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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

Original statement: "Whenever I do yoga, I feel calm"
Diagram: Yoga right arrow calm
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 right arrow Yoga
Does this follow from the original statement Yoga right arrow calm?
No.
Here's another version:
If I’m not doing yoga, then I’m not feeling calm.
Diagram: not Yoga right arrow not Calm
Does this follow from the original statement Yoga right arrow calm?
No.
Last version:
If I’m not feeling calm, then I’m not doing yoga.
Diagram: not Calm right arrow not Yoga
Does this follow from the original statement Yoga right arrow 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

Original statement: If I’m in civics class, then I’m in school today.
Diagram: Civics right arrow School
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 right arrow Civics
Does this follow from the original statement Civics right arrow 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 right arrow not School
Does this follow from the original statement Civics right arrow 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 right arrow not Civics
Does this follow from the original statement Civics right arrow 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

Original statement: Whenever there’s a puppy in my house, I feel happy.
Diagram: Puppy in house right arrow Happy
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 right arrow Puppy in house
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 right arrow B is not logically equivalent to B right arrow A

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 right arrow B statement and make it not A right arrow 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 right arrow 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 right arrow B
    is not equivalent to
    not A right arrow not B

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 right arrow no Puppy
Yes! This follows from the original statement!
A right arrow B
is logically equivalent to
not B right arrow 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 right arrow 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 right arrow 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 right arrow 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 right arrow 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?

  • duskpin seedling style avatar for user Noelia.Vasquez1993
    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)
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    • primosaur tree style avatar for user Scott Chrisman
      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!
      (61 votes)
  • marcimus pink style avatar for user Devorah Workman
    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)
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    • blobby green style avatar for user Ilyas Khamis
      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)
  • blobby green style avatar for user Daniel Sykes
    Is there a specific LSAT question example you could show us where these will apply?
    (7 votes)
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  • blobby green style avatar for user Shannon Jackson
    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)
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  • blobby green style avatar for user Mina Ghabrai
    I have difficulties applying the steps to the examples, all of my answers were wrong, PANIC TIME
    (2 votes)
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  • blobby green style avatar for user brodyvolpe40
    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?
    (2 votes)
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    • blobby green style avatar for user estebanb3d
      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)
  • blobby green style avatar for user sophie stimac
    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)
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  • blobby green style avatar for user Maanyaa Gupta
    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
    (1 vote)
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  • blobby green style avatar for user praj8900
    If I live in Mexico City then I live in North America.
    (1 vote)
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  • blobby green style avatar for user dire.ezeh
    Is it necessary to add "does/did" to the 3rd question? Would "If your score improves then you studied" also work?
    (1 vote)
    Default Khan Academy avatar avatar for user