Necessary and Sufficient Conditions for the Case



§1. Necessary Conditions

To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y. A necessary condition is sometimes also called "an essential condition". Some examples :

  • Having four sides is necessary for being a square.
  • Being brave is a necessary condition for being a good soldier.
  • Not being divisible by four is essential for being a prime number.

To show that X is not a necessary condition for Y, we simply find a situation where Y is present but X is not. Examples :

  • Being rich is not necessary for being happy, since a poor person can be happy too.
  • Being Chinese is not necessary for being a Hong Kong permanent resident, since a non-Chinese can becoming a permanent resident if he or she has lived in Hong Kong for seven years.

Additional remarks about necessary conditions :

  • We invoke the notion of a necessary condition very often in our daily life, even though we might be using different terms. For example, when we say things like "life requires oxygen", this is equivalent to saying that the presence of oxygen is a necessary condition for the existence of life.
  • A certain state of affairs might have more than one necessary condition. For example, to be a good concert pianist, having good finger techniques is a necessary condition. But this is not enough. Another necessary condition is being good at interpreting piano pieces.

§2. Sufficient Conditions

To say that X is a sufficient condition for Y is to say that the presence of X guarantees the presence of Y. In other words, it is impossible to have X without Y. If X is present, then Y must also be present. Again, some examples :

  • Being a square is sufficient for having four sides.
  • Being divisible by 4 is sufficient for being an even number.

To show that X is not sufficient for Y, we come up with cases where X is present but Y is not. Examples :

  • Loving someone is not sufficient for being loved. A person who loves someone might not be loved by anyone perhaps because she is a very nasty person.
  • Loyalty is not sufficient for honesty because one might have to lie in order to protect the person one is loyal to.

Additional remarks about sufficient conditions :

  • Expressions such as "If X then Y", or "X is enough for Y", can also be understood as saying that X is a sufficient condition for Y.
  • Some state of affairs can have more than one sufficient condition. Being blue is sufficient for being colored, but of course being green, being red are also sufficient for being coloured.

§3. Four Possibilities

Given two conditions X and Y, there are four ways in which they might be related to each other:

  1. X is necessary but not sufficient for Y.
  2. X is sufficient but not necessary for Y.
  3. X is both necessary and sufficient for Y. (or "jointly necessary and sufficient")
  4. X is neither necessary nor sufficient for Y.

This classification is very useful in when we want to clarify how two concepts are related to each other. Here are some examples :

  • Having four sides is necessary but not sufficient for being a square (since a rectangle has four sides but it is not a square).
  • Having a son is sufficient but not necessary for being a parent (a parent can have only one daughter).
  • Being an unmarried man is both necessary and sufficient for being a bachelor.
  • Being a tall person is neither necessary nor sufficient for being a successful person.
Critical Thinking Web
https://philosophy.hku.hk/think/

Everyone is familiar with the concept of a Necessary Condition. For example, we all know that air is necessary for (human) life. Without air, there is no (human) life. Similarly, a microscope (or some other instrument) is necessaryfor human beings to see viruses. (Viruses are too small to be seen by the naked eye.) 

Similarly, everyone is familiar with the concept of a Sufficient Condition. For example, it suffices (i.e. it is sufficient for) an object's having four sides that it is a square. Or, again, it is sufficient for your having something to drink that you have a glass of Coca-Cola®.


The Concepts of Necessary Conditions and Sufficient Conditions
https://www.sfu.ca/~swartz/conditions1.htm



Necessary Conditions - If we say that "x is a necessary condition for y," we mean that if we don't have x, then we won't have y. Or put differently, without x, you won't have y. To say that x is a necessary condition for y does not mean that x guarantees y.


Some examples will help here.


Having gasoline in my car (I have a gasoline engine) is a necessary condition for my car to start. Without gasoline (x) my car (y) will not start. Of course, having gasoline in the car does not guarantee that my car will start. There are many other conditions needed for my car to start.


Having oxygen in the earth's atmosphere is a necessary condition for human life. Certainly, having oxygen will not guarantee human life. There are many other conditions needed for human life other than oxygen in the atmosphere.


Being 18 years of age is a necessary condition for being able to buy cigarettes legally in North Carolina. Of course, being 18 does not guarantee that a person will buy cigarettes. There are many other conditions that lead to a person buying cigarettes than being 18 years of age.


Sufficient Conditions - If we say that "x is a sufficient condition for y," then we mean that if we have x, we know that y must follow. In other words, x guarantees y.


Consider the following examples.


Earning a total of 950 points (95%) in this Critical Thinking class is a sufficient condition for earning a final grade of A. If you have 950 points for the course, then it must follow that you will have a final grade of A.


Pouring a gallon of freezing water on my sleeping daughter is sufficient to wake her up. If I pour the gallon of freezing water on her then its guaranteed that she will wake up.


Rain pouring from the sky is a sufficient condition for the ground to be wet.

Please note that in none of these example is the sufficient condition also a necessary condition.


For example, it is not necessary to earn 950 points to earn an A in this course. You can earn 920 points to earn an A. (We cannot say that if you do not have 950 points then you can't have an A.)


It is not necessary to pour a gallon of freezing water on my daughter to wake her up. (A wrecking ball against the wall will do it as well.)


Similiarly, it is not necessary for rain to be pouring from the sky for the ground to be wet. The sprinkler could be on as well.


Necessary and Sufficient Conditions

http://faculty.uncfsu.edu/jyoung/necessary_and_sufficient_conditions.htm 


Confusion of Necessary with a Sufficient Condition


A causal fallacy you commit this fallacy when you assume that a necessary condition of an event is sufficient for the event to occur. A necessary condition is a condition that must be present for an event to occur. A sufficient condition is a condition or set of conditions that will produce the event. A necessary condition must be there, but it alone does not provide sufficient cause for the occurrence of the event. Only the sufficient grounds can do this. In other words, all of the necessary elements must be there.


Examples:

  1. Juan: "How do you think you'll do on our philosophy exam tomorrow?" Monique: "Great, I read all the books."  Juan: "Yeah but do you understand this stuff?" Monique: "I said I read all the books, didn't I?"
  2. Don't let all the talk about the necessity of exercise to a long life mislead you. Jim was a jock all his very short life.
  3. Who said food keeps us alive? Tom died a few days ago and he was not short of good food.
  4. I don't know why the car won't run; I just filled the gas tank.
  5. Why don't you want to spend your life with me? I love you, and am I not good to you?
  6. The counselor told me that if I wanted to graduate I must have at least 128 credit hours. Well, I've got that, but they're saying I won't walk this semester. How misleading!
  7. The job description said that they were looking for someone with a Master's degree. I've got my MA, but I cannot understand why they did not hire me.
  8. My high school English teacher told me successful people have extensive vocabularies. So, I've tried to learn a new vocabulary word every day since then. I should be a successful person soon.
  9. "If you want the chairman to like you," he said, "you'll have to show how impressed you are with his ideas." Well, I turned myself into an ego-stroking machine, but it's pretty clear I have yet to win him over.
  10. I've heard this ever since puberty: in order to fall in love with another, you have first to love yourself. This is such a crock! I have no trouble loving myself, bit I can't say I've ever fallen in love in my life.


Confusion of Necessary with a Sufficient Condition 

https://www.txstate.edu/philosophy/resources/fallacy-definitions/Confusion-of-Necessary.html 





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 guaranteesthat 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  ball over net
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  no point
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 notnecessary 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…



Minding Your Ps and Qs on the LSAT: Necessary and Sufficient Conditions

If you’ve ever been told to “mind your Ps and Qs”, you know that the expression equates to being instructed to mind your manners. That is, of course, unless you’re studying for the LSAT, where Ps and Qs have nothing to do with being polite. In fact, seeing Ps and Qs may inspire some LSAT takers to feel particularly impolite: they generally signify a conditional reasoning problem, which can be stressful and confusing for those not familiar with how conditional reasoning works. However, with a little effort we can easily demystify the basics of conditional reasoning, so that you’re able to mind all types of Ps and Qs.

Introduction to Conditional Statements

A conditional statement is composed of an ‘if-then’ proposition and its equation is written as p → q (if p, then q), where p is a hypothesis and q is a conclusion. It’s important to remember that p and q are just placeholders and you won’t normally see those letters on the actual LSAT. Instead, you are likely to be given an if-then statement such as “If William cooks dinner, then Erin washes the dishes.” You can write that statement out in the p → q equation format: William cooks dinner → Erin washes dishes. 

Now that we have laid out the basic format of a conditional statement, we need to discuss the differences between sufficient and necessary conditions. The ‘if’ or p part of a conditional statement is a sufficient condition, while the ‘then’ or q part of a conditional is a necessary condition. It’s easiest to explain the difference between sufficient and necessary conditions through examples. 

Examples - Sufficient Conditions

Let’s say you walk outside and see that a car parked on the street is wet. A sufficient condition for that car to be wet would be that it just rained. Another sufficient condition would be that your neighbor is washing their car and sprayed it down with a hose. Both of these sufficient conditions explain how the car could have become wet - but neither of them are necessary for the car to have become wet, as the car could have gotten wet in other ways.

Another example would be that microwaving water is a sufficient condition to heat the water, but it is not a necessary condition, as there are other ways water can be heated that do not involve microwaving. A sufficient condition fully explains how a given outcome could occur, without being the only possible explanation for how that outcome could have resulted.

Examples - Necessary Conditions

If something is a necessary condition it means it absolutely must take place for a given outcome to occur; the outcome cannot occur without that condition being met. In order to attend a certain concert, you need tickets. Thus, having tickets is a necessary condition for attending that concert.

However, just because an outcome requires a necessary condition doesn’t mean that the necessary condition is the only thing required for the outcome to occur. For example, it may be a necessary condition for Student A to study in order to pass their math class. This means that there is no way that Student A can possibly pass math without studying. But even if Student A studies, that may not guarantee they pass their class. They may also have to take additional steps, such as going to tutoring, reading their textbook, and completing all their homework.

Examples - Sufficient + Necessary Conditions

In our if p then q equation, p represents a sufficient condition, while q represents a necessary condition. Let’s go back to our earlier example: if William cooks dinner, then Erin washes the dishes. William cooking dinner is a sufficient condition for Erin washing the dishes. There may be other circumstances in which Erin washes dishes, but William cooking dinner is one sufficient reason why Erin will wash up. However, Erin must wash the dishes in order for William to cook dinner. Since the only way William cooks dinner is if Erin washes the dishes, Erin washing the dishes is a necessary condition.

How to Master Sufficient and Necessary Conditions

Keep an eye out for ‘trigger words’ that can signify the introduction of either a sufficient or a necessary condition. Words usually associated with sufficient conditions include: if, when, whenever, every, all, any, people who, and in order to. Words commonly linked with necessary conditions are: then, unless without, until, only/only if, must, and required.

The only way to really get the hang of sufficient and necessary conditions is to practice them. In fact, you could say that practice is a necessary condition for understanding sufficient and necessary conditions. Review the examples presented in this blog post and then try writing out some examples of your own. You practice identifying sufficient and necessary conditions → you’ll be an expert at them in no time!

Stay tuned for the next post where we will begin to manipulate conditional statements with a little bit of help from renowned rapper Missy Elliott...

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