Sherlock1122 wrote:Hey Mike,

First off, thank you for writing this book. I just started it a few weeks ago, but finally feel like someone is writing to those of us trying to bump up from the high 160's to the 170s. In my experience over the past few months, it's harder than I anticipated especially with cookie cutter courses and other materials.

For some reason I am really struggling on a particular type of complex "or" rule: the if and only if.

As an example: H will go before J if and only if H is after M.

I keep breaking it down to:

IF M-H, THEN H-J

and

H-J ONLY IF M-H

The answer to the drill is MHJ or JHM.

It has to do with a contrapositive right? I run through the rule in my head and continually end up with:

M-H-J/M-H-J

Sorry for the seemingly simple question I just can't seem to wrap my mind around it. Thanks in advance!

Thanks for the thanks and happy to try to help --

I think what you are feeling w/these rules is what pretty much everyone feels at first -- they seem simple enough, but are very difficult to "nail" conceptually -- however, once you do, I think you'll be glad to see such rules come up --

I'll give u a short and long explanation -- see if either of them help make it click better for you --

1) Short --

H will go before J if and only if H is after M:

H - J if M - H : this translates to: M-H --> H-J; contrapositive: J-H --> H-M.

Meaning: When M is before H, H must be before J. When J is before H, H must be before M.

M - H - J; J - H - M

H - J only if M - H: this translates to H-J --> M-H; contrapositive H-M --> J-H.

Meaning: When H is before J, M must be before H. When H is before M, J must be before H.

M-H-J; J-H-M

(I think that maybe your translation of "only if" is perhaps what tripped you up)

2) Longer Explanation:

The if-and-only-if is really just a complicated way of giving you an either-or-but-not-both scenario -- if you can think about it on those terms, I think it's much easier to translate them correctly.

To illustrate why -- let's imagine a super-simple scenario -- all people in a group are assigned to either team a or team b but not both --

Imagine two different types of rules:

1) Either/or but not both: "Marc and Tom are assigned to different teams."

This gives us two possibilities: M on A, T on B, or T on A, M on B. Those are the only two options we have. Simple enough.

2) Conditional: "If Marc is assigned to Team A, Tom will be assigned to Team B."

This gives us three possibilities: M on A, T on B; T on A, M on B, or both M and T on B.

It is the last of those options (both being assigned to team B) that differentiates the either/or (but not both) from a conditional, and it's the last of those options that makes conditional statements a bit more difficult to think about.

On a conceptual level, you can think of either/or (but not both) as offering guarantees in "two directions" (know about A, can figure out about B, know about B and can figure out about A) whereas the conditional only gives guarantees in one direction (know about A, can figure out about B, but we don't know anything if told M is on team B). The conditional is actually harder to think about correctly, and the fact that conditionals only go one way is probably the most important thing to know about them. Since either/or but not both rules go in "both directions" they offer the test writers a counterpoint to conditional rules.

Why am I mentioning all this? Because "M and T will be assigned to opposite teams" can be written in a harder to understand way like this:

"Marc will be assigned to team A if and only if Tom is assigned to team B."

Again, this is a more complicated way of saying that M and T will be assigned to opposite teams.

Does that make sense? If so, let's apply it to the more complex ordering situation you brought up --

Imagine if, instead of the rule we were given, we were given a simple straight up conditional:

H will go before J if H is after M.

What could we get from this?

M-H --> H-J; contrapositive J-H --> H-M

Seems similar to what we had before, but this conditional actually gives us different possible outcomes -- from it, we could get:

M-H-J, J-H-M, or H before both M and J.

It's the last possibility that people can forget and that makes conditional statements tricky.

Now let's imagine they gave us a different rule:

"Either M or J, but not both, go before H."

This only gives us two options -- M-H-J or J-H-M.

Notice this rule acts as a simpler counterpoint to it's conditional relative.

What's another, more complex way to write this same rule?

H will go before J if and only if H is after M.

Again, per the reasoning I discussed above, it will also give us the exact same outcomes as the either/or but not both rule.

Does that make sense? Again, I think the bi-conditional is a classic and typical LSAT component -- seems simple enough but acts like a mental tongue-twister -- it drives me nuts just trying to explain it -- so, if you still have some concerns let me know and i'll be happy to discuss further -- MK