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# Conditional reasoning and logical equivalence

## How do we recognize logically equivalent conditional statements?

Conditional (or “if-then”) statements can be difficult to master, but your confidence and fluency on the LSAT will improve significantly if you can recognize the various equivalent ways that a true conditional statement can be expressed.

**Note:**Many students find it helpful to diagram conditional statements, and we encourage you to do so whenever you find it useful.

Let’s start with a few basic statements, and consider their logical implications:

*Example 1*

*Example 1*

**Original statement:**"Whenever I do yoga, I feel calm"

**Diagram:**Yoga

In other words,

**yoga**is sufficient to trigger guaranteed**calm**.Let's assume the original statement to be true.
Now, consider this variation:

*If I’m feeling calm, then I’m doing yoga.*

**Diagram:**Calm

Does this follow from the original statement

**Yoga**$\to $ calm?**No.**

Here's another version:

*If I’m not doing yoga, then I’m not feeling calm.*

**Diagram:**

**not**Yoga

**not**Calm

Does this follow from the original statement

**Yoga**$\to $ calm?**No.**

Last version:

*If I’m not feeling calm, then I’m not doing yoga.*

**Diagram:**

**not**Calm

**not**Yoga

Does this follow from the original statement

**Yoga**$\to $ calm?**Yes – it does!**

**Why?**If doing yoga is sufficient to make me calm, as the original statement asserts, if I’m not calm, I couldn’t possibly be doing yoga—because every time I do yoga, I feel calm.

* Example 2*

*Example 2*

**Original statement:**

*If I’m in civics class, then I’m in school today.*

**Diagram:**Civics

Let's assume this to be true, and now consider a version that flips the order:

*If I’m in school today, then I’m in civics class*

**Diagram:**School

Does this follow from the original statement Civics $\to $ School?

**No.**

**Why not?**Well, I could be in school, and eating lunch in the cafeteria. Knowing that I’m in school isn’t enough—it isn’t

**sufficient**—to conclude that I’m in civics.

Now let's consider a version that makes the

**if**part and the**then**part negative—Does this follow from the original statement?*If I’m not in civics class, then I’m not in school today.*

**Diagram:**

**not**Civics

**not**School

Does this follow from the original statement Civics $\to $ School?

**No.**

**Why not?**I could be sitting in geometry class and still be in school. Knowing that I’m not in civics isn’t sufficient to conclude that I’m not in school.

Finally, let's consider the version that results when you reverse the direction and negate both conditions:

*If I’m not in school today, then I’m not in civics class*

**Diagram:**

**not**School

**not**Civics

Does this follow from the original statement Civics $\to $ School?

**YES.**

**Why?**The original statement asserts that if I’m in civics class, then I must be in school. So there’s no way I could attend civics class unless I’m in school.

This logically equivalent statement is sometimes called the

**contrapositive**of the original statement.*Example 3*

*Example 3*

**Original statement:**

*Whenever there’s a puppy in my house, I feel happy.*

**Diagram:**Puppy in house

Let's assume this statement to be true! Consider these questions:

**If we reverse the order, will the new statement be logically equivalent to the original statement?**

**If**I’m happy,**then**there’s a puppy in my house.- Happy
Puppy in house$\to $

No. This is not equivalent. This version is sometimes called the

**converse**of the original statement. The converse of any true if-then statement is**not necessarily true**.In this case: if I'm happy, you don’t know why—it could be because of a puppy, but it could also be because of something else!

**Another way of putting it:**the converse does not follow logically. It is not a supportable deduction.

**Takeaway:**

**A** B is not logically equivalent to B$\to $ A$\to $

**If we make both things negative, will the new statement be logically equivalent to the original statement?**

When you negate both parts of a conditional statement and keep them in the same order—in other words, you take a true A $\to $ B statement and make it $\to $

**not**A**not**B — you create a statement that is**not logically equivalent**and therefore not necessarily also true.*If there isn’t a puppy in the house, then I’m not happy.***no**Puppy$\to $ **not**Happy

This version is sometimes called the

**inverse**of the original conditional statement. The inverse of any true if-then statement is**not necessarily true.**If there's no puppy, that fact doesn't guarantee that I'm not happy. Maybe my guinea pig is making me happy.**Takeaway:**

**A** B$\to $

is*not*equivalent to**not A** not B$\to $

**If we reverse the order, AND make both parts negative, will the new statement be logically equivalent to the original statement?**

*If I’m not happy, then you know for sure that there isn’t a puppy in the house.*

**not**Happy

**no**Puppy

**Yes! This follows from the original statement!**

**A**$\to $ B

**is logically equivalent to**

**not B**$\to $ not A

This version is sometimes called the

**contrapositive**of the original conditional statement.**That’s it!**These are the two, and only two, definitive relationships that we can be sure of. You

*don’t*know anything if I simply tell you that I feel happy. Maybe I’m happy because I had an extra delicious doughnut. Maybe I’m happy because someone got me flowers. Similarly, you don’t know anything about my emotional state if I tell you that there are no puppies in my house. A puppy in my house

*guarantees*my happiness, but other things could make me happy, too.

**But**if I tell you that I'm not happy, you can say with certainty that there isn't a puppy in my house; if there were, then I would be happy!

**Top Tip:**In essence, the contrapositive is when you take away a

**guaranteed result**from a certain

**trigger**. If that guaranteed result isn’t there, then that trigger must not be there either!

### Why is the contrapositive important on the LSAT?

On the LSAT, you’ll often be asked to infer a result. And many times, the trigger you’re given won’t be the trigger that’s explicitly stated in the text, but rather the trigger of the (implicit) contrapositive. This will happen most often in Analytical Reasoning and Logical Reasoning, and being fluent in recognizing a rule’s logically equivalent contrapositive will help you gain speed and accuracy on the test.

### How do we form a contrapositive?

Let’s practice forming a contrapositive, with the following conditional statement:

*If I’m skateboarding, I will wear a helmet and protective gloves*

**Skateboarding**$\to $ helmet and gloves

#### Step 1: Flip the terms

Your first step is to

**flip**the statement, but keep the arrow pointing in the same direction; in other words, take*everything*on the left and place it on the right, and take*everything*on the right and place it on the left, like this:**Helmet and gloves**$\to $ skateboarding

We're just getting started—this is definitely not a logically equivalent statement, because it tells us that if I’m wearing a helmet and gloves then I

*must*be skateboarding. But maybe I wear that stuff when I mountain bike, too. So we’re not done yet.#### Step 2: Negate *every* term

The second step is to negate every single term in the chain, no matter how many terms there are. If the term was positive before, then we make it negative. If it was negative before, we make it positive:

**If not helmet and not gloves**$\to $ not skateboarding

It might

*look*like we’re done now, but we actually aren’t. Look at the conditions carefully: The statement as it currently stands tells us that if I am wearing**neither**helmet**nor**gloves, then I’m not skateboarding. But that’s not quite right—if I was wearing gloves, but no helmet, you could still know that I wasn’t skateboarding. Similarly, if I was wearing a helmet, but no gloves, you could know that I wasn’t skateboarding. I said that I would always wear both—both are**necessary**. We need a final step.#### Step 3: Change every instance of “and” to “or”, and change every instance of “or” to “and”

This step isn’t always applicable, but it is here. If we change every instance of “and” to “or” and change every instance of “or” to “and”, we end up with this statement:

**If not helmet OR not gloves**$\to $ not skateboarding

*Now*we have a statement that is logically equivalent to the original statement! The original statement was that if I’m skateboarding, then I’m definitely wearing both helmet and gloves! So if I’m not wearing

*either*of those two things, then I’m not skateboarding.

### Your turn

Try forming some contrapositives on your own!

Write down the contrapositives for the following statements:

**If**I live in New York City**then**I live in North America.

**If**you are human**then**you are a vertebrate.

**If**you play outside in the rain today**and**you don’t use your umbrella**then**you'll be cold**and**wet when you come inside.

**If**you don't study**then**your score won’t improve.

## Want to join the conversation?

- what do you do if you have something that says for example: if M is chosen then N nor L can be chosen? Specifically, how do you handle the word "nor" ?(14 votes)
- Good question!

First, I think you have to add the word neither to your sentence, so that it's correctly worded.

If M is chosen, then neither N nor L can be chosen.

Our first step here is to understand what neither/nor is saying exactly. The word neither addresses both N/L. Neither/nor states that both terms are excluded. It's like saying N cannot be chosen and L cannot be chosen. Notice the "and" here.

If M, then neither N nor L

this is the same as:

If M, no N and no L

Flipping this gives us:

If no N and no L, then M

Then changing the terms:

If no N and no L, then M

changes to:

If N or L, then no M

It might be useful to remember that the flip-side of neither/nor is either/or. "Neither" combines terms, and "either" singles them out. Nor is basically "and", and contrasts directly with the "or" from either/or.

Flipping your sentence, using either/or should make some sense now.

If either N or L are chosen, then M is not chosen.

If N or L, no M

Hope this helps!(83 votes)

- I would just like to state a short cut method for everyone's convenience.

If A then B= If not B then not A

If not A then B= If not B then A

If A then not B= If B then not A

If not A then not B= If B then A

I hope it helps(17 votes) - I have two questions about the content please. First, it states that step 3 "Step 3: Change every instance of “and” to “or”, and change every instance of “or” to “and” doesn't always apply. So to clarify does the latter mean that step 3 only applies in cases when a conditional statement contains the word "and" or the word "or" ?

Also, could you please provide a step by step break down of the following example prior to arriving at the result: If you play outside in the rain today and you don’t use your umbrella then you'll be cold and wet when you come inside.

Thank You Kindly,

Devorah(8 votes)- So, from my understanding, and/or statements add conditions, and are only to be changed into each other when already present (and -> or , or -> and).

For the example, I find using "not" for negatives helpful because it's a binary choice. So:

If you play outside in the rain today and you don't use your umbrella then you'll be cold and wet when you come inside.

Becomes:

If play outside and not use umbrella then cold and wet

Or, in it's core components:

If P and not U then C and W

The above is just restating the question in simple terms, the next step is to flip the positives/negatives and invert the criteria:

If not W and not C then not P and U

Finally, we flip the ands to ors, and we're left with:

If not W or not C then not P or U.

Which, if we add in the other words, becomes:

If you're not Wet or not Cold then you did not Play outside or you did use an Umbrella.

Hopefully this makes sense,

Ilyas(8 votes)

- Is there a specific LSAT question example you could show us where these will apply?(7 votes)
- Hey y'all! So i just worked a grouping analytical reasoning question that doesn't make much sense to me. It seems to be using the exact kind of logic the above say is not equivalent.

Basically the grouping question is asking you to assort 4 different people to move 3 different objects and each object requires only 2 people to move.

One of the rules is that Grace moves the sofa "if and only if" Heather moves the recliner. The guide says that the rule Gsof then Hrec has the deduction Not Gsof then Not Hrec. Isn't that deduction an inverse and thus not equivalent to the statement Gsof then Hrec? Or should i not even be thinking of conditionals during the analytical reasoning section?

The question also says that since if Jtable then Mrecliner then the deduction Not Jtable then Not Mrecliner. Isn't that deduction also an inverse and not equivalent?(3 votes)- When dealing with an "if and only if" statement, the inverse is correct. However, if it had only said if Grace moves the sofa then Heather moves the recliner, then the inverse would not be logically equivalent. So, always look out for if and only if statements which may be diagrammed as an arrow with two heads between both elements meaning that it works in both directions.(1 vote)

- This is difficult for me to tie it all together. I got the format, but I don't no what question's it apply to.(2 votes)
- I have difficulties applying the steps to the examples, all of my answers were wrong, PANIC TIME(2 votes)
- Thank you so much for taking the time to explain such a difficult concept.(2 votes)
- what about "both" --> "if the stand carries watermelons, then it carries figs or tangerines or both." specifically I'm trying to diagram that in a conditional sorting diagram and cant figure out how to map the arrows.(2 votes)
- One way to do it can be: If the stand doesn't carry at least one of fig and tangerine, then it doesn't carry watermelons.(0 votes)

- If I live in Mexico City then I live in North America.(1 vote)