Please explain sufficiency vs necessity :(

ToniT220

An example in this post is from a live class so it MAY BE A SPOILER****

Hi! I am continuously running into issues with conclusions regarding sufficiency and necessity. I completely understand the structure of Lawgic, and I can chain conditionals with no issues using Lawgic, my issue is when sufficiency and necessity lead to a conclusion, and I cannot conclude the argument is valid or draw a conclusion. I can write it out correctly, I just don't understand what it really means..

Example:
Exercise 2: Evaluating Argument Validity
Is the following argument valid?

The vote to grant Chancellor Palpatine emergency powers will not pass if Senator Amidala delivers her speech. Amidala cannot deliver her speech unless the attempt to assassinate her fails. Her assassins planted a bomb on her starship but unbeknownst to them, she was not on the ship when the explosive detonated. Therefore, the vote to grant the Chancellor emergency powers will not pass.

The argument is not valid because of the Lawgic: (I have the structure down)
SAS → /P
SAS → AAF
AAF

---

/P

Where I am getting confused is the explanation that is provided: "Satisfying a necessary condition yields no valid conclusions." So when can we yield a valid conclusion?? What condition should I be looking at to conclude whether an argument is valid or not?

Another example:

Biologist: We know the following things about plant X. Specimens with fuzzy seeds always have long stems but never have white flowers. Specimens with curled leaves always have white flowers, and specimens with thorny seedpods always have curled leaves. A specimen of plant X in my garden has a long stem and curled leaves.
Q: From the biologist's statements, which one of the following can be properly inferred about the specimen of plant X in the biologist's garden?

I have all of the Lawgic correctly written down:
fuzzy seeds-> long stems
fuzzy seeds -> /white flowers
curled -> white flowers
thorny seedpods -> curled leaves

x has a long stem and curled leaves

The answer: it has white flowers but lacks fuzzy seeds.
HOW??
I understand it has white flowers, but how is it not "It has white flowers and thorny seedpods."

Is it because if there are curled leaves, then there are white flowers (curled leaves -> white flowers), the fact that having curled leaves is in the sufficient means that white flowers has to follow?
And thorny-> curled means nothing because curled is not in the sufficient?
If something is satisfied in the necessary, you can't conclude anything from that?

I have literally spent HOURS trying to understand this (and understanding other examples further down LR). I don't want to move past chaining conditionals until I can completely understand this, so I'm stuck in my studying. I'm actually struggling so hard. Also is it clear what I'm getting confused on... ?? I can re-edit if this is too all over the place sorry

MattyCzar

When it comes to a lawgic chain, you can never draw conclusions going from right to left, everything must go left to right. You seem to have trouble understanding sufficiency and necessity, so I'll include my own examples that might help.

"Bob lives in Japan. Anyone that lives in Tokyo must live in Japan."
So, Bob -> Japan, Tokyo -> Japan.

Can we conclude anything about Bob and his relationship to Tokyo here? Nope! Living in Japan is necessary for living in Tokyo, as in we cannot live in Tokyo without living in Japan. And living in Tokyo is sufficient for living in Japan. If someone tells you 'I live in Tokyo,' you can assume that they live in Japan. But if someone tells you 'I live in Japan,' you cannot assume that they live in Tokyo. That's what we have here with Bob. He very well may live in Tokyo, but he could also live in Osaka or Kyoto, we don't have the info to know for certain and therefore cannot draw any conclusions.

So going to your biology example, I wrote down:
FS -> LS
FS -> /WF
(contrapositive: WF -> /FS)
TS -> CL -> WF

Given LS and CL, we can conclude WF from (CL -> WF), but nothing from LS. Now with WF, we can use the contrapositive of FS -> /WF to conclude that the plant additionally cannot have fuzzy seeds. Therefore, based on what we're given, we can conclude WF, /FS. You cannot read right to left, but you can take the contrapositive and read that from left to right. 'If you live in Tokyo, you live in Japan' becomes 'If you do not live in Japan, you cannot live in Tokyo' just as 'If you have fuzzy seeds, you do not have white flowers' becomes 'If you have white flowers, you do not have fuzzy seeds.'

Yes, you're correct in saying that you cannot conclude anything from the necessary. To use an additional example, it is necessary to have a high GPA, high SAT, and be in many clubs to attend Harvard. Attend Harvard -> high GPA and high SAT and many clubs. If someone tells you 'I had a high GPA, high SAT, and was in many clubs in high school,' can you conclude that they attended Harvard? Not at all! Maybe they didn't even apply, and that's certainly a necessity to attend Harvard!

To sum up, I'll give you one final run down.

In X -> Y, if X then Y must follow. If you live in NYC, you must live in the US. If you attend college, you must have graduated from high school. Living in NYC or attending college are sufficient to lead us to conclude that you live in the US or have graduated high school. But if I asked you to list out everything that is necessary to attend Harvard Law School, you could spend all day. High uGPA, high LSAT, letters of rec, fill out the application, be admitted, be eligible to attend a US law school, etc. We can't take just one of these and conclude that, based on just that, they attend Harvard Law School. That's what you're trying to do when you go from right to left in lawgic.

Arcana

I like to think about it in terms of cats and mammals whenever I get confused.

Bear with my analogy for a moment - I will answer your question about affirming the necessary further down in this, but stick with the cat metaphor for a bit. At the end, I will help to apply the cat analogy to the problem you're stuck on.

Please note - for this entire post, when I say CAT, I really mean "House Cat."

OK- Being a cat is sufficient for being a mammal.

Conversely, being a mammal is necessary for being a cat.

the relationship is written like this.

cats --> Mammals

we call the left thing the sufficient condition, because affirming it is SUFFICIENT for the condition on the right to be true. (that is to say - if you are indeed a cat, its enough to make you a mammal, but its not guaranteed to be the only way to be a mammal)

we call the right thing the necessary condition, because affirming it is NECESSARY for the condition on the left to be true.

If you put an animal in a bag and tell me its a mammal, I can't, however, logically deduce "Oh, it's a cat!".

That would just be flipping the relationship around arbitrarily.
mammal --> cat
You can't do this! Our general knowledge tells us why - There are tons of mammals that are NOT cats!

As you know, The conclusion of "all mammals are cats" is not implied from "all cats are mammals" . you'd be insane to conclude this. You just, simply put, can't work right to left. It leads you to saying pants on head stupid things like "Because it's a mammal, it's a cat." It is also extremely easy to do on accident, especially when under pressure or when there are deliciously tempting answer choices that APPEAR to give you what you are looking for (this is what happened in your problem, by the way. We've all done it a dozen times.)

OK, but, if you did want to go and work in the other direction, that is the contrapositive. Flipping and negating lets us preserve the truth of the original statement.
/mammal --> /cat
If you're not a mammal, you're for sure not a cat. Ok, yeah, that still works. Cool. This passes the reality check of our analogy.

Let's go deeper - here is that original, proper relationship expanded, so we have a bit more to work with.

All cats (again, house cats) are member of the family Felidae, and thereby also members of the order Carnivora, and thereby also members of the class of Mammal.

cat --> felidae --> carnivora --> mammal

Everything to the right of the cat is necessary for cat-ness. If it's a cat, its gotta be all of those. If I disprove any of those, ie, I say "Hey this animal in the bag is NOT a member of carnivora" then it's not a cat, because it HAS TO BE ALL THOSE things to be a cat.

If I tell you "the animal in the bag is a mammal" it is called "affirming the necessary". It doesn't tell us anything. It's a mammal? Cool, I guess, but in this relationship we have, it doesn't let me conclude anything. It could be a mammal, but not a member of Carnivora. Deer exist, and they are mammals, but not carnivores.

Saying "The animal in the bag is not a mammal" however, lets us validly conclude things. I know, 100% for a fact, that it is not a carnivora, not a felidae, and not a cat. This is called denying the necessary.

Let's consider sufficiency -

Being a cat (again, housecat) is SUFFICIENT for the stuff on the right. It doesn't guarantee that its the ONLY way, though. In fact, you and I happen to have real world knowledge that it is NOT the only way. There are other members of the genus felidae (tigers, lions, etc), other members of carnivora, and other kinds of mammals, or put differently in most arguments, there are other ways of achieving that end. Being a cat (housecat), however, is SUFFICIENT for being a member of felidae, being a member of carnivora, or being a member of mammal.

What if I did this?
What if I said the animal in the bag is not Felix the cat (housecat)?
Ok, well, let's look at our chain.

Felix (who is a cat) --> cat --> felidae --> carnivora --> Mammal

Ok, the animal in the bag isn't Felix. What can I say about the animal in the bag?

Uhhh..... Nothing, really. Just that it isn't Felix, but we already said that. It could still be a cat, just not Felix. This is denying the sufficient. It doesn't let us conclude anything.

If I say that the animal in the bag IS Felix the cat, though, I can validly conclude that the animal in the bag IS a cat, IS a felidae, IS a carnivora, and IS a mammal. This is affirming the sufficient.

important things to remember: Seriously, write these down and keep wrestling with them until you understand them. The rest of my post will help to illustrate a bit more, but you MUST feel comfortable with these concepts.

cats -> mammals
this statement says "Being a cat requires being a mammal."

we call the left thing the sufficient condition, because affirming it is SUFFICIENT for the condition on the right to be true.

we call the right thing the necessary condition, because affirming it is NECESSARY for the condition on the left to be true.

Denying the necessary and affirming the sufficient let you LOGICALLY CONCLUDE things. This is the "must be true" type of shit.

Affirming the necessary doesn't let us conclude anything: This if you tell me the animal in the bag is a mammal. Ok, so what? Yes, being a mammal is necessary for being a cat, but tons of animals are mammals. This doesn't let us make any valid conclusions about the animal in the bag. I have no fucking clue which mammal it could be. All I can say is "it's a mammal." But you already told me that.

Denying the necessary Lets Us Conclude Things: This is if you tell me the animal in the bag is NOT a mammal. Well, dang, now I know a lot! I know there's no fucking way it's a cat! (or a Carnivora, or a Felidae, and definitely not Felix!)

Affirming the Sufficient Lets Us Conclude Things: "The animal in the bag is definitely a cat." Whoa - I know a lot now! I know it's also a Felidae, Carnivora, and a Mammal! It HAS TO BE those things! (careful though - it doesn't tell us it's Felix!)

Denying the sufficient doesn't let us conclude anything: "That animal in the bag isn't a cat." Again, I can't conclude much of anything from this. It could still be a bunch of other things. You might already be starting to say something about Felix.

Hold that thought about Felix. We'll come back to it.

If you look at
felix --> cats --> felidae --> carnivora --> mammal

Consider that some things are sufficient to some parts, but necessary to others.

Let's say I have two animals in two bags.
One, i say, is a member of carnivora, the other, I say, is not a member of carnivora. What can I VALIDLY conclude about each animal?

think about it.

ok - let's check your thinking.
felix --> cats --> felidae --> carnivora --> mammal

Animal A, who I've affirmed membership in carnivora:

I can validly conclude that this animal is a mammal. I have no clue about anything to the left of that - is it a felidae, is it a cat, is it felix? No idea. This is because carnivora is SUFFICIENT for membership in mammals. However, it is NECESSARY to the rest of the stuff on the left. The only way I can work in that direction is with the contrapositive, which requires /carnivora (which isn't the case for this animal)

Animal B, who I've denied membership in carnivora:
Well - if it's not a carnivora, I know that it CAN'T be a member of the felidae family, I know it's not a cat, and I know it's not Felix the cat. These are all logically true statements. It could still be a mammal, though (or it could not!) I can't say anything one way or the other about that. Again - this is for the same reason. It is necessary to the stuff on the left, so denying it lets us conclude things on that side. It is sufficient to the stuff on the right, so saying "It's not a carnivora" doesn't tell us anything about if its a mammal.

Lastly: on the note of Felix.

You hopefully said that denying membership in Cats (again, remember house cats) earlier doesn't tell us anything about felidae (what if its a lynx or an ocelot), doesn't tell us anything about carnivora, doesn't tell us anything about being a mammal. Denying membership in cats, however, DOES tell us that this animal is not Felix.

OK - hopefully that helped.

The "cats and mammals" test is something I use every time I take a practice test, so I hope it helps you too.

now - for your example problem.

Cats -> mammals is a relationship.
The test will then sometimes give you an example to consider, ie, your example of "a speciment of plant x has blah blah blah". This is the animal in the bag.

for your example:
these are the relationships they've given us to work with:

1. fuzzy seeds-> long stems
2. fuzzy seeds -> /white flowers
3. curled -> white flowers
4. thorny seedpods -> curled leaves

given about our specimen (the animal in the bag):
x has a long stem and curled leaves

OK - we affirm long stems and curled leaves.

1. Fuzzy seeds -> long stems(we affirmed the necessary, and can't conclude
anything from this. If we tried to, we would be saying "all mammals are cats.)

2. Fuzzy seeds -> /white flowers (Can't conclude anything yet! We have nothing yet that could interact with this statement. Let's come back to if we learn more.)

3. Curled --> white flowers (We have affirmed the sufficient. the relationship says that if we have curled leaves, we MUST have white flowers. Our speciment does have curled leaves, so it must have white flowers. OK, We now know our mystery specimen has whiteflowers.)

Wait! We just learned we have white flowers! This might interact with our previous statement.

Let's go back to 2.

2A. fuzzy seeds -> /white flowers. Hmm.... If it's got fuzzy seeds, it must NOT have white flowers. We can also say "If it has white flowers, it must NOT have fuzzy seeds." (contrapositive)

2B. (Contrapositive) thus, white flowers --> /fuzzy seeds. AH hah! We can do something here! We know we have white flowers, we affirm the sufficient. We now KNOW our mystery plant does not have fuzzy seeds.

4:. thorny seedpods -> curled leaves
This is where you went wrong - you did an "All Mammals are Cats" here. We have affirmed the necessary - we know that our specimen has curled leaves. However, this relationship is only saying that curled leaves are necessary for something to have thorny seedpods. Thorny seed pods, on the other hand, are merely SUFFICIENT for having curled leaves.

There may be other ways to get curled leaves, just like the statement
cats -> mammals
leaves room for other ways to be a mammal other than being a cat (which we happen to also know is true from our general knowledge.) In short - affirming the necessary doesn't let us conclude anything.

OK - here's what we concluded.

We started with knowing that it has curled leaves and a long stem.

We concluded that our mystery specimen has white flowers, and it does not have fuzzy seeds.

There is an answer choice that perfectly matches:

"it has white flowers but lacks fuzzy seeds."

ToniT220

@MattyCzar Thank you so SO SO MUCH! You are so so sweet for explaining this so simply for me, I really appreciate you

Ok so we cannot conclude anything from the necessary condition, but we can use a sufficient condition (if we are given that) and see what necessary condition it leads to?

For the biology example: (because we are given 2 things: long stem and curled leaves), the rule is that if you have fuzzy seeds, you will always have a long stem (which means nothing for the conclusion because it does not say we have fuzzy seeds so long stem could come from something else) and if you have curled leaves, then you must also have white flowers and since we have curled leaves, we know for sure we have white flowers. (Is this reasoning all correct so far?)

So since we cannot conclude anything else based off the information provided, we then use the contrapositive of what we DO know: if you have fuzzy seeds -> /white flowers.

So basically, we can only conclude something if we are GIVEN information that satisfies the sufficient condition which then GUARANTEES the necessary condition? Because for the first example I included, just because AAF happened, it doesn't mean anything unless the sufficient condition (SAS) happened? The sufficient condition HAS to happen for the necessary condition to happen, correct? And that's when we can validly conclude something?

Thank you SO SO much again, I've written down your examples and they have helped so much you're the best!!!

ToniT220

@Arcana
Omg, the example with the cats (in different classes) just FINALLY clicked for me!!!! I actually want to cry, I feel like my brain just unlocked a new power or something bc you literally broke down every single ELEMENT that I have been confused on for WEEKS. I want to give you a hug rn.

Denying the necessary Lets Us Conclude Things
Affirming the sufficient Lets Us Conclude Things I've literally been searching for an explanation like this wow.

So in the biology example, (copying it here so I can see it while typing lol):
**4: thorny seedpods -> curled leaves
This is where you went wrong - you did an "All Mammals are Cats" here. We have affirmed the necessary - we know that our specimen has curled leaves. However, this relationship is only saying that curled leaves are necessary for something to have thorny seedpods. Thorny seed pods, on the other hand, are merely SUFFICIENT for having curled leaves.

There may be other ways to get curled leaves, just like the statement
cats -> mammals
leaves room for other ways to be a mammal other than being a cat (which we happen to also know is true from our general knowledge.) In short - affirming the necessary doesn't let us conclude anything.**

So just because the necessary condition of curled leaves is affirmed, it means nothing for a conclusion because it did not give us "plant x has thorny seedpods" ? For example, if it had said that plant x had thorny seedpods (but did not say anything about curled leaves) then we could conclude it had curled leaves right? Because we are affirming the sufficient? (This is just me thinking out loud so i can make sure my reasoning is correct now)

I cannot thank you enough for this, you made me finally understand this and I am so so so so grateful for your extensive explanation and taking the time out of your day to do that I'm still mindblown rn at how that finally clicked for me... god i feel dumb lol

MattyCzar

@ToniT220 Yep! It seems like you've got it now! Lawgic is a one way street, you can't go backwards (except for 'some' statements, but if you haven't reached that part yet, don't worry about it)

Arcana

@ToniT220 said:

@Arcana
Omg, the example with the cats (in different classes) just FINALLY clicked for me!!!! I actually want to cry, I feel like my brain just unlocked a new power or something bc you literally broke down every single ELEMENT that I have been confused on for WEEKS. I want to give you a hug rn.

Denying the necessary Lets Us Conclude Things
Affirming the sufficient Lets Us Conclude Things I've literally been searching for an explanation like this wow.

So in the biology example, (copying it here so I can see it while typing lol):
**4: thorny seedpods -> curled leaves
This is where you went wrong - you did an "All Mammals are Cats" here. We have affirmed the necessary - we know that our specimen has curled leaves. However, this relationship is only saying that curled leaves are necessary for something to have thorny seedpods. Thorny seed pods, on the other hand, are merely SUFFICIENT for having curled leaves.

There may be other ways to get curled leaves, just like the statement
cats -> mammals
leaves room for other ways to be a mammal other than being a cat (which we happen to also know is true from our general knowledge.) In short - affirming the necessary doesn't let us conclude anything.**

So just because the necessary condition of curled leaves is affirmed, it means nothing for a conclusion because it did not give us "plant x has thorny seedpods" ? For example, if it had said that plant x had thorny seedpods (but did not say anything about curled leaves) then we could conclude it had curled leaves right? Because we are affirming the sufficient? (This is just me thinking out loud so i can make sure my reasoning is correct now)

I cannot thank you enough for this, you made me finally understand this and I am so so so so grateful for your extensive explanation and taking the time out of your day to do that I'm still mindblown rn at how that finally clicked for me... god i feel dumb lol

You've got it my friend!
You nailed it in the example you gave (i bolded for ease of reference)

for the relationship

thorny seedpods --> curled leaves
affirming the sufficient has the power for us to logically conclude that our specimen has curled leaves.

Affirming the sufficient and denying the necessary let us conclude things.

If you ever find this stuff feeling foggy again (it happens to me more often than I'd care to admit), you can go back to cats and mammals during your studies so that you can feel like you are on solid ground again.

It is my pleasure! I spent a long time trying to figure out what this stuff means in the past few months, and I had to turn it over in my head for a while to be able to define it. Also... typing it all out helped me, as it forced me to put some of this stuff into words and know it a bit better.

Good luck studying - you can do it!

ToniT220

@MattyCzar Just got to some today... not loving this section AT ALL lol