Question on contrapositive for some and iff

hihihi9993

Hi all!

So I need some clarifications...! Please help!

Contrapositive for "some" doesn't exist because it's reversible.
A <-s-> /B (Some A are not = /B <-s->A (Some not B are A)

Right?

Then why do we have contrapositive for "if and only if" even though it's reversible? A <-> B = /A <-> /B
I know that A <-> B = A -> B + B -> A, so the contrapositive has to be /B -> /A + /A -> /B.

So my questions are as follows:
(1) Since the reasoning proves that contrapositive exists, does that mean that "Contrapositive for "some" doesn't exist because it's reversible" is wrong?
(2) When I am taking the contrapositive A <-> B (= A -> B AND B -> A), I say /A <-> /B because /B -> /A AND /A -> /B. In this case, why don't I change And to Or? I know I did the contrapositive, but I am starting to get confused...!

Thank you so much in advance!!!

Freddy_D

You are on the right track with the first part of your question. The reason you can't contrapose the some statement is NOT because it is reversible, it is because doing so would not create a logical sentence that is equivalent to the original. Contrapositives are inherently equivalent. None of the existential quantifiers (some and most) can be contraposed. The reason we can contrapose the if and only if is because it is using universal quantifiers and doing so would create a logically equivalent sentence.

tldr version: you can't contrapose when existential quantifiers (some and most) are present
you can contrapose when universal quantifiers are present

hihihi9993

@Freddy_D Right!!! I think everything that I learned got mixed up in my head Thank you for clearing things up for me!! Thank you

hihihi9993

@Freddy_D said:
You are on the right track with the first part of your question. The reason you can't contrapose the some statement is NOT because it is reversible, it is because doing so would not create a logical sentence that is equivalent to the original. Contrapositives are inherently equivalent. None of the existential quantifiers (some and most) can be contraposed. The reason we can contrapose the if and only if is because it is using universal quantifiers and doing so would create a logically equivalent sentence.

tldr version: you can't contrapose when existential quantifiers (some and most) are present
you can contrapose when universal quantifiers are present

One more question, why do you think

The contrapositive of a <-> b becomes /b -> /a AND /a -> /b, and not /b -> /a OR /a -> /b?????? Isn't the contrapositive of AND, OR?

Freddy_D

@d931n027h The reason it would be an AND instead of an OR is because an OR can be inclusive or exclusive. If it was an OR that would allow for one of the possibilities to not occur which would be incorrect in this case. The AND is necessary because it tells us that this is the case EVERY time.

B --> A (Every "not B" is a "not A")

AND

A --> B (Every "not A" is a "not B")

Hope that helps (I'm new at this, too haha)

hihihi9993

@Freddy_D Thank you so much!!!

Freddy_D

@d931n027h said:
@Freddy_D Thank you so much!!!

You're welcome. Good luck