Advanced Conditional Statements

[mikeyjacobs]

Hey,

My question is general, I'm working in the LG workbook pg. 133. I'm confused as to why some statements with only 2 variables have two conversions and how this impacts the logic chain.

For example, if a question were to say:

John is chosen if, and only if, Billy is selected

From the LG workbook there would be two ways to do this??
J ---> B ***** Contra -B--->-J
B ---> J ***** Contra -J--->-B

My main question is when you create a logic chain with a statement like this, which version would you use? Or will it ultimately cancel itself out, by rewiring the chain to have the same net affect? Also, if there are only two variables then why are there two versions of the logic. I hope my question makes sense.

Thanks,

Mike

[noah]

Hi Mike,

Thanks for the question. Because this is a big issue, I have a suspicion I'm not going to completely resolve this -- so let's keep the conversation going if my hunch is right!

Let's think about "If, and only if" statements for a sec:

Rob is going to the party if, and only if Sarah does not go. (they just broke up?)

So, if Rob goes, we know that Sarah did not go. R -> ~ S

And if Sarah goes, we know Rob will not. S --> ~ R

And if Sarah does not go, we know that Rob will go (we're told he'll go if she does not) ~ S --> R

And if Rob does not go, Sarah did (otherwise he would have gone). ~ R --> S.

So this results in 4 logical statements:

R --> ~ S
S --> ~ R
~ R --> S
~ S --> R

And all of those need to be represented in the logic chain. One option is to do two-sided arrows (think about it, R in leads to S out, and S out leads to R in). However, I find those arrows hard to use, as I've trained myself not to read "up" an arrow. So, I put a circle at the end of each side of the line, to remind myself it's a special situation. Some people do "X"s at each end.

So, first wrap your head around why "if and only if" creates these 4 statements. Then memorize that fact.

The other time that a statement creates more than 2 conditional diagrams (and note that for many of the drills on page 132, nothing new is found through creating contrapositives) is when the statement involves 3 elements. The issue to think about is when can you break up the pair of elements into separate statements and when do you have to leave them together.

If you're facing something like "If J and R are in, so is G", then you cannot break that up. You cannot say that J leads to G, since you need both to "trigger" the chain.

But, if you're dealing with "If K is in, so is M and N", then you can break that up (and you should). K leads to M and K leads to N. The result is the same with these split up statements and the original combined one: K leads to both M and N.

So, you can break up statements when the "and" is on the necessary (right) side of the condition, but not when it's on the sufficient (left) side.

(There's a whole other issue of how to make the contrapositive of statements with pairs -- the basic rule is: the contrapositive of "and" is "or" and vice versa)

With "or", it's reversed.

If you're facing "If either X or Y is in, so is Z", you could say that X leads to Z and that Y leads to Z.

But if you're dealing with "If S is in, either T or V is too", you cannot say that S leads to T, nor can you say the S leads to V. One of those is true, but not necessarily both. And by the way, it could be that S leads to both T and V being in; the LSAT would say "but not both" if it wants to restrict that.

So, you can break up "or" statements when the "or" is on the sufficient (left) side, but not when it's on the necessary (right) side.

Something to keep in mind is that the chain is not designed to represent every constraint in every game. Sometimes it's useful to simply write out a complex rule to the side, and remember to refer to it when using the chain. This is particularly true with contrapositives of statements that have an "and" or "or" pair that you cannot split. I put a star next to elements for which there's a special rule I need to remember.

There are also some games where you might want to write two elements as a combined trigger. The CD game is an infamous example: june-2000-pt31-s1-q7-13-a-music-store-carries-ten-t209.html

The chain is the base from which you want to feel comfortable adapting to deal with whatever the LSAT throws at you. Getting fast and accurate with diagramming the basic and somewhat complex rules is a must-do (including mastering statements with "if and only if" or "unless" or "except").

Tell me if you have any follow-up questions.

[mikeyjacobs]

Hey Noah,

Thanks for the speedy reply. If I am understanding you correctly you are saying for IF AND ONLY IF questions you can use a double arrow which would leave you with 2 conditions? Since they always have to be on the opposite side of each other you can represent it as:

X<--->-Y
Y<--->-x

I understand this is not the way you said you do it, but I think this is clearer to me. Let me know if my thinking is correct.

Is the IF AND ONLY IF similar to the DIFFERENT logic, whereby you can also use the double arrow as they both have to be on opposite sides at all times.

Thanks,

Mike

[noah]

If I am understanding you correctly you are saying for IF AND ONLY IF questions you can use a double arrow which would leave you with 2 conditions? Since they always have to be on the opposite side of each other you can represent it as:

X<--->-Y
Y<--->-x

I understand this is not the way you said you do it, but I think this is clearer to me. Let me know if my thinking is correct.

You're almost completely correct. The elements do not need to be on opposite sides of the chain. For example, if a rule said "X is in in, if and only if Y is in" then you'd have a double-sided arrow (or whatever form you use) going between X and Y on the in side and between X and Y on the out side. Make sense?

Is the IF AND ONLY IF similar to the DIFFERENT logic, whereby you can also use the double arrow as they both have to be on opposite sides at all times.
Mike

I'm not sure what you mean by DIFFERENT logic. Are you referring to a rule like "X and Y can never be together"?

- Noah

[mikeyjacobs]

I think I understand, you wrote:

You're almost completely correct. The elements do not need to be on opposite sides of the chain. For example, if a rule said "X is in in, if and only if Y is in" then you'd have a double-sided arrow (or whatever form you use) going between X and Y on the in side and between X and Y on the out side. Make sense?

I get this! In this example you can also use a double sided arrow as well as you indicated.

As for the DIFFERENT logic, I am referring to this type of logic statement:

N is on a Different Team than Mike
N<----->-M
M<----->-N

Is this correct, I originally meant that both these statements(IF AND and Diff) have similar logic, just worded differently.

Thanks,

Mike

[noah]

N is on a Different Team than Mike
N<----->-M
M<----->-N

Is this correct, I originally meant that both these statements(IF AND and Diff) have similar logic, just worded differently.

Generally, when you get a rule like that, it is part of a game in which everyone must be assigned to one or another group, in which case you're right on the money.

If the game is such that there's a possibility of not being selected at all, then you could not say that.

Good questions.

[hovaLSAT]

Just for a clarification on only vs. the only:

The phrases "the only" and "if only" will ALWAYS introduce the sufficient condition, correct? I had heard about "the only" but am not sure about "if only"...

The phrase "only if" will ALWAYS introduce the necessary condition, correct?

And the phrase "only" will ALWAYS signify the necessary condition but we need to sometimes figure out which phrase it is referring to...

Thanks for the help!

[noah]

Just for a clarification on only vs. the only:

The phrases "the only" and "if only" will ALWAYS introduce the sufficient condition, correct? I had heard about "the only" but am not sure about "if only"...

The phrase "only if" will ALWAYS introduce the necessary condition, correct?

And the phrase "only" will ALWAYS signify the necessary condition but we need to sometimes figure out which phrase it is referring to...

Thanks for the help!

You're a bit turned around on some of these.

These rules can be messed with on advanced questions.

For example:

The only reason for falling in love is money. (love --> money)
I often fall in love, but the only reason is for the money. (love --> money).

So your rule that "the only ALWAYS introduces the sufficient" can be broken.

The basic story is:

"only if" and other non-"the only" forms of "only" generally come right before the necessary
"the only" - be careful as often something is separating it and the necessary that it indicates

Give me a sentence with "if only" and I'll help you decode -- it's a rare thing, not worth your concern.

[hovaLSAT]

Thanks for the post

To be honest I have never heard of or seen an "if only" but..

If only it would rain, then John will be happy.

Rain ---> Happy (?)

For your Example

"The only reason for falling in love is money. (love --> money)
I often fall in love, but the only reason is for the money. (love --> money)."

What is your process here? You just think about what makes sense?

Would it be safe to say that most of the time when you see a "the only" it introduces a sufficient?

Because many times I see examples such as:

"The only way to become strong is by lifting weights."

Strong ---> weights

Thanks

[noah]

If only it would rain, then John will be happy.

Rain ---> Happy (?)

Right, this would translate to rain --> John happy. We don't know it's the only reason. In this case, the "only" is an idiomatic expression indicating emphasis.

For your Example

"The only reason for falling in love is money. (love --> money)
I often fall in love, but the only reason is for the money. (love --> money)."

What is your process here? You just think about what makes sense?

I focus on identifying what is necessary--what's the thing that has to happen if or for something else to happen. Then I work "backwards" to find the sufficient.

Would it be safe to say that most of the time when you see a "the only" it introduces a sufficient?

Because many times I see examples such as:

"The only way to become strong is by lifting weights."

Strong ---> weights

Yes, that's generally how it it works out, but it's a breakable rule: My goal is to become strong and the only way is to lift weights.