.
What does this mean though that the contrapositive has the same truth value as
p -> q? Like, the truth of table of
p -> q was just a fact that was given to me. How do I know what the truth value for each possibility of
¬q -> ¬p is though? Is it simply that we can always convert the contrapositive back to the
p -> q statement by "un-negating" the
¬q -> ¬p and then know the truth value based on original truth table for
p -> q where we know it's only False when
p is True and
q is False?
Just generally confused. I hope my rambling question makes some sense.
Thanks in advance.