A Complete LSAT Conditional Reasoning Lesson

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Conditional reasoning is a common feature of the LSAT, tested heavily in both the logic games and logical reasoning sections. While the term ‘conditional reasoning’ is a little intimidating, it’s important to realize that you already understand these logical relationships intuitively and use them in your daily speech.

In this post we deal with every aspect of conditional reasoning needed for the LSAT, including the basics of if-then statements, making simple inferences, avoiding common mistakes, how to deal with conjunctions in a conditional statement, and how to spot and diagram conditional statements that are often deliberately obscure or confusing on the LSAT. We also provide a handy rule chart for quick reference. Bookmark this page as you may find it necessary to come back and study these rules repeatedly as they sink in.

INTRODUCTION TO CONDITIONAL STATEMENTS

To illustrate how you already understand conditional reasoning, let’s look at an example:

Your name is Tiffany and you live in Southern California. Your just found out from another pal that your friend Valerie went to the mall (without you, no less!)

You’ve known Valerie, like, forever and know that she doesn’t drive. She would never be caught dead on a bus and considers herself too cool to bike or walk. Also, Valerie’s parents are out of town in Palm Springs. No one else likes Valerie besides you and your other friend Mandi. So if Valerie went to the mall, you know the only way that happened is that Mandi must have picked her up and taken her.

You tell your friend, “Oh, then Mandi must have taken her.”

Tiffany in our example just properly employed conditional reasoning to make a logically valid inference. If vapid teenagers can do it, so can you. The conditional relationship that Tiffany understood can be expressed as a conditional statement as follows: if Valerie went to the mall, then Mandi drove her. This statement can be diagrammed as follows:

Long way:

Valerie at mall ——> Mandi drove her

Shorter way:

Vm —–> Md

Now that we’ve seen one of these conditional statements, let’s try to break down what we are looking at exactly. A word on definitions: you don’t have to memorize any of these terms here to ace the LSAT. You will not be quizzed on the names of these logical relationships. You *only* have to be able to understand how conditional logic works and be able to make valid inferences using information you are given.

CONDITIONAL STATEMENTS: DEFINITIONS

At their base, conditional statements are statements consisting of two conditions that have a relationship to each other. Let’s look at the basic conditional statement:

If A then B

The relationship is this: if the first condition A occurs, then the second condition B must occur as well. Note time and order do not apply: B could occur before, during or after A, or could be a permanent fact about the world. Nor does B need to cause A. Another way to think about it is that if A is true, B must be true as well.

This example of a conditional statement consists of two separate parts: A is the sufficient condition (the information that follows ‘if’) and B is the necessary condition (the part that follows ‘then’).

So in our earlier example, Valerie going to the mall is the sufficient condition and Mandi driving her is the necessary condition. Let’s take a look at these separately:

Example conditional statement: If Valerie went to the mall, then Mandi drove her

Sufficient condition: (Valerie going to the mall)

This part of the statement is known as the sufficient condition because if you know it has occurred, then this information is sufficient to know that something else (the necessary condition) also occurred. If we know Valerie went to the mall, then that is all we need to be told to also know that Mandi must have driven her.

Necessary condition: (Mandi driving her)

This part of the statement is called the necessary condition because it is absolutely necessary for it to occur in order for the sufficient condition to occur. If Valerie went the the mall, we know it was necessary that Mandi also drove her; she couldn’t possibly have gotten to the mall any other way.

Extra note: some sources might refer to the sufficient condition as the ‘antecedent’ and the necessary condition as the ‘consequent’.

ASSUMING THE TRUTH OF A STATEMENT- VALIDITY AND SOUNDNESS

We used a slightly preposterous example to illustrate a point: generally on the LSAT you are required to assume the truth of a conditional statement. If the LSAT tells you, “if Valerie went to the mall, then Mandi must have drove her,” then for the time being you just accept that as the absolute, unassailable truth. Yes we know that there are possibly 100 other ways Valerie might have gotten to the mall- aliens could have brought her, she could have walked there in her sleep after taking Ambien, etc. Thinking like this gets you absolutely no where when you have been told to assume the truth of a statement. Just accept it as an immutable fact and move along. For the rest of that question, if you are told Valerie is at the mall, then you know Mandi drove her (unless of course the question tells you otherwise, such as by introducing new information).

The distinction to be made here is one of validity versus soundness. A conclusion drawn from premises can be considered logically valid even if it’s premises are not necessarily true. Here is an example of a valid argument but one which contains a clearly false premise, leading us to a clearly false conclusion.

Premise 1: If you fly, then you are bird

Premise 2: Barack Obama recently flew to China

False conclusion: Barack Obama is a bird

Because we know in real life that our current president is not a bird, the argument is clearly not sound. However, it is by it’s form valid. If we are told to assume the two premises are true, the conclusion follows naturally from them and is true as well. It is good to get used to preposterous arguments like this now, because you are going to encounter them plenty as you tackle the LSAT. Don’t be distracted: when you are told to assume something is true, you should generally be looking at the form of the argument rather than questioning the content.

SIMPLE INFERENCES: MODUS PONENS

Now it’s time to make an inference. We will start with the same one Tiffany made already in the earlier example. Say you are told again in an LSAT question, “if Valerie went to the Mall then Mandi drove her.” We now understand that this is a true statement about the world for the purposes of answering that question. Each and every time Valerie happens to be at the mall, then Mandi drove. Now the question gives you another premise, telling you, “Valerie is in fact at the mall.” What do we know now? What inference can we make about what else must have happened? By now answering this should be easy. We know that Mandi drove her!

This inference is known as modus ponens. It means that when the sufficient condition did indeed occur or is true, the necessary condition also must occur or be true as well. Here is the rule in abstract:

Premise 1: If A then B

Premise 2: A occurred

Valid inference: Therefore B occurred

The shorter way of writing this is:

A —> B

A

So B

SIMPLE INFERENCES: MODUS TOLLENS

At this point let’s switch to another example sentence just to keep you on you toes. Let’s say an LSAT question tells us “If Valerie is happy, then Brad must have taken her on a date.” The question further tells us, “Brad couldn’t have taken Valerie out, because he went to the movies with Mandi.” Well what do we know about Valerie? Is she happy right now? Not a chance. She can’t be happy unless Brad takes her out. That’s just where she’s at in her life right now. Therefore, she is not happy. The inference that we just made is called modus tollens, which means that if a necessary condition does not occur or is not true, then the sufficient condition can’t possibly have occurred or be true either.

Let’s look at it in simple form so that you can see it again:

Premise 1: If Valerie is happy, then Brad must have taken her on a date.

Premise 2: Brad did not take her on a date

Valid inference: Valerie is not happy

Again, in abstract form:

A ——> B

Not B

Therefore not A

THE CONTRAPOSITIVE AND THE “FLIP AND NEGATE” TECHNIQUE

As you may have figured out from modus tollens, anytime we have a conditional statement we know that when the necessary condition does not occur, the sufficient condition cannot occur either. This is just an inverted way of saying that when the sufficient condition occurs, the necessary condition occurs as well. Any conditional statement can be inverted to be expressed this negated form, which is ‘logically equivalent,’ meaning it expresses the same relationship. This flipped around rule is called the contrapositive. In case I just lost you there, let’s just look at some example of contrapositives. It’s nothing more than a different way to express exactly the same logical relationship. First let’s look at it in the abstract:

Original conditional statement: If A then B

Contrapositve of that statement: If Not B then Not A

Another, faster way you write it is with slashes meaning ‘not,’ so B —–> A

Remember that on the LSAT you always want to use the quickest means of diagramming anything that you can understand. Speed things up by using the short hand that we show you here.

Now let’s take a look at forming the contrapositive in a natural language example so you can get a feel for how this inversion plays out in everyday speech.

Original conditional statement: If Valerie is happy, then Brad took her on a date ( Vh ——> Bd)

Contrapositive of that statement: If Brad did not take Valerie on a date, then Valerie is not happy. (Bd ——-> Vh)

Basically, to get a contrapositive what you do is you simply flip the necessary and sufficient condition around around and negate them both. (Don’t flip the ‘if’ and ‘then’ or other indicator language that you learn later in this post. That stays where it is.) Always do that and you will never mess up the contrapositive of a simple conditional statement. Flip, then negate.

This trick of flipping and negating works for all simple conditional statements, regardless of whether things are negated to begin with or not. If you started in the information “if not B then not A”, you always can flip and negate to form a contrapositive “A then B.” Remember that like in a algebra, the negation of a negative is a positive. So if you negate the statement “not B” to “not not B,” that can be written more simple as the positive, just “B.”

Original: B ——> A

Contrapositive: A ——-> B

The term “contrapositive” just means a flipped version which is logically equivalent.  So A —> B and B —> A are contrapositives of each other. Again, the terminology is not too important. Just remember to flip and negate to get another valid statement.

Modus ponens and modus tollens work with the contrapositive of a statement just as it does with the original statement.

MORE CONTRAPOSITIVE EXAMPLES:

Here are some further examples to help drill in this “flip and negate” technique:

Example 1:

Statement: If Valerie isn’t cheerleading, then brad must not be playing a football game

Diagram: VC ——-> BF

Contrapositive: If Brad is playing football, then Valerie is cheer-leading

Diagram: BF ——–> VC

Example 2:

Statement: If Valerie and Brad are dating, then Brad isn’t hanging with his bros at all

Diagram VBD ——-> BHB

Contrapositive: If Brad is hanging with his bros, then he and Valerie aren’t dating

Diagram: BHB ——> VBD

No matter which statement are or aren’t negated to begin with, you just flip and negate. Again, the negation of a negative is a positive. So if you negate the statement “not B” you come up with just “B.”

Here is some more even in abstract so you can get fully used to this concept:

Original: A ——> B    Contrapositive:  B ——> A

Original: X ——> Y    Contrapositive:  Y ——> X

Original: L ——> M   Contrapositive:  M ——> L

Original: G ——> H   Contrapositive:  H ——> G

COMMON CONDITIONAL REASONING MISTAKES

Now let’s address some common logical errors. For one reason or another, it’s easy to get tripped up when handling a lot of these conditional statements. Looking at some examples will help you learn to avoid these mistaken inferences when doing problems:

Fallacy #1: Denying the sufficient condition

Premise 1: If Mandi went to Forever 21, she bought new earrings.

Now say you know that Mandi did not go to Forever 21. This is premise 2-

Premise 2: Mandi didn’t go to Forever 21

Can we make any proper inferences about Mandi from these two premises? The common one that people try to make is to infer that Mandi did not buy earrings. However, we simply don’t know either way from these premises. She may indeed have not bought earrings. She may have bought them somewhere else. We don’t know anything beyond that she didn’t physically go buy earrings at Forever 21. Here’s the invalid conclusion again-

Invalid conclusion: Mandi didn’t buy earrings

In abstract this fallacy looks like this:

Fallacy #1 in abstract:

P1: A —–> B

P2: A

Invalid conclusion: B

Another common way this happens if someone negates without flipping necessary and sufficient conditions. So they start with A —–> B and might hastily write down A —> B, thinking they have a valid contrapositive, when they do not.

Fallacy #2: Affirming the necessary condition

The next common LSAT logic error is mistakenly reversing the necessary and sufficient conditions. Say we know absolutely Mandi bought earrings, do we know she went to Forever 21? No we do not. Just because a necessary condition occurs, this does not tell you whether the sufficient condition occurred as well.

P1: If Mandi went to Forever 21, then she bought earrings

P2: Mandi bought earrings

Invalid conclusion: Mandi went to Forever 21.

In abstract:

P1: A —–> B

P2: B

Invalid conclusion: A

Another way this happens is if you mistakenly reverse the necessary and sufficient conditions without properly negating them, so you start with A —–> B and flip to B ——-> A. This is invalid. Don’t do it.

LINKING INFERENCES:

On the LSAT, you don’t often get just a couple of simple premises; conditional statements tend to travel in packs, especially in the logic games section. Let’s take a look at two premises:

P1: A —–> B

P2: B —–> C

We know whenever A occurs, B does as well. We also know that whenever B occurs, so does C. This can be expressed as a chain:

Combined rule: A —-> B —–> C

Following the chain, we could also just cut out B entirely and form a new rule:

Proper inference: A —–> C

In logic, this is called ‘syllogism’, but all you need to remember is that you can cut out links in the chain if need be. Let’s look at a more complicated example of linking:

P1: G —–> H

P2: H —–> K

P3: M —–> K

P4: J ——> K

Here you might have seen right away that you can link G —–> H ——> K. What a little harder to spot is that you can make the contrapositive of P3 and then link that with the others as well: the contrapositive M —-> K is K —- > M. This then links up to form G —-> H —-> K ——> M

While in might appear that you could link p4, you actually cannot link it with the main chain. You can however link it with the contrapositive of P3 to form J —–K —–> M

You can also do longer contrapositive chains if you wish to. Here is one you can properly form: M —-> K —–> H —–> G

This chain was made with P2 and the contrapositives of P2 and p1. It looks complicated, but just remember that all you are doing is forming the individual contrapositives with your flip and negate technique and then seeing where they can link up.

Our three fully linked rules are now:

G —-> H —-> K ——> M

J —–K —–> M

M —-> K —–> H —–> G

When doing a logic game, these linked rules are going to be a lot easier to handle than looking at them all separately. Say this is a grouping game and I know that G is definitely in the group. I just look at the chain and I know that H and K are in their as well. Voila!

When you get good at this, you may often be able to see the implications without writing out every possible chain. Practice practice practice. Grouping games that test this skill are common throughout the LSAT preptests that you will use in practice. Get out there and start finding some.

CONTRAPOSITIVES WITH CONJUNCTIONS

Adding conjunctions creates an extra step when forming contrapositives. Let’s look at a conditional statement with a conjunction so you can see this in action:

Conditional statement: If Brad likes a girl, then it’s either Valerie or Mandi (BLG ——-> LM or LV)

One might be tempted to do the contrapositive out as LM or LV ——-> BLG (If Brad does not like Mandi or does not like Valerie, then he does not like a girl). However, this would be incorrect. Think about it: Brad might like Mandi and not like Valerie or vice versa, in which case he clearly like some girl. All we know is that when he doesn’t like both, then he must not like any girl at the moment.

Therefore the proper way to write the contrapositive is: If Brad does not like Mandi and does not like Valerie, then he does not like a girl (LM and LV ——-> BLG).

The rule is this: when forming the contrapositive of a conditional statement with a conjunction, flip, negate, then change the ‘and’ to ‘or’ and vice versa.

So the rule in abstract is:

Example 1:

A —–> B or C          Contrapositive: B and C ——-> A

Example 2:

A ——> B and C      Contrapositive: B or C ——-> A

Example 3:

A or B —–> C         Contrapositive: C ——> A and B

Example 3:

A and B —–> C         Contrapositive: C ——> A or B

RECOGNIZING CONDITIONAL STATEMENTS

Okay, now you have all the rules in place that you need to handle conditional reasoning on the LSAT. The only problem that remains is spotting conditional reasoning when you see it. The problem is that the LSAT logical reasoning section doesn’t always write conditional statements in simple if-then statements: rather they write them out in all kinds of tricky ways.

Let’s just start this off by showing you some of the tons of different ways that you can write out the same damn conditional statement:

Original if-then statement:

If Mandi got mononucleosis, then she must have been exposed to the Epstein-Barr Virus ( MM ——> EBV)

Alternative ways of writing it:

1. To get mono, Mandi had to have been exposed to EBV

2. Mandi got mono only if she was exposed to EBV

3. Only if she was exposed EBV could mandi have gotten mono.

4. When Mandi gets mono, you know she must have been exposed to EBV

5. Unless Mandi was exposed to EBV, she couldn’t have gotten mono

6. Exposure to EBV must have been necessary if Mandi were to get mono.

7. If the lifeform known by the name Mandi contracted mono, she was without a doubt exposed to the member of the virus family Herpesviridae known as Epstein Barr virus.

etc.

The important thing for you to do is cut through all the nonsense and hone in on the relationship between the two parts. All on these sentences say the same thing, which is that exposure to EBV is absolutely necessary in order for Mandi to have caught mono. If you are ever stuck, do that trick. Say to yourself “which one is necessary for the other one to have occurred.” Once you nail that down, you place that thing in the necessary condition spot (spot ‘B’ in the statement A —-> B), then put the rest in the sufficient condition spot (spot ‘A’ in the statement A —-> B).

The other shortcut is to just recognize various words that indicate either a necessary or sufficient condition.

Words that indicate a sufficient condition: If, every, any, all, whenever, and when

Words that indicate a necessary condition: Then, must, unless, until except, only, only if, without, requires/required

DEALING WITH UNLESS, UNTIL, WITHOUT, EXCEPT

To me, conditional statements with unless are the trickiest. Take this sentence: “Unless I was skiing, I can’t have done a rodeo flip.” I found sentences like this very hard to visualize in diagram form until I learned a speacial technique. The technique is:

Step 1: just put whatever goes immediately after the unless and put it into the necessary condition:

——-> I go skiing

Step 2 is to put whatever is left in the sufficient condition and negate it. So “I can’t have done a rodeo flip” becomes “I did a rodeo flip.”

I did a rodeo flip ——>

final sentence: If I did a rodeo flip, then I must have gone skiing (RF ——> S).

This trick works with ‘except,’ ‘until,’ and ‘without as well, so the sentences “without skiing, I can’t have done and rodeo flip,” and “except for when I’m skiing, I can’t do a rodeo flip” would create the same diagram.

Return to the this technique over and over until diagramming these unless-type sentences becomes natural.

CONDITIONAL REASONING RULE CHART:

Now, we got all the rules you need on paper, let’s put them all in one place:

Modus Ponens:

P1: A ——> B

P2: A

Valid inference: Therefore B

Modus Tollens:

P1: A ——> B

P2: Not B

Valid inference: Therefore Not A

Contrapositive: Valid Forms

Original: A ——> B    Contrapositive:  B ——> A

Original: A ——> B    Contrapositive:  B ——> A

Original: A ——> B   Contrapositive:  B ——> A

Original: A ——> B   Contrapositive:  B ——> A

Syllogism:

P1: A —-> B

P2: B —-> C

Valid inference: A —–> C

Contrapositives with Conjunctions:

Example 1:

A —–> B or C         Contrapositive: B and C —–> A

Example 2:

A —–> B and C      Contrapositive: B or C —–> A

Example 3:

A or B —–> C         Contrapositive: C ——> A and B

Fallacy #1: 

P1: A ——> B

P2: A

Invalid inference: Therefore B

Also often written as A ——> B

Fallacy #2

P1: A ——> B

P2: B

Valid inference: Therefore A

Also often written as B ——> A

Words that indicate a sufficient condition: If, every, any, all, whenever, and when

Words that indicate a necessary condition: Then, unless, except, only, only if, without, requires/required, must

 

FINAL THOUGHTS:

We hope this has been helpful. If you need clarification on any topic here please let us know in the comments. This is a difficult topic the first time you encounter. Often the best way if to have it explained from a lot of different sources. I learned this first from a logic course in college, which was a great way to slowly familiarize oneself with all of these concepts. We highly recommend getting an LSAT prep book that further drills you on these topics.

The best concise teaching of the topic I have seen is in the Powerscore Logical Reasoning Bible. It has plenty of drills to enforce what you have learned. By the time you learn it here then relearn and drill with the Logical Reasoning Bible, I promise you will be a master of conditional statements! Best of luck and sign up for our e-mail list for more instructional posts straight to your inbox.

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11 Comments

  1. A quick question about Contrapositives with conjunctions:

    As you have written it: A —–> B or C Contrapositive: -B and -C —–> -A, which I understand.

    However, recently I encountered this kind of reasoning on a logic games question (grouping) that kind of threw me for a loop. The question was from the LSAT Superprep book, preptest B, game 2, if you are familiar.

    It has a rule that basically states of: -Y –> L or O. This rule states that if Y is “out,” the L and O cannot be together, which is to say that if L is “out,” the O is “in,” and vice versa.

    The contrapositive of this rule would be: -L and -O –> Y. This rule states that when both L&O are “out,” Y is “in.”
    Based on what your post says and what the Powerscore LG book says, this is as far as the contrapositive should take me.

    However, it seems to also follow that when L&O are both “in,” the Y is also “in.” The reasoning being that when -Y, the L&O simply cannot be together, but when L&O are together, whether they are “in” or “out,” then Y is also “in.”

    So… back to the example from this post: A —–> B or C Contrapositive: -B and -C —–> -A

    Would another possible contrapositive for this example be B and C –> -A?

    Hopefully I am making sense, and hopefully you guys can help me clear some of this up…

    Also, I really appreciate your site, it has been a great resource and your 3 month study schedule has been a big help for me!

    – Jordan

    • Hi Jordan, I accidentally deleted your response to my response, so I’m just thinking let’s start fresh. I have the superprep at home so I’ll take a look when I get home. I think my response had mistakes so not your fault that you were confused.

      • Okay, so it looks like the game indeed says “not Y then L or O but not both” It’s an in or out game so -Y means Y is out.
         
        So I’m pretty sure you were right on your analysis in the response today. As far as diagramming this, it’s really two separate rules so it’s hard to capture that meaning in one diagram. I always wrote something like -Y —-> L or O (but not both). The contrapostitive of that is Both L and O —-> Y
         
        That’s kind of scrappy though. It might be better to represent it visually. That might look like this:
         
        Y out —-> in: O/L | out: O/L
         
        The contrapositive would be:
         
        O and L together —–> Y in (doesn’t matter which side they are on, in or out, as long as they are together, then you have Y)
         
        So your analysis of the game is spot on.
         
        However, you ask: “A —–> B or C Contrapositive: -B and -C —–> -A
        Would another possible contrapositive for this example be B and C –> -A?” That’s not the case because this is a simpler rule, lacking the extra “not both” requirement. Therefore you can’t infer B and C –> -A in my example. I hope that makes sense. The “not both” changes things. I can explain further if this isn’t clear, but I think you’ve got a handle on it.

  2. I have conditional reasoning down, except when I encounter “either X or y, but not both” types. Can you explain statements like these in more detail? E. G. If A is selected, then either b or c, but not both, will be selected.

    Thanks

    • In total, this statement “If A is selected, then either b or c, but not both” means you have three options (three things that can happen with the ABC relationship):

      1) AB together or
      2) AC together or
      3) Never ABC together

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