2.3: Necessary and Sufficient Conditions

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It is so easy to confuse Necessary and Sufficient Conditions. Knowing the difference can help you avoid this fate. After introducing the distinction, we’ll discuss why it is important to be aware of it.

First, some vocabulary:

A state-of-affairs or event or condition is something that can happen. My going to the mall, or your passing your exam, or the wall’s turning red in color, Chris’s getting a DUI, an electron’s being annihilated, etc. are all examples of states of affairs or events.

When one of these in fact occurs or happens or is true, we say that it obtains. So, I can say that

“the state of affairs where Andrew Lavin is an instructor at FRC has obtained.” This means that Andrew Lavin is in fact an instructor at FRC. ‘To obtain’ is simply the verb that attaches to states of affairs and events and the like. It means something like “happens.”

• For Chris getting a DUI to obtain, he must have been caught driving while intoxicated.
• The wall turning red will obtain if we paint it red.

These are unnatural ways to talk, but this verb can be helpful for defining Necessary and Sufficient Conditions.

Necessary Conditions

A necessary condition is something that must obtain if something else is to obtain.

It means if that for which it is a necessary condition obtains, then it must also obtain/have obtained.

If X is a necessary condition for Y, then any time Y obtains, X must also obtain.

Y requires or needs X in order to obtain. (X is necessary for Y)

If peanut butter is necessary for making a PB&J, then you must have peanut butter if you are going to make a PB&J. Any time you make a PB&J, you must have peanut butter. A PB&J requires or needs peanut butter to be a PB&J.

Examples

Example $$\PageIndex{1}$$

Peanut Butter is a necessary condition for making a PB&J sandwich.

Jelly is also a necessary condition for making a PB&J.

Air being present is a necessary condition for breathing. It’s also necessary for a flame.

Paying for a product is a necessary condition for walking out of a store with that product without stealing (and without that product being free or given to you by the store owner).

Proving someone’s guilt beyond a reasonable doubt is a necessary condition for convicting them of a criminal offense in the American criminal trial system.

Being an animal is a necessary condition for being a mongoose.

More intuitively, a necessary condition is something you need if you are going to do something else. It’s whatever must happen if something else is to happen. You can’t be a mongoose without being an animal, so being an animal is a necessary condition for being a mongoose. If you didn’t steal something, then normally it follows that you did the necessary things: you paid for it before walking out of the store.

Sufficient Conditions

A Sufficient Condition is something that is enough for something else to obtain.

It means that if the sufficient condition obtains, then that for which it is a sufficient condition is sure to obtain as well.

If X is a sufficient condition for Y, then any time X obtains, Y will also obtain.

X causes or is enough for Y to obtain. (X is sufficient for Y)

If you break your leg, it will hurt, so breaking your leg is sufficient for being in pain. Breaking your leg is enough (it’s all you need to do) to be in pain. If you break your leg (any time you break your leg), you will be in pain. Breaking your leg causes you to be in pain.

Sufficient Conditions are a bit trickier when it comes to examples. They always rely on certain unstated background assumptions and so it feels like most of them have relatively easy-to-find counterexamples. We have to practice a bit of charity in interpreting sufficient conditions.

Examples

Example $$\PageIndex{2}$$

Having Peanut Butter, Jelly, Bread, and a knife is a sufficient condition for making a PB&J sandwich (all else being equal).

• Note that these are individually necessary and jointly sufficient, meaning taken together, they form a sufficient condition.

Taking air in through your mouth into your lungs and extracting the oxygen is a sufficient condition for breathing.

Walking out of a store with one of their products without paying is a sufficient condition for stealing (assuming it wasn’t free or given to you by the owner).

Being caught by police in the middle of committing a crime is generally a sufficient condition for being arrested for that crime.

Being a rabbit is a sufficient condition for being a mammal.

If a thing is a rabbit, then it is a mammal. All that is needed to be a mammal is to be a rabbit. If you want to breathe, all that you need to do is take air into your lungs through your mouth and extract the oxygen. That’s not the only way to breathe (you can breathe through your nose, for instance), so we know that this is a sufficient but not necessary condition.

Necessary and Sufficient Conditions

Note that a condition can be:

1. Necessary but not Sufficient

2. Sufficient but not Necessary

3. Necessary and Sufficient

4. Neither Necessary nor Sufficient

Necessary and Sufficient conditions are things that are both enough for and required for something else.

If X is a necessary and sufficient condition for Y, then:

If X obtains, then Y must obtain (so any time X obtains, Y also obtains)

And

If Y obtains, then X must also obtain (so any time Y obtains, X also obtains)

Examples

Example $$\PageIndex{3}$$

Putting peanut butter on one slice of bread and putting jelly on another slice of bread and putting them together (PB&J sides inwards) is a necessary and sufficient condition for making a PB&J sandwich.

Taking sandwich ingredients and sandwiching them between two slices of bread or two halves of a roll is a necessary and sufficient condition for making a sandwich.

Having a belief that is true and isn’t accidentally true is a necessary and sufficient condition for having knowledge (some think).

Earning a final grade of C or better is a necessary and sufficient condition for passing a General Education course.

Being 21 years of age or older is a necessary and sufficient condition for being legally allowed to drink in the US (barring special legal constraints and not counting states that allow certain cases of underage drinking).

Neither

Of course, there are loads of conditions that are neither necessary nor sufficient for other things to occur. Here are some examples

Example $$\PageIndex{4}$$

Claude pouring his espresso too fast is neither necessary nor sufficient for the sun to be in the sky.

Eating BBQ is neither necessary nor sufficient for being vegetarian (in fact, depending on the protein, it’s probably sufficient for not being a vegetarian!)

Paying for a milkshake after ordering it for yourself is neither necessary nor sufficient to get a ride to Phoenix with your cousin for Christmas.

Background Assumptions

Every conditional claim makes certain background assumptions. For instance, driving is a sufficient condition to get to work on time. But it’s not going to be sufficient if your car breaks down, or if you get in an accident, or if a meteor destroys your workplace, or if Thanos from Marvel’s Avengers turns you into dust while you’re on the middle of the drive. Every time we claim something is a condition for something else—especially a sufficient condition—we are making the assumption that everything else basically proceeds as normal. If weird stuff starts happening, then my condition might not be sufficient anymore—but this doesn’t mean that what I said originally was false.

Who Cares?

Why study Necessary and Sufficient conditions?

First, it helps us understand the dependencies between events and things in our world. Getting clearer about these distinctions helps us to reason more clearly about conditions, conditionals (if...then... propositions), cause and effect, evidence and implication.

Second, it helps us get clear on what is involved in definition. What are we trying to do in defining something? Generally, we’re trying to give necessary and sufficient conditions for that thing (though this is a problematic account of definition, it’s good enough for our purposes). When we define what a dog is, we’re listing what all dogs must have to be dogs (dog DNA, Dog parents, etc.) and what features are such that if anything has those features, then that thing is a dog (what features make something a dog). It might be necessary and sufficient for being a dog that one has a certain canine genetic signature. All and only dogs have this signature. This is almost certainly false, since biology is a wild and confusing place, but we get the idea of what we’re at least trying to do when we define a term: identify what all and only those things that fall under that concept have.

Third, it helps us avoid common fallacies (errors in reasoning). Confusing necessary conditions for sufficient conditions is all-too-common in everyday reasoning and (as you’ll see in the videos from Wireless Philosophy) can lead to really bad consequences—like wrongful convictions!

Here’s an example:

Example $$\PageIndex{5}$$

If we find the murder weapon in Scott’s apartment, then we know Scott committed the crime.

Is finding the murder weapon necessary or sufficient? What happens if we don’t find the murder weapon? Is Scott off the hook? If we know that either Scott or Mohinder committed the crime, but we don’t find the murder weapon in Scott’s apartment, does that mean Mohinder did it?

If we don’t get clear on the relationship between necessary and sufficient conditions, then we won’t be able to answer these questions properly and we’ll potentially convict the wrong person or let the wrong person go.

In this case, finding the murder weapon is a sufficient condition for knowing that Scott committed the crime.

If we don’t find it, that doesn’t mean anything at all when it comes to whether or not Scott committed the crime.

Let’s think about an analogy: If I poison you, you’ll get sick. If I don’t poison you, then who knows if you’ll get sick? We don’t know that you won’t get sick! You might eat bad mayonnaise or get an infected tick bite!

So, we don’t know that Scott didn’t commit the crime if we didn’t find the murder weapon in his apartment.

Here are some other examples of necessary and sufficient condition confusion:[1]

Example $$\PageIndex{6}$$

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?"

Monique is confusing the fact that reading all of the books is necessary for understanding the material of the course with the false idea that reading all of the books is sufficient. It’s not sufficient, you have to read and understand (and probably some other stuff too!) in order to do well on the philosophy exam.

Example $$\PageIndex{7}$$

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.

Jim dies tragically short, but not from exercise! The speaker here confuses the fact that regular exercise is necessary for a long, healthy life with the false claim that regular exercise is sufficient. It’s not sufficient, as Jim’s tragic example shows. It takes a lot more than regular exercise to get you a long and healthy life since failing to exercise isn’t the only thing that can go wrong.

Example $$\PageIndex{8}$$

I don't know why the car won't run; I just filled the gas tank.

Again, a full gas tank is necessary, but it isn’t sufficient for a car to run. There is a lot more that can go wrong with a car other than an empty gas tank. Having gas isn’t the only condition that needs to be in place.

Example $$\PageIndex{9}$$

She didn’t have any diseases, I don’t understand why she died!

Having a disease is sometimes sufficient for dying (tragic, though it may be). But it isn’t necessary. There are many more ways to die (sadly) than getting a disease.

Example $$\PageIndex{10}$$

My cat didn’t get taken away by an eagle, so I don’t understand why it’s missing.

This is a kind of comical example. It’s silly, right? Of course, if your cat did get taken away by an eagle, then it would be missing, but it would also be missing if it was hiding under the porch, or hitched a ride on a semi-truck to Tucson! There are lots of ways for a cat to go missing (so it’s not necessary that it be taken away by an Eagle), but it isn’t indeed sufficient for it to be missing that an Eagle swooped in and grabbed it.

[1] I got most of these from <http://www.txstate.edu/philosophy/resources/fallacy-definitions/Confusion-of-Necessary.html>

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