Yesterday, I spent some time discussing the difference between "if" and "only if." We learned that "I'll go to your party IF Miguel goes" is not the same as "I'll go to your party ONLY IF Miguel goes," for LSAT purposes. Today we'll discuss the statement "I'll go to your party IF AND ONLY IF Miguel goes," because this means something different than both of the prior statements. You might need to memorize this. "If and only if" means that BOTH rules ("I'll go IF Miguel goes," and "I'll go ONLY IF Miguel goes") are simultaneously in effect. What this means is that Miguel is both a sufficient condition AND a necessary condition for my attendance. As we discussed yesterday,
"I'll go IF Miguel goes" indicates that Miguel is a sufficient condition for my attendance. If Miguel is there, then I'm definitely going to be there.
"I'll go ONLY IF Miguel goes" indicates that Miguel is a necessary condition for my attendance. If Miguel is not there, then I'm definitely not going to be there.
"I'll go IF AND ONLY IF Miguel goes" indicates that both of these rules are in effect. So you'll either see both of us at your party or neither of us at your party--you definitely won't see either of us there alone.
To diagram this statement, you can do one of two things. First, you could simply put both rules in effect:
M-->N (I'll go if Miguel goes.)
N--> M (Contrapositive)
N-->M (I'll go only if Miguel goes.)
M--> N (Contrapositive)
If you write all that, you'll have correctly captured all the implications of the "if and only if" rule.
Or, you can simply make the arrow go both ways. Note that "If and only if" is the ONLY time you can allow the arrow to go both ways. The diagram would look like this:
M<-->N (I'll go if and only if Miguel goes.)
N<--> M (Contrapositive)
Take a deep breath--we're almost done with our discussion of the basics of LSAT arcana. Fortunately for people with bad memories like mine, there aren't that many things to memorize. Once you've got these basics down, it'll just be a matter of practice.
In later posts, I'll talk about the use of "unless" as it pertains to the LSAT, and the practice of linking conditional statements together. As always, I promise that it's easier than it looks at first blush. Send me an email, or pick up the telephone--I'm always happy to provide additional examples. And for tons more written examples--using actual LSAT questions--of everything I discuss on the blog, check out my book: Cheating the LSAT.