Logic Fundamentals: A Lesson In Conditional Reasoning

The following article was written by a TLS user who scored a 180 on the September 2009 LSAT and who tutors pre-law students in LSAT preparation. He also authored How I Scored a 180 for Top-Law-Schools.com.

In this LSAT lesson, I will explore conditional reasoning and its relevance to the LSAT. By studying the concepts within this article, I hope your understanding of conditional statements will improve. Because conditional statements occur often in both the logic games and logical reasoning sections of the LSAT, I have no doubt that an improved understanding of them will enable you to score higher on the test. On these grounds, conditional reasoning is often the first topic I cover with the LSAT students I tutor.

Please note that I intend for this article to be studied, not simply read. You will get more from it if you take your time and participate in the learning experience. This is a long article. If this material is new to you, I would not suggest trying to digest it all in one sitting. You have to let the concepts settle in to some degree. Scattered throughout the article, you will find downloadable worksheets with accompanying answer keys. Completing these should help solidify many of the concepts. Let’s begin.

What is a Conditional Statement?

A conditional statement is, in its most easily recognized form, an “if…then…” statement. The following is, for example, a conditional statement:

If I bang my shin on the table, then I will immediately scream in pain.

A conditional statement is composed of two parts: the antecedent (which follows “if”) and the consequent (which follows “then”). In our example statement, the phrase “I bang my shin on the table” is the antecedent, and “I will immediately scream in pain” is the consequent.

Conditional statements are also described in terms of sufficient and necessary conditions. While I don’t believe you will ever encounter the antecedent/consequent terminology on the LSAT, you may encounter a question where you need to understand the meaning of sufficient and necessary conditions.

Something (we’ll call it “A”) is said to be a sufficient condition for something else (“B”) when the occurrence of A is all that is needed for the occurrence of B. The antecedent is the sufficient condition for the consequent.

B is said to be a necessary condition for A whenever A cannot occur without B also occurring. The consequent is the necessary condition for the antecedent.

This language can be somewhat confusing, so let’s look at our example statement to clarify these definitions. Banging my shin on the table is all that is needed for me to scream in pain (i.e. it is sufficient), so banging my shin is considered the sufficient condition. I cannot bang my shin on the table without screaming in pain (screaming necessarily follows the banging of my shin), so screaming in pain is the necessary condition.

You should be fine if you can simply remember that the antecedent (the phrase following the “if”) is the sufficient condition for the consequent (the phrase following the “then”) and the consequent is the necessary condition for the antecedent.

Diagramming Conditional Statements

Diagramming conditional statements is absolutely necessary for attacking certain types of logic games, especially in/out games. Thankfully, it is very straightforward. With practice, you should be able to do so without thinking. Let’s take our original conditional example:

If I bang my shin on the table, then I will immediately scream in pain.

To begin, take the antecedent and represent it with a letter (or two). So “I bang my shin on the table” becomes “BS”. It really doesn’t matter how you represent it as long as you remember what it stands for. Let’s do the same with the consequent: “I will immediately scream in pain” becomes “SP”. Now write the abbreviated antecedent, followed by an arrow pointing to the abbreviated consequent:

BS → SP

Contrapositive

The contrapositive of a conditional statement is logically equivalent to the original conditional, but provides a different visualization of it. You find the contrapositive by first switching the positions of the antecedent and consequent. Then you negate each term. (Negation can be symbolized by a tilde or by crossing out the letters.)

~SP → ~BP
or
SP
BP

The contrapositive would read, “If I will not immediately scream in pain, then I did not bang my shin on the table.” If the original conditional is true, then so is the contrapositive.

Diagramming with And/Or
Occasionally you will encounter a conditional that has more than one sufficient or necessary condition. For example:

If I bang my shin on the table, then I will immediately scream in pain and begin to cry.

This would be diagrammed as follows:

BS → SP + C

The proper contrapositive is diagrammed as follows:

~SP or ~C → ~BS

If I am not screaming in pain or crying, then I did not bang my shin.

Notice that the “and” becomes “or”. Similarly, “or” always becomes “and”. This makes sense if you think it through. Banging my shin results in me screaming in pain and crying. So if I am screaming in pain but not crying, I couldn’t have banged my shin. And if I am crying but not screaming in pain, I also couldn’t have banged my shin. In other words, I only need to be refraining from at least one (not necessarily both, like “and” would imply) of these two for you to know that I did not bang my shin.

Let’s start practicing:

Working backwards may help:

Proper Inferences and Mistakes in Conditional Reasoning

I see the same mistakes in conditional reasoning over and over again with my LSAT students. Their unwarranted assumptions often lead to many missed points. I will do my best here to explain, by making use of some hypothetical scenarios, what you can and cannot legitimately infer when confronted with a certain conditional statement.

Let’s use the same conditional statement above, and let’s assume it is universally known to be true:

If I bang my shin on the table, then I will immediately scream in pain.

Hypothetical #1

You walk into the room and witness me banging my shin on the table. What can you legitimately and definitively infer given that I have just banged my shin (i.e. the antecedent of this conditional statement has been realized)?

[Pause…]

Hopefully you said that I will immediately scream in pain. This is called modus ponens in classical logic. We can illustrate it thusly:

Premise One: If I bang my shin on the table, then I will immediately scream in pain.
Premise Two: I banged my shin on the table.
Conclusion: I will immediately scream in pain.

Or, using our shorthand developed above:

P1: BS → SP
P2: BS
C: SP

This is a valid argument form. In other words, when given two premises, one of which is a conditional and the other is the fulfillment of the antecedent of that conditional, you are allowed and encouraged to infer the consequent. Hopefully, this one is obvious.

Hypothetical #2

Now let’s say that you come into the room, and I am just sitting on the sofa casually sipping some tea. There are no primal screams emanating from my body. Can you conclude anything about shin banging?

[Pause…]

Yes, you can. You can conclude that I did not bang my shin immediately preceding the moment you entered the room. Why? Because had I banged my shin, I would be screaming. But I am not screaming. I cannot both be screaming and not be screaming simultaneously (so far as I know). Therefore, it is impossible for me to have banged my shin on the table immediately before you entered the room. This is called modus tollens and can be represented in the following way:

Premise One: If I bang my shin on the table, then I will immediately scream in pain.
Premise Two: I am not now screaming in pain.
Conclusion: I did not immediately prior to this moment bang my shin on the table.

Remembering the contrapositive can help us understand why this works:

P1: BS → SP (and its logically equivalent contrapositive: ~SP → ~BS)
P2: ~SP
C: ~BS

So we now have two valid argument forms concerning conditional reasoning. In the first, modus ponens, an affirmation of the antecedent (or sufficient condition) results in the logically valid assertion of the consequent. In the second, modus tollens, a denial of the consequent results in a logically valid denial of the antecedent.

Hypothetical #3

But what if you come into the room and I just happen to be screaming in pain? You didn’t see what happened, but you remember the truth of our conditional statement: If I bang my shin on the table, then I will immediately scream in pain. So what can you say about what happened right before you came in, knowing that I am now screaming in pain?

[Pause…]

Perhaps you said that I banged my shin on the table. While this is certainly possible, it is not necessarily the case. I could have dropped a jar of olives on my toe. Or maybe I tore my ACL jumping over the couch. There are any number of reasonable explanations for why I am screaming in pain. To assert that I must have banged my shin on the table ignores these other possibilities. This is the fallacy of affirming the consequent:

Premise One: If I bang my shin on the table, then I will immediately scream in pain.
Premise Two: I am screaming in pain.
Conclusion (invalidly drawn): I just banged my shin on the table.

P1: BS → SP
P2: SP
No conclusion concerning BS can validly be drawn from these premises (except that it is possible).

To show more clearly why you cannot do this, take the following conditional statement:

If Genghis Khan committed suicide by jumping off a skyscraper, then Genghis Khan is dead.

This is clearly a true statement. And we know that Genghis Khan is dead. Does that mean he committed suicide by jumping off a skyscraper? Don’t be ridiculous. The Mongols may have been amazing, but it’s tough to simultaneously be nomadic and build skyscrapers, especially in the 13th century. In this case, affirming the consequent results in a ludicrous conclusion. And yet it is logically equivalent to the mistake made above.

Premise One: If Genghis Khan committed suicide by jumping off a skyscraper, then Genghis Khan is dead.
Premise Two: Genghis Khan is dead.
Conclusion (invalidly drawn): Genghis Khan committed suicide by jumping off a skyscraper.

If you are given a conditional statement and given that the consequent has occurred, do not make the mistake of assuming that the antecedent is what must have led to the consequent’s occurrence. If the consequent has occurred, this says nothing about whether or not the antecedent has also occurred.

Hypothetical #4

Finally, let’s say that you and I are sitting together on a couch. There is no table in this room, and so there is no prospect of me banging my shin on one. Can you say whether or not I will soon begin screaming in pain?

[Pause…]

No, you cannot. As we said earlier, I could scream in pain for any number of reasons. Maybe I have appendicitis and could begin screaming from that. Asserting that I will not begin screaming in pain because I did not bang my shin is an instance of the fallacy of denying the antecedent:

Premise One: If I bang my shin on the table, then I will immediately scream in pain.
Premise Two: I did not immediately preceding this moment bang my shin on the table.
Conclusion (invalidly drawn): I am not now screaming in pain.

P1: BS → SP
P2: ~BS
No conclusion concerning SP can validly be drawn from these two premises.

To show what ridiculous conclusions you might come to if you make a habit of employing this invalid argument form, let’s apply the same flawed logic to our Genghis Khan example:

Premise One: If Genghis Khan committed suicide by jumping off a skyscraper, then Genghis Khan is dead.
Premise Two: Genghis Khan did not commit suicide by jumping off a skyscraper.
Conclusion (invalidly drawn): Genghis Khan is not dead.

Synopsis

To review, there are two valid argument forms and two common mistakes in conditional reasoning. It is not important that you memorize what they are called, as this knowledge will never be tested on the LSAT. However, make sure you know what each move consists of and whether it is valid or invalid.

Modus ponens: A valid argument form where the antecedent is affirmed, leading to the affirmation of the consequent.
P1: A → B
P2: A
C: B

Modus tollens: A valid argument form where the consequent is denied, leading to the denial of the antecedent.
P1: A → B
P2: ~B
C: ~A

Fallacy of affirming the consequent: A common mistake whereby the consequent is affirmed and the antecedent is (incorrectly) affirmed as well.
P1: A → B
P2: B
C (invalidly drawn): A

Fallacy of denying the antecedent: A common mistake whereby the antecedent is denied and the consequent is (incorrectly) denied as well.
P1: A → B
P2: ~A
C (invalidly drawn): ~B

Validity and Soundness

A distinction needs to be made between a valid argument and a sound one. A valid argument is a “good” argument from a purely logical standpoint. More precisely, a valid argument is one in which it is impossible for the conclusion to be false if the premises are true. As already mentioned, modus tollens and modus ponens are valid argument forms. However, whether the premises are true or not has no bearing on whether or not an argument is valid. In fact, that question cannot be answered within the domain of logic.

A sound argument is one which has true premises that necessarily lead to the (true) conclusion. In other words, it is a valid argument that also has true premises.

Examples

Here is an example of a valid argument that has a false premise, leading to a false conclusion:

P1: If you are a movie star, you are a woman.
P2: Brad Pitt is a movie star.
C: Therefore, Brad Pitt is a woman.

This is an example of modus ponens, so we know it is a valid argument. However, the first premise is obviously false as there are many male movie stars (Tom Hanks, Morgan Freeman, George Clooney, etc). This leads to a false conclusion: Brad Pitt (as far as I know) is not a woman.

Now let’s explore a sound argument:

P1: If you are a professional golfer, you have swung a golf club.
P2: Tiger Woods is a professional golfer.
C: Therefore, Tiger Woods has swung a golf club.

For this trivial argument, we’ve again employed modus ponens and know that it is logically valid. In addition, all of the premises are true. So this argument is sound in addition to being valid.

Why Do We Care?

You may be wondering why I’ve made the distinction between sound and valid arguments. Sometimes my students read an argument in the logical reasoning section and get caught up in trying to determine whether the premises are true or not. This almost never matters on the LSAT. We are not concerned with soundness. We are only concerned with the logical strength of the arguments. We care about validity. You are not on a fact-finding mission. You are not in any way being tested on your knowledge. You are simply being tested on your ability to think logically.

In life, we care about truth in additional to logical validity. On the LSAT, we care much more about logical validity.

Conditional Reasoning in Logic Games

Now we get to apply what we’ve learned above to the LSAT. When I took the LSAT in September of 2009, there were two in/out grouping games for which an understanding of and ability to manipulate conditional statements was key. It seems like these types of games are becoming more and more popular, so doing them quickly and accurately may be crucial for your own LSAT score.

I have created the following game setup which closely resembles the type you would find in an actual LSAT. I will guide you through how to best set up such a game. (Because of copyright issues, I cannot reprint any actual LSAT questions or games here. If you want additional practice on actual LSAT questions (and you should), you can find more in/out grouping games in PrepTests 20, 33, 45, and 58, among others.)

An In/Out Logic Game

It is Bar Review night at Stalevard Law School, and a group of students are heading out for the night. The group will include some of the following seven students: Tanya, Ulysses, Vincent, Wilhelmina, Xerxes, Yolanda, and Zahir. The composition of the group is subject to the following:

If Yolanda goes to the bar, then Tanya does not go.
If Xerxes does not go to the bar, then Zahir does go.
If Vincent goes to the bar, then so does Yolanda.
If Wilhelmina does not go to the bar, then Ulysses does go.
If Wilhelmina does go to the bar, then so does Vincent.

Always create a fairly complete setup of the rules before diving into any questions. I view the setup stage as an investment: it may feel like you are taking too much time, but if done properly, you will be saving time down the road.

These rules are clearly all conditional statements. Begin by diagramming each conditional statement and its contrapositive. You’ll find the answers below, but give it a shot yourself before moving on.

If Yolanda goes to the bar, then Tanya does not go.
Y → ~T
Contrapositive: T → ~Y (If Tanya goes to the bar, then Yolanda does not.)

If Xerxes does not go to the bar, then Zahir does go.
~X → Z
Contrapositive: ~Z → X (If Zahir does not go to the bar, then Xerxes does.)

If Vincent goes to the bar, then so does Yolanda.
V → Y
Contrapositive: ~Y → ~V (If Yolanda does not go to the bar, then neither does Vincent.)

If Wilhelmina does not go to the bar, then Ulysses does go.
~W → U
Contrapositive: ~U → W (If Ulysses does not go to the bar, then Wilhelmina does.)

If Wilhelmina does go to the bar, then so does Vincent.
W → V
Contrapositive: ~V → ~W (If Vincent does not go to the bar, then neither does Wilhelmina.)

Let’s put all these together in a more palatable format:

 Original Conditional: Contrapositive: Y → ~T T → ~Y ~X → Z ~Z → X V → Y ~Y → ~V ~W → U ~U → W W → V ~V → ~W

That may seem somewhat daunting, but with practice it should take no longer than 60 seconds to diagram all the conditionals and their contrapositives. By test day, this should be incredibly simple for you.

Now we can link some of these conditional statements together. You can link two conditional statements together anytime the necessary condition of one statement reappears as the sufficient condition of another statement. For example, the conditional statements
A → B
and
B → C
can be linked to produce a third conditional statement
A → C.

In the first statement (A → B), B appears as the necessary condition. B reappears as the sufficient condition in the second conditional (B → C). If A occurs, then B occurs. And if B occurs, C occurs. So if A occurs, C occurs.

On the LSAT, I do not suggest removing the intermediate step(s) from your diagram (in this case B). It is easier to keep track of one or two longer strings of conditionals as opposed to several short ones. So, if the above example were on the LSAT, I would suggest you diagram it as A → B → C. This encompasses all three conditional statements in one.

Take a minute to solidify these skills:

Let’s return to the original setup and see if we can produce some linkages. Again, try to come up with them yourself and continue reading once you have finished or hit a roadblock.

The contrapositives of the first and third rules link together:
T → ~Y
~Y → ~V
T → ~Y → ~V

We can then add the contrapositive of the fifth rule to the end of this chain:
~V → ~W
T → ~Y → ~V
T → ~Y → ~V → ~W

Finally, lets add the fourth rule to the chain:
~W → U
T → ~Y → ~V → ~W
T → ~Y → ~V → ~W → U

Excellent! You can, of course, create a (logically equivalent) contrapositive of this entire chain. You can do this the same way we created the first chain (by finding small bits to link together from our original table), or you can create it from our first chain. For the latter method, simply negate every variable and reverse the direction of the arrows, just as you would with a conditional statement with only two variables. You should end up with this:

~U → W → V → Y → ~T

We can now create our final setup. Unfortunately, we have one rule which couldn’t be combined with any of the others. But this is still a much more manageable setup than our original table of disparate conditionals and their contrapositives.

T → ~Y → ~V → ~W → U
~U → W → V → Y → ~T
~X → Z
~Z → X

Now we are armed with all the information we’ll need to attack the questions. Let’s go through some questions.

Question One

Which one of the following could be a complete and accurate list of law students who go to the bar?

A. Tanya and Xerxes
B. Vincent and Wilhelmina
C. Ulysses and Yolanda
D. Xerxes, Wilhelmina, and Zahir
E. Ulysses, Xerxes, Yolanda, and Zahir

Try to answer this question on your own using the setup we came up with above. Do not move on until you’ve found the answer or gotten stuck.

Let’s go through each of the answers:

A) By looking at the first long chain (T → ~Y → ~V → ~W → U), we see that if Tanya goes to the bar, then so does Ulysses. Since Tanya is included in this answer choice but Ulysses is not, this cannot be a complete list of those going to the bar.

B) By looking at the second long chain (~U → W → V → Y → ~T), we see that if Vincent goes to the bar, so does Yolanda. Since Vincent is included in this answer choice but Yolanda is not, this also cannot be the correct answer.

Furthermore, one or both of Xerxes and Zahir must go to the bar (according to the rule that didn’t fit into our chain). If Xerxes does not go, Zahir does. If Zahir does not, Xerxes does. If both go to the bar, the rule is not violated. But if neither does, the rule is violated. Neither Xerxes nor Zahir goes to the bar in this answer choice, so it cannot be correct.

C) Neither Xerxes nor Zahir appears in this answer choice either. Therefore, this is also an incorrect choice (see B for explanation).

D) Let’s look again at the second long chain: ~U → W → V → Y → ~T. If Wilhelmina goes, then so do Vincent and Yolanda. Wilhelmina appears in this answer choice without either of them, so this cannot be the right answer.

E) Since we’ve eliminated all the other answers, we could confidently select this one and move on. But let’s just make sure there are no violations. We have at least one of Zahir and Xerxes, so we are fine there. The inclusion of Ulysses does not imply the inclusion of anyone else. And the inclusion of Yolanda only implies the exclusion of Tanya, who does not appear in the answer choice. So this could be a complete and accurate list of the students going to the bar. Therefore, it is the correct answer.

Perhaps you are saying to yourself (but hopefully not), “But if Ulysses goes to the bar, then Yolanda does not!” Some of my students make this error because they like to move back across the arrows. They take the first chain (T → ~Y → ~V → ~W → U) and start at the right side, moving to the left. Just as you will pop your tires on spikes if you try to exit a parking lot by going against the arrows, you will ruin your LSAT score if you go against the arrows in a conditional chain. This is because you are committing the fallacy of affirming the consequent. Please scroll up to Hypothetical #3 above to remind yourself why this is a bad idea. (Hint: Genghis Khan never saw a skyscraper, let alone committed suicide by jumping off one.)

Similarly, some might protest, “If Yolanda goes to the bar, then so does Vincent!” They see the second chain (~U → W → V → Y → ~T) and want to move backwards to Vincent from Yolanda. Here you’re facing the same fallacy as above, but for some reason it seems more appealing because you’re already in the middle of the chain. Try to think of the conditional chain as a raging river: if you are dropped into the middle of it, you can’t swim against the flow (i.e. move backwards against the arrows). Furthermore, you can’t help but go with the flow, i.e. there is no way to avoid going with the arrows.

Question Two

If Tanya goes to the bar, which of the following must be true?

A. Ulysses goes to the bar.
B. Xerxes goes to the bar.
C. Zahir does not go to the bar.
D. Exactly three students go to the bar.
E. Exactly four students go to the bar.

Give this one a shot before moving on.

A) If Tanya goes to the bar, it sets in motion the entire first chain of conditionals (T → ~Y → ~V → ~W → U). Clearly, this includes Ulysses going to the bar. So this is the correct answer. See how easy things can become once you learn how to work with and diagram conditionals?

The rest of the answer choices could be true, but we are looking for something that must be true.

Question Three

Which of the following pairs cannot go to the bar together?

A. Tanya and Xerxes go to the bar.
B. Ulysses and Vincent go to the bar.
C. Ulysses and Yolanda go to the bar.
D. Tanya and Wilhelmina go to the bar.
E. Xerxes and Yolanda go to the bar.

A) Remember that Xerxes (and Zahir) are not part of the longer conditional chains, and so they really have no bearing on the others. (Apparently they are the life of the party – everyone gets along with them, and no one is going to the bar without at least one of them coming.) Xerxes and Tanya can go the bar together, so this is not the correct response.

B) This pairing also presents no problems. Although Ulysses and Vincent both appear in the conditional chains, nothing in the chains stipulates that they can’t go to the bar together.

Again, I hear the protests: “But if Ulysses goes to the bar, then Vincent can’t!” One might get this impression from the second conditional chain (~U → W → V → Y → ~T). Because Ulysses is going, Wilhelmina isn’t, and so neither is Vincent, etc. This is the fallacy of denying the antecedent. Scroll back up to Hypothetical #4 to see why this is a problem. (Hint: Genghis Khan is, in fact, dead.)

C) Again, this presents no problems, unless you fall into the same trap of denying the antecedent.

D) If Tanya goes to the bar, Wilhelmina cannot. See the first conditional chain (T → ~Y → ~V → ~W → U). This is the correct answer.

E) Like we said for response A, the inclusion of Xerxes provides no problems for anyone else.

Closing Thoughts on Conditional Statements in In/Out Logic Games

Most in/out logic games become much easier if you can do two things:

1. Practice enough so that you can quickly and accurately transcribe the rules and link any conditionals.
2. Avoid making mistakes like denying the antecedent and affirming the consequent.

I cannot emphasize this enough: practice, practice, practice! You will get faster at diagramming conditionals, you will see the linkages more easily, you will make fewer mistakes, and you will finish these types of games more quickly with fewer wrong answers.

Conditional Statements in Logical Reasoning

In the logic games section, conditional statements are not disguised in any way. They most often appear in the “if…then…” construction. However, it is generally not so simple in logical reasoning. You will often see what I call “embedded conditional statements.” These are sentences which are not obviously conditional statements, but can be translated into the “if…then…” format without losing their meaning.

Take the following sentence:

All little boys like cartoons.

Because we are so used to thinking that conditional statements are “if…then…” statements, at first glance this may not appear to be a conditional statement. However, this sentence is exactly the same for our purposes as the following:

If you are a little boy, then you like cartoons.

Or, perhaps more formally:

If an entity is a little boy, then that entity likes cartoons.

Thinking of the sentence in this way enables us to apply all of the rules and methods we covered above. We quickly realize that if you don’t like cartoons, you must not be a little boy (modus tollens, or the contrapositive). And we’re reminded that we can’t make the mistake of thinking that just because someone is not a little boy, they don’t like cartoons (fallacy of denying the antecedent). Similarly, we know that just because someone likes cartoons, it doesn’t necessarily mean they are a little boy (i.e. we can avoid the fallacy of affirming the consequent). This is not to say, of course, that you could not know this without translating the sentence into the “if…then…” format. But for most people, putting it in a more recognizable format simplifies that process.

Indicator Words

For many, the awareness of embedded conditionals is enough to begin recognizing them. You should be able to simply think through the meaning of a sentence and recognize the sufficient and necessary condition(s). If you are struggling with this task, familiarizing yourself with a list of indicator words may help. We already know, of course, that if introduces a sufficient condition. Other words that may introduce a sufficient condition include every, any, all, and when. In addition to then, the following words can introduce a necessary condition: only, unless, except, and must.

There are other words that introduce sufficient and necessary conditions. It dangerous to rely solely on a memorized list; try your best to truly understand the meaning of the sentence. After translating a sentence into the “if…then…” construction, think it over for a minute to ensure that the original meaning has not been distorted.

At this point, let’s try to get some practice recognizing and diagramming less obvious conditional statements:

Diagramming Unless, Except, and Without Statements

As mentioned above, unless, except, and without usually introduce a necessary condition. As such, the term introduced by any of these words is placed after the arrow in a diagram. The remaining term must be negated before being placed in front of the arrow as the sufficient condition. For example:

Unless I get into a top six law school, I will become a professional bowler.

“I get into a top six law school” is introduced by “unless” and therefore becomes the necessary condition. We can abbreviate it “T6”. “I will become a professional bowler” is negated and then becomes the sufficient condition. Let’s abbreviate it PB. Our diagrammed conditional statement should look like this:

~PB → T6 (If I will not become a professional bowler, then I got into a top six law school.)

Contrapositive: ~T6 → PB (If I do not get into a top six law school, then I will become a professional bowler.)

That is the formulaic way to work with such statements. However, as I mentioned above, it is best to simply understand the meaning of a sentence and work from there.

Logical Reasoning Example Question

Hopefully you can now pick out conditional statements that were not so clear to you before. Let’s try out your skills on a hypothetical LSAT logical reasoning question:

Doctor: Eating healthy foods is essential to living a good life, for without a healthy diet you will not receive all the necessary vitamins. Without these vitamins your body will deteriorate.

Which one of the following, if assumed, allows the conclusion of the doctor’s argument to be properly inferred?

A. Anyone who eats a healthy diet will live a good life.
B. At least some people who receive the necessary vitamins still have bodies that will deteriorate.
C. Everyone who has a healthy diet gets all the vitamins they need.
D. If your body deteriorates, you cannot live a good life.
E. Anyone who does not eat healthy foods should take supplemental vitamin pills.

The question asks what must be assumed in order for the conclusion to be properly inferred. We must first recognize what that conclusion is. The conclusion is the statement that is (allegedly) supported by the premises. In this case, our conclusion is the first statement: “Eating healthy foods is essential to living a good life.” Let’s diagram this conditional statement by first translating it into an if…then… statement:

If you live a good life, then you eat healthy foods.
GL → EHF

Or…

If you do not eat healthy foods, then you do not live a good life.
~EHF → ~GL

Now on to the premises which are supposed to support this conclusion:

“Without a healthy diet you will not receive all the necessary vitamins” becomes “if you do not eat healthy foods, then you will not receive all the necessary vitamins.” We should look like this:

~EHF → ~V

And “without these vitamins your body will deteriorate” can be expressed “if you do not receive these vitamins, then your body will deteriorate.”

~V → BD

So now we have premises and a conclusion. Our answer is the missing premise which allows us to legitimately infer the conclusion.

P1: ~EHF → ~V
P2: ~V → BD
P3: ?
C: GL → EHF or ~EHF → ~GL (remember, these are two ways of expressing the same thought)

Now we’re well prepared to take a shot at the answer choices.

A) This is basically taking the conclusion and applying the fallacy of affirming the consequent (or denying the antecedent, depending on whether you’re looking at the contrapositive). It does nothing to legitimize the inference of the conclusion.

B) If we add this statement as a premise in our argument, does it help us get to the conclusion? No.

C) This, again, doesn’t help our argument at all.

D) Let’s diagram this one: BD → ~GL. If we plug that in as our third missing premise, we come up with this:

P1: ~EHF → ~V
P2: ~V → BD
P3: BD → ~GL
C: GL → EHF or ~EHF → ~GL

Clearly, those three premises can be linked to produce a long conditional chain:

~EHF → ~V → BD → ~GL

Now we see that if you do not eat healthy foods, you will not live a good life. This is the conclusion we wanted to be able to properly infer. So (D) is the correct answer.

E) The doctor is not talking about what should be done. He/she is talking about what is the case. This is reason enough to avoid this response.

I can imagine that some readers might be inclined to cross out (D) as a potential right answer simply because they don’t think it’s true. After all, Stephen Hawking’s body deteriorated, and by many measures he has led a good life. But remember, we are not interested in what is true! We are interested in logical validity, not soundness.

Closing Thoughts on Conditional Reasoning in Logical Reasoning Questions

An ability to recognize conditionality is crucial to your success on the logical reasoning section. With practice, it will not take you two pages of notes to do one problem. In fact, it will not always be necessary to diagram conditional statements at all if you can hold that information in your head without becoming prone to errors in reasoning. That said, don’t be afraid to jot some notes down; you should eventually be able to do so very quickly and without thinking too hard about it.

LSAT Language vs. Everyday Language

When people use “if…then…” statements in real life, they often are not used in the strictly logical sense. Perhaps this is one of the reasons why so many people make mistakes in conditional reasoning on the LSAT. As far as the LSAT is concerned, everything is to be taken in its logical sense.

For example, my high school principal would often say things over the intercom like, “All seniors must report to the auditorium.” This an embedded conditional, the meaning of which is logically equivalent to, “If you are a senior, then you must report to the auditorium.” If you are a junior, you assume that you do not have to report to the auditorium. And this was generally a correct assumption because I think that the principal is really trying to say two things: (1) if you are a senior, you must report to the auditorium; (2) if you are not a senior, then you do not need to report to the auditorium. The second statement is simply understood to be implied, and for this reason you are justified in your assumption that you, as a junior, do not need to report to the auditorium.

But if you encountered a similar sentence on the LSAT, that is not an assumption you can make. The principal’s statement doesn’t explicitly address juniors and so you could not correctly assume that you do not need to report to the auditorium. The issue simply wasn’t addressed.

You could only correctly assume two things if this appeared on the LSAT: (1) if you are a senior, then you must report to the auditorium; and (the contrapositive) (2) if you are not required to report to the auditorium, then you are not a senior. (Of course, the knowledge that you are not required to report to the auditorium cannot be assumed, but would need to be introduced in an additional premise.)

This confusion is natural. This is one reason why I think it is so crucial – especially if you are someone who often makes these mistakes – to get into the habit of diagramming conditionals. Diagramming a conditional demystifies it and helps you realize that it is subject to the same rules as any other conditional. Let’s diagram “All seniors must report to the auditorium,” aka “If you are a senior, then you must report to the auditorium.”

S = you are a senior
RA = you must report to the auditorium

S → RA
(Contrapositive) ~RA → ~S

This should help you recognize that there are only two conditions that, if met, imply anything else (see just above: “You could only correctly assume…”). So if the condition that you are a junior is met, this truly implies nothing (on the LSAT). Only if the condition that you are a senior or if the condition that you are not required to report to the auditorium is met can you infer anything else.

You should approach conditional statements as if you are a machine. You cannot assume anything is implied and you must subject all conditional statements to the same rules. I find that diagramming them aids in this task; perhaps this is because it forces you to focus on the form of the relationship rather than on the content.

Closing Thoughts

Well, you’ve made it to the end. Congratulations to those who’ve put in the work to get this far – I’m sure you will be rewarded. However, you must continue to practice. As I have said many times already, practicing is the best way to get better at this type of thinking. Some of you might not be ready to put in the work because you think you understand conditional reasoning. But it is not enough just to understand it. You need to know it so well that it takes almost no time to diagram and link conditionals on an in/out game. You need to be converting sentences into their “if…then…” format at will.

But at the same time, remember that conditional reasoning is not the only thing you need to be an expert at to do well on the LSAT. Do not neglect other parts of your study.