This is an explanation of the first logic game from Section IV of LSAT Preptest 36, the December 2001 LSAT.
A fruit stand sells at least one of these fruits: figs, kiwis, oranges, pears, tangerines, and watermelons (F, K, O, P, T, W). You need to determine what fruit or fruits it sells based on the rules.
Game Setup
Update: The “note to advanced students” below is now how I recommend all students approach games like this. I no longer draw the rules individually, I just add them one by one to a larger diagram.
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This is a grouping game. There are only two options for the fruit. They can be in the cart, or not in the cart.
As with many in/out grouping games, you can combine all of the rules into two diagrams that cover every variable. This can seem difficult, but it isn’t so hard if you take it step by step. Just keep adding little pieces until you’ve built a big diagram.
It’s important to be able to take the contrapositives. Sometimes we can’t add a rule onto a diagram the way it’s given, but we can add the contrapositive. Taking the contrapositive is easy:
- Reverse the terms.
- Put a line through them, or remove the line if they already had one.
- Change any “and” to “or”, and vice versa.
So: A ➞ B becomes “not B” ➞ “not A”.
Or, a plain English example: If you have a pet, then you have a cat or a dog (pretend it’s true)
P ➞ C or D
Not C and not D ➞ no pet
The first step for most people on an in-out game should be to diagram all of their rules and contrapositives.
Note For Advanced Students: I personally just add each rule to a single large diagram, but that’s an advanced tactic. If you want to try that yourself, just skip the step where I draw each of the rules individually. Instead, attach each new rule on to your existing diagram.
The advanced method is a lot faster and more efficient, but it can lead to error if you’re not 100% confident with sufficient-necessary statements.
The Four Rules + Their Contrapositives
Here are the four rules drawn with their contrapositives:
Combining the Rules Into a Larger Diagram
Now I’ll show you how to combine these into a single large diagram. Start with one rule and draw it in the middle of your page. The first rule is as good as any.
Now look for another rule that you can match up to it. You need a rule that has “K” as the necessary condition, or that has “not P” as the sufficient condition. If that sounds complicated, think of it like playing dominoes: you need to make both sides match to join something together.
We can add in the second rule: “Not T leads to K”
The next rule tells us that oranges have two necessary conditions. Pears are one. So without pears, we can’t have oranges.
Therefore, not P leads to not O.
The next rule mentions W. W requires figs or tangerines. The contrapositive is “no F” and “no T” ➞ no W.
(the “not both” in the rule is useless. “Or” already implies that we could have both.”)
We already have no T on this diagram. We can add no F in above.
I’ve also connected no W to no O. It may seem superfluous, since no T also leads to no O.
But, it’s possible to have T but still not have W. In that case, it’s important to know that O is still out.
Drawing A Contrapositive Of The Main Diagram
This diagram is done, so now you can flip it around and do the contrapositive. Remember to cross out (or remove the line from) everything you flip. Lastly, change “and” to “or” and “or” to “and.”
It’s also a very good idea to be looking over the rules again when you’re drawing the contrapositive. It’s easy to make a mistake, and a mistake is often disastrous.
The new diagram starts from O. O leads to P and W (rule 3).
P leads to not K (contrapositive of rule 1). Not K leads to T (contrapositive of rule 2). And we can simply draw in rule 4 to show that W leads to T or F.
(It may seem superfluous to draw W’s rule. If O is in, then we already have T, which satisfies the rule. But…O doesn’t have to be in. Neither does P. So we should know what happens if W is in but not O or P. )
Two Important Rule Types On In/Out Games
A quick note about these two rules:
They look very similar, but they mean different things.
If K is in, P is out. And vice versa. So one of K and P is always out. And they could also both be out. This type of rule means we can’t have both of the variables in together.
The next rule says that if T is out, K is in, and vice versa. One of T and K is always in. And they could both be in.
The reason both T and K could be in (and P and K could both be out) is that you can only follow the arrows left to right. So K being in doesn’t tell us anything about T.
To sum up: an important deduction here is that one of K and T is always in, and one of P and K is always out. We have at least 1 fruit always in and at most 5 in.
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Christion Walker says
“If K is in, P is out. And vice versa. So one of K and P is always out. And they could also both be out. This type of rule means we can’t have both of the variables in together.
The next rule says that if T is out, K is in, and vice versa. One of T and K is always in. And they could both be in.
The reason both T and K could be in (and P and K could both be out) is that you can only follow the arrows left to right. So K being in doesn’t tell us anything about T.
To sum up: an important deduction here is that one of K and T is always in, and one of P and K is always out. We have at least 1 fruit always in and at most 5 in.”
How is this not a contradiction?
FounderGraeme Blake says
You mean the “not both” and “at least one” rules?
Let’s give a couple of examples:
“You have a cat or a dog”
“If you are in america, you are not in Australia”
The first one is at least one, and maybe both. (Or can include both). You draw this as: not cat –> Dog and no dog –> cat
Why? Because we know you have one. So if you lack one type of pet, you must have the other
In America/Australia, you can’t be in two places at once. So, if you are in one, you aren’t in the other: not both. E.g. American –> Not australia
But you could be in neither place. For example, I’m writing this from Canada, so I’m not in either country.
Hope that helps!