If you’ve been working on Logic Games, you know that making deductions is key, whether in the setup stage or while working through the questions. Today I’m going to share with you a key source of deductions for in/out games: the reserved spot deduction. It’s a type of deduction that is often central to in/out games, so you need to know it if you want to reach your full potential on the games section.
Just as a heads up: this is going to be pretty technical. But I’m going to do my best to explain in everyday English.
Nevertheless, learning a deduction like this often takes time. You might feel a headache coming on the first time (although I hope not!) Or you might follow along perfectly well, and then later on find yourself wondering “Wait, wait. How do we know that again??” That’s all perfectly normal as you learn something new. If that happens, reread. Or take a break and then reread. Or come back the next day to see if you can connect the dots faster. And of course, reach out to me if you need a little more help.
In/out games: The basics
For in/out games, our job is to sort entities into one of two groups. Either a bird is IN the forest, or it is OUT of the forest. Either someone IS interviewed, or they are NOT interviewed. We’ll use the example of a book being IN the library or OUT of the library.
An important thing to notice here is that if something isn’t in one of the groups, it must be in the other group. If a book isn’t in the IN THE LIBRARY group, in other words, it must be in the NOT IN THE LIBRARY group. Something also can’t be in both groups. Sounds obvious, but it’s absolutely essential.
The four possible scenarios
Now, before any rules come into play, let’s think through what could happen with two books, book A and book B.
We really have four distinct possibilities for how A and B could pan out:
- A and B are both in the library.
- A and B are both NOT in the library.
- Only A is in the library, while B is not.
- Only B is in the library, while A is not.
What are reserved spots?
Suppose you have a rule that states that “Either A or B, or both, must be in the library.”
In that case, we’d want to make sure to reserve a spot for them in the library, which we can indicate by listing them in the IN group. We’ll use a slash between them to indicate that we don’t care which one is there, as long as at least one of them is.
Note: It’s important to learn not to misinterpret the slash as meaning we only get to have one of them. Having both of them in the library would also be ok.
We’ll call this a “reserved spot” because essentially, we’re reserving a spot in one of the groups to make sure there’s always room for at least one of two entities in the group.
To put it in concrete terms, if we’re reserving a spot in the IN group for either book A or book B, we’re essentially making sure that we always have room to put A or B (or both) in the group. We’re preventing ourselves from filling up the library with other books to the extent that there’s no longer enough room for A or B.
Kind of like how you might RSVP with a +1 to a wedding even if you aren’t sure yet who you’ll go with, just so that you make sure that the table you’re assigned to at the reception isn’t entirely full of random people you don’t know.
Reserved spot deductions: Formal logic rules
It’s nice when the LSAT comes right out and tells us to reserve a spot in one of the groups with a rule stating that “either A or B, or both, must be in the library.” But it’s not usually that straightforward.
Reserved spot deductions can also come from formal logic rules, although it takes some thinking and training in order to recognize them. Consider this rule:
“If A is not in the library, then B is in the library.”
We would diagram this rule as not A → B. (If not A, then B.) It’s worth noting here that “not A” means “A is in the OUT group” and “B” means “B is in the IN group.”
Let’s also write out the contrapositive, which we can form by switching the order of the elements and negating them. (Note: Just like two negatives in math made a positive, “not A” negates to “A.”)
Contrapositive: not B → A
In everyday English, here’s why the contrapositive is true. We know that if A is not in the library, that will guarantee that B IS in the library. So if B is NOT in the library, we’ll know that A IS in there. (Because if it wasn’t, then B would be.) (It’s everyday English, but still might require another read through and careful thinking if you aren’t crystal clear on your contrapositives yet.)
Rethinking the possible scenarios
Earlier, we mentioned the four scenarios that are theoretically possible with two books:
- A and B are both in the library.
- A and B are both NOT in the library.
- Only A is in the library, while B is not.
- Only B is in the library, while A is not.
But if we apply the rule that states not A → B or the contrapositive that not B → A, then we can see that the second possibility doesn’t really exist anymore. There’s no way to have both of these books not in the library, because having one of them out forces the other one in.
The reserved spot deduction
If we think about what the remaining three scenarios all have in common, it’s that they all have something in the IN group. Scenario 1 has two things in the IN group, and scenarios 3 and 4 each have one thing in the IN group.
That means that we have to have at least one of them in the IN group. In other words, we’d better reserve a spot for one (or both) of them in there. That’s our reserved spot deduction.
Thinking through the “or both”
We’ve been talking about the fact that these reserved spots are for A or B “or both” but it’s time to really focus on that “or both.” This is another place where students have a hard time thinking through the rule at first. So let’s break it down.
The rule states what happens when we have “not A,” and the contrapositive states what happens when we have “not B.”
But neither rule states what happens when we have A in the IN group. So if we place A in the IN group, and scan the sufficient conditions (the part before the arrow) in our rules, we see that neither rule applies. That means B is free to be wherever it wants, including possibly in the IN group.
Likewise, neither rule states what happens when we have B in the IN group. So if we place B in the IN group, A is still unrestricted. It doesn’t matter whether A is in the IN group or the OUT group, which means that both A and B could be in the IN group together.
Reserved spot deduction shortcut
That’s a lot to think through, but thankfully there’s a quick shortcut to making reserved spot deductions.
Let’s write out our rule and our contrapositive again:
- not A → B
- not B → A
The three possibilities we mentioned earlier happen in these circumstances:
- The original rule applies (because we have not A)
- The contrapositive applies (because we have not B)
- Neither rule applies
When neither rule applies, it’s because we have the opposite of the triggers. This would happen when instead of not A, we have A. And instead of not B, we have B.
And here’s the cool thing. That situation is represented by the necessary conditions (the parts after the arrow) in both rules.
So essentially, here are our three possibilities:
When we scan those three possibilities, we can see that they all have something to do with the IN group, and so that’s where we should put our reserved spot.
Applying the shortcut
Consider another rule, that states that “If C is in the library, then D is not in the library:
- C → not D
- Contrapositive: D → not C
A rule with no reserved spots
Not every formal logic rule is going to result in reserved spot deductions. Here’s an example of one that doesn’t: “If E is in the library, then F is also in the library.”
- E → F
- Contrapositive: not F → not E
If we use the shortcut, we can see that the first possibility is entirely about the IN group, while the second is entirely about the OUT group. The third possibility includes both groups.
That means we don’t get to reserve a spot in either group.
(Side note: That third possibility is still useful because it reminds us not to interpret this rule as “E and F must be together.” In fact, they don’t have to be together. F could be IN while E is OUT.)
Why reserved spot deductions matter
If you’ve read this far, congrats! This was pretty technical!
The last thing I’d like to share with you is how to actually make use of these reserved spot deductions. Essentially, why all of this even matters.
In/out games: When groups get full
I love teaching my students to think through where deductions usually come from in each game type.
For in/out games, full groups are going to be a super helpful source of deductions. When we know that one of the groups is full, it means that ALL the other entities are forced into the other group. That’s powerful.
Sometimes the LSAT will tell you how many entities are in the groups. Sometimes they won’t specify in the setup, but particular questions might stipulate a limit. Either way, we’ll want to always keep an eye out for a group that’s full or almost full.
A reserved spot means that you can get to a full group faster.
Consider the scenario in which we have a reserved spot for A/B (or both) in the IN group. And further imagine a question like the following:
If E and one other book are the only ones in the library, which of the following could be true?
In this situation, as soon as we put E in the IN group, we can already see that the “one other book” must be either A or B. After that, the group is full. And that means that everyone else must be OUT. We would then be ready to scan our answer choices with confidence.
Your turn
Consider these rules and ask yourself if there would be a reserved spot deduction from them. I’ve included one especially tough one for you to take a stab at. If you want to check your answers, send me a quick message.
- If radishes are in the soup, then noodles are not in the soup.
- Sophia is taking the train if Tonia is not taking the train.
- If Robert takes Spanish, then he will also take Algebra.
- If foxes are in the zoo, then neither manatees nor dolphins are in the zoo.
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