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LG Rule with 'otherwise'

Alum Member
edited June 2014 55 karma
Hello!

Simple question + looking for confirmation.

If a rule says: If Sarah is cool, she will go to the party with her sister; otherwise, she will not.

Sarah cool ---> Party with sister

~ Sarah Cool ---> ~ Party with sister

Yes? I read somewhere that this is like an 'if and only if' rule... can someone comment?

Their contrapositives:
~Party with sister ---> ~Sarah cool
Party with sister --> Sarah cool
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• Member
24 karma
Good question.. would also love to see an answer!
• Alum Member
edited June 2014 32 karma
TL;DR:
If X then Y; otherwise not Y.
If and only if X occurs will Y occur.

^Statements written like those two are always biconditionals. They should be written out like this: X <---> Y (or the contrapositive, ~X <---> ~Y)

---

If Sarah is cool, she will go to the party with her sister; otherwise she will not. If you take that statement apart, it actually says two things:

1.) Sarah is cool --> Sarah goes to party with sister
2.) ~Sarah is cool (what the "otherwise" means here) --> ~Sarah goes to the party

If you take the contrapositive of 2, it turns into this:

If Sarah goes to the party with her sister then she is cool.

So now we know that if Sarah is cool she goes to the party and if she doesn't go to the party then she's not cool. The two ideas are *directly* tied to each other. Either it is the case that Sarah is cool *and* going to the party OR it is the case that Sarah is not cool and not going to the party. In other words:

Sarah is cool <---> Sarah goes to party with sister

Making one true forces the other to be true and making one false forces the other to be false. That statement is logically equivalent to an "if and only if" statement.

If and only if Sarah is cool will she go the party with her sister.

That statement means that (1) If Sarah is cool then she will go the party with her sister (X --> Y) and (2) Only if Sarah is cool will she go to the party with her sister (Y --> X). Statement 2 is the contrapositive of the second statement from the first translation up above.