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If X, then Y | Sufficiency and necessity
What is a conditional statement, and what are sufficient and necessary conditions?
In this lesson, you will learn how to recognize arguments that contain conditional statements, and learn the difference between sufficient and necessary conditions.
Let's start with an example:
Imagine that Willie and Lola are creating a game to play in the garage. What are the rules?
Willie: I have to hit the ball over the net in order to score a point during my shot.
Lola: Right, but if your ball doesn’t hit the table, then you lose the point.
Willie: Okay, but during my serve, I need my served ball to hit my side of the table first before going over the net in order for the serve to count.
Lola: Agreed! Then, if my returned ball goes into the net, then I lose the point.
You may recognize this game as ping pong (or table tennis), but it doesn’t matter what the game is—the point is that every game is made up of rules. We could also call these rules conditions—if one thing happens or doesn’t happen, then another thing happens or doesn’t happen.
In this article, we’ll show you how logical statements that have sufficient and necessary conditions act just like game rules, and we’ll talk about how Willie’s rules are of a different nature than Lola’s rules.
If you understand this distinction and its implications on the LSAT, then you will be rewarded with a higher score!
Necessary conditions
Let’s think about what it means for something to be necessary. You may have noticed that both of Willie’s rules named a necessary component:
- Willie has to hit the ball over the net in order to score a point during his shot.
- During Willie’s serve, he needs his served ball to hit his side of the table first before going over the net in order for the serve to count.
In both of these rules, there is a goal (following the signal words “in order to”), and a requirement for that goal (signaled by the italicized words). Here is arguably the most important thing you need to know about necessary conditions:
Necessary conditions don't guarantee any kind of result.
What does this mean? Well, Willie has to hit the ball over the net in order win the point, but that doesn’t mean that the ball going over the net guarantees that Willie gets the point. Perhaps if Willie hits the ball over the net but the ball never hits the table, then Willie doesn’t score. In other words, Willie hitting the ball over the net is just one element (among many, perhaps) that’s necessary in order for him to win the point.
The same is true with the second rule. Willie could meet the necessary condition—and his served ball hits his side of the table first before going over the net—but that doesn’t guarantee that the serve counts! The ball could go over the net and never hit the other side of the table (for instance)—in that case, the serve would not count (according to actual table tennis rules).
Necessary conditions, symbolically
Many students like to diagram conditional statements to help them observe the conditions more cleanly; the first rule above might look like this when diagrammed:
- If Willie wins a point during his shot then Willie hit the ball over the net
or
- Willie wins a point
Here’s a logically equivalent way of saying the same thing:
- If Willie doesn’t hit the ball over the net then Willie doesn’t win the point.
or
- Ball NOT over net
Top tip: Note how the conditional statement if X, then Y is logically equivalent to the statement if NOT Y, then NOT X. This logically equivalent version of a statement is sometimes called its contrapositive.
You’ll notice that the necessary condition in these diagrammed statements is always on the right. That’s because the right-hand statement doesn’t lead to another result. This makes sense because a necessary condition doesn’t guarantee any event. It’s necessary to meet the condition on the right in order for the condition on the left to occur, but meeting that right-hand necessary condition doesn’t guarantee that the left-hand condition occurs.
Sufficient conditions
Question: How are Lola’s rules different than Willie’s?
Answer: They name a sufficient condition instead of a necessary condition:
- If Willie’s ball doesn’t hit the table, then Willie loses the point.
- If Lola’s returned ball goes into the net, then Lola loses the point.
In both of these rules, there is an event (following the signal word “if”), that, if met, guarantees another event. Here are two important things you need to know about sufficient conditions:
- A sufficient condition, if met, guarantees another event with no exceptions. But
- A sufficient condition is not necessary for that event to happen, since there could be many other conditions that are also sufficient for the resulting event to happen.
What does this mean, in the context of our examples? In the first example, if Willie’s ball doesn’t hit the table, then Willie is guaranteed to lose the point. But that’s not the only way for Willie to lose the point! For example, if Willie puts his palm on the table, he’ll lose the point, according to the rules of table tennis. That’s another condition that’s sufficient to bring about the same result (of Willie losing the point).
In the second example, Lola’s returned ball going into the net guarantees that Lola loses the point. But if we were to learn that Lola just lost a point, we would not be able to infer that her returned ball must have gone into the net! There could be many other conditions that were sufficient for Lola to lose the point. In other words, it’s not necessary for Lola to hit into the net in order for her to lose a point.
Sufficient conditions, symbolically
When mapping a conditional rule, the sufficient condition is generally put on the left. Our first Lola example might look like this:
- If Willie’s ball doesn’t hit the table, then Willie loses the point
It’s good practice to also note the rule’s logically equivalent contrapositive:
- If Willie doesn’t lose the point, then Willie’s ball did hit the table
There is only one direction in which you can logically read the events described by this rule: once the “trigger” (sufficient condition) on the left is pulled (or true), the event on the right is certain to occur—100% of the time.
- If X, then Y does not logically imply If Y, then X—We cannot say that if Willie loses a point then his ball must not have hit the table; there could be so many other reasons that Willie loses a point!
- If X, then Y does not logically imply If NOT X, then NOT Y—We cannot say that if Willie’s ball hits the table then he will win the point.
Takeaways
- Conditional rules are just like game rules, with events that can be true “only if” something else is true, or “if” something else is true (to name just two examples of signals).
- A sufficient condition guarantees the truth of another condition, but is not necessary for that other condition to happen.
- A necessary condition is required for something else to happen, but it does not guarantee that the something else happens.
Necessity and sufficiency in the Logical Reasoning section of the LSAT
One flaw that’s commonly committed in Logical Reasoning arguments is a confusion of necessity versus sufficiency, so let’s examine what that looks like.
Mistake 1: Thinking that a condition is sufficient when in fact it is necessary
Consider this basic example of this classic pattern of flawed reasoning:
Mistake 2: Thinking that a condition is necessary when in fact it is sufficient
Consider this flawed argument:
In summary, look for relationships of necessity and sufficiency in Logical Reasoning and verify that the arguer isn’t confusing one for the other.
Your turn
Now’s your chance to practice recognizing necessary and sufficient conditions!
Note: This idea can show up in many different Logical Reasoning question types.
Take your time with these!
Want to join the conversation?
The reasoning in Question 2, Answer A doesn't make sense to me. I eliminated this answer because it contained the necessary condition signal "only if." Am I missing something about A that makes the conditional rules sufficient instead?
And let's say they use the word "if" in place of "only if." I still don't understand the reasoning, since the turn out (I use Y) is the goal, and following the recipe (S1) as well as using high-quality ingredients (S2) follow the same pattern as answer C and the original passage. The goal (Y) is met, and one of the sufficient conditions (S1) is said to be true. Therefore the other sufficient condition (S2) must also be true. There appear to be two sufficient conditions.(17 votes)- I'm wrestling with this, too, but I think I've got it. In the prompt, we have an outcome--the book tour being successful--and two conditions which we're told are enough to guarantee that the outcome will occur. If the tour is well-publicized and the author is an established writer, we are told that for sure the tour will be successful. Okay, but there could be other ways of achieving a successful book tour, too. Nothing in the formula says that you have to be a well-established author or that you have to publicize the tour in order to have success. Maybe a bunch of people just happened to be at the book tour locations for completely unrelated reasons, and this caused the book tour to be successful. The two conditions are sufficient to guarantee a successful book tour, but they aren't necessary to any successful book tour.
Answer A, on the other hand, introduces an outcome--the recipe turning out--which happens "only if" two other conditions are met. This means that there isn't another way for the condition to be met without fulfilling both of these conditions. You have to follow the recipe exactly and you have to use high-quality ingredients. The wording allows for no exceptions.
The prompt under answer A after you push "check" was a little confusing to me at first. It says that we can eliminate it "because the passage’s evidence contains two sufficient conditions (X and Y), whereas this choice contains only one (X)." In other words, we can eliminate choice A because while the passage (ie the original prompt) contains two sufficient conditions answer A only includes one. The outcome in choice A--the recipe turning out--actually is a sufficient condition. It functions the same way. If the recipe turns out, it's sufficient for us to determine that the two necessary conditions must have been met. Does that make sense?(34 votes)
- If the explanation in Example 2, Answer A is confusing, it's super helpful to read the next article: The Logic of "If" vs. "Only if"(16 votes)
- To me, choice A and C in question 2 seem the same... I don't understand why C is correct but A isn't. Can someone explain this to me in a little more detail please?(5 votes)
I am having trouble knowing when to group variables together. For example, in example 2, answer choice D, "suffer from dry rot and poor drainage" is one variable (X), whereas in answer choice C, "kept in the shade and watered more than twice weekly" is considered two variables (X and Y). How can I figure out that there is more than one variable present and not two parts to one variable?
Thanks!
Dom(1 vote)
I would say it's in the wording. "suffer from dry rot and poor drainage" is one thing, because the house SUFFERS. The fact that is suffers from two things isn't relevant, only that it suffers. "kept in the shade and watered more than twice weekly" is two things. It is kept in the shade, and it is watered. Two things occurred, whereas in the first example, one thing occurred (suffering). The fact that the suffering was comprised of two parts is what's confusing you, but the suffering is what happened, not the dry rot and poor drainage.(8 votes)
- Original Q.: X (well publicized) and Y (established writer) right arrow→right arrow Z (successful book tour)
Julia = Y and Z, therefore X must be true.
A: This choice has the same structure:
X (kept in shade) and Y (watered more than twice weekly) right arrow→right arrow Z (die)
This cactus was X and Z, so Y must be true.
THE CONCLUSION IS DIFFERENT (Even though same structure?)
Okay. I get it. Even though the conditions were (X,Y, Z) were rearranged in the conclusions of both, they still have the same structure. Okay.(2 votes) - Is the disqualifier in answer A the term "only if"? The rest of the answer seems to match. (X if Y +Z) to (Y + X thus Z)
Second question: Answer C does not appear to be a complete match because the answer starts with "if Condition + Condition then Result". The set up starts with 'Result' followed by the qualifying conditions. IS IT OK TO FLIP THE CONDITIONS AND RESULTS IN THIS MANNER?(2 votes) - Also, aside from the 3 articles listed here under the lessons, where can I find more information/instruction on necessary vs. sufficient conditions in general? The article is a good start, but I need something more in depth.(2 votes)
Some of the signal words for sufficient/necessary conditions are confusing to me. Specifically:
Signal of Sufficiency:
"In order for S to happen, Y must happen"
Signal of Necessity:
"In order for Y, N must happen"
Could someone please explain how these are derived or how they make sense? In the sufficient condition statement "If Lola's ball goes into the net, Lola loses a point" for instance...wouldn't this be translated to "Lola loses a point if Lola's ball goes into the net" (which per the signal words above would mean S = Lola loses a point?) Please let me know if I need to clarify - hopefully this makes sense!(1 vote)- In the "Necessity and sufficiency in the Logical Reasoning section of the LSAT. Mistake 1" section, what does the author mean by "evidence"?(0 votes)
- What would be the correct conclusion for the argument in Example 1 if it weren't flawed?(0 votes)
