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Since this is a board where careful logical reasoning is important, I thought it might be good to have a thread for practicing this skill. Please feel free to contribute any exercises you might have. I'll start with one, below:

Consider the following proposition P: "An outlet for population overflow is required for a country's economy to prosper."

Now the consider the following argument A against proposition P: "We need only look to our own shores to find counter-evidence: Cuba has long been able to discharge its surplus population by sending people here, and yet its economy has done quite poorly."

Why is A an invalid argument against P?

Consider the following proposition P: "An outlet for population overflow is required for a country's economy to prosper."

Now the consider the following argument A against proposition P: "We need only look to our own shores to find counter-evidence: Cuba has long been able to discharge its surplus population by sending people here, and yet its economy has done quite poorly."

Why is A an invalid argument against P?

>>3236

>is required

Required to know all causative requirements.

Exercise:

This tripped up some MIT students so, let's see how you do:

In China, parents want boys. The government wants to limit births, so they compromise: parents may have children until they get a boy. For simplicity, we'll say every family has one boy and however many girls preceded that boy. What is the sex ratio of Chinese children?

>is required

Required to know all causative requirements.

Exercise:

This tripped up some MIT students so, let's see how you do:

In China, parents want boys. The government wants to limit births, so they compromise: parents may have children until they get a boy. For simplicity, we'll say every family has one boy and however many girls preceded that boy. What is the sex ratio of Chinese children?

>>3236

>an outlet is required

a requirement for a solution is not the same as a solution. A is invalid because it assumes that meeting a requirement for a solution must be the same as having a solution, which is incorrect. For example: If having a gasoline budget is a requirement for owning a car. Having a gasoline budget does not mean you have a car, but rather the ability to.

>>3249

>parents want boys

at all or as opposed to girls?

>The government wants to limit births, so they compromise

limiting births does not address the original concern, that parents want boys, and it cannot, until the parameters of the original concern are better defined.

>parents may have children until they get a boy

that's not a compromise, and it will do absolutely the opposite of what they intend.

>What is the sex ratio of Chinese children?

This requires one to know not only the percentage of families that had 1 girl before they had a boy and 0 girls before they had a boy, but also the number of families that had 2 girls before they had a boy, and 3 girls before they had a boy, and 4, and so on. The question assumes that the student already knows this, or knows a method for extrapolating this incomplete data.

For simplicity's sake, I will assume perfect statistics of 50/50 on boys and girls being born, 50% of the parents would have had 1 girl and then 1 boy, and 50% of the parents would have had 1 boy, so we are left with 1 girl to every 2 boys, for a sex ratio of 1:2

>an outlet is required

a requirement for a solution is not the same as a solution. A is invalid because it assumes that meeting a requirement for a solution must be the same as having a solution, which is incorrect. For example: If having a gasoline budget is a requirement for owning a car. Having a gasoline budget does not mean you have a car, but rather the ability to.

>>3249

>parents want boys

at all or as opposed to girls?

>The government wants to limit births, so they compromise

limiting births does not address the original concern, that parents want boys, and it cannot, until the parameters of the original concern are better defined.

>parents may have children until they get a boy

that's not a compromise, and it will do absolutely the opposite of what they intend.

>What is the sex ratio of Chinese children?

This requires one to know not only the percentage of families that had 1 girl before they had a boy and 0 girls before they had a boy, but also the number of families that had 2 girls before they had a boy, and 3 girls before they had a boy, and 4, and so on. The question assumes that the student already knows this, or knows a method for extrapolating this incomplete data.

For simplicity's sake, I will assume perfect statistics of 50/50 on boys and girls being born, 50% of the parents would have had 1 girl and then 1 boy, and 50% of the parents would have had 1 boy, so we are left with 1 girl to every 2 boys, for a sex ratio of 1:2

>>3249

>we'll say every family has one boy and however many girls preceded that boy. What is the sex ratio of Chinese children?

Half of families have a boy for their first child (and therefore no girl).

1/4 of families have (girl, boy).

1/8 of families have (girl, girl, boy).

1/16 of families have (girl, girl, girl, boy).

Etc.

So the average number of girls per family is:

1*1/4 + 2*1/8 + 3*1/16 + ... = approximately 1.000

So the gender ratio is 1:1 (approximately).

>we'll say every family has one boy and however many girls preceded that boy. What is the sex ratio of Chinese children?

Half of families have a boy for their first child (and therefore no girl).

1/4 of families have (girl, boy).

1/8 of families have (girl, girl, boy).

1/16 of families have (girl, girl, girl, boy).

Etc.

So the average number of girls per family is:

1*1/4 + 2*1/8 + 3*1/16 + ... = approximately 1.000

So the gender ratio is 1:1 (approximately).

>>3254

>a requirement for a solution is not the same as a solution. A is invalid because it assumes that meeting a requirement for a solution must be the same as having a solution, which is incorrect. For example: If having a gasoline budget is a requirement for owning a car. Having a gasoline budget does not mean you have a car, but rather the ability to.

Yes, that is right! The argument A confuses necessary conditions with sufficient conditions.

>at all or as opposed to girls?

Presumably at all?

>limiting births does not address the original concern

It is not intended to. Those are two separate, competing objectives. Parents want boys. The government wants population control. Those two concerns are in tension with one another.

>that's not a compromise,

It is a compromise between the parents wanting to have many male children and the government wanting to limit the number of children the parents can produce.

>a requirement for a solution is not the same as a solution. A is invalid because it assumes that meeting a requirement for a solution must be the same as having a solution, which is incorrect. For example: If having a gasoline budget is a requirement for owning a car. Having a gasoline budget does not mean you have a car, but rather the ability to.

Yes, that is right! The argument A confuses necessary conditions with sufficient conditions.

>at all or as opposed to girls?

Presumably at all?

>limiting births does not address the original concern

It is not intended to. Those are two separate, competing objectives. Parents want boys. The government wants population control. Those two concerns are in tension with one another.

>that's not a compromise,

It is a compromise between the parents wanting to have many male children and the government wanting to limit the number of children the parents can produce.

>>3256

I suppose I missed that step, but I think it's still incomplete. Assuming that reduction rate (which seems reasonable)half of the families would have a boy first, but since every family accounted for eventually has a boy, we assume two halves each with 1 boy average. so ?:2

if 1*1/4+2*1/8+3*1/16 . . . _ 1.000 then that accounts for a 1:1 ratio in the half that has girls, so after you add back in the half that has no girls, you are still left with a ratio of girls to boys at 1:2

I suppose I missed that step, but I think it's still incomplete. Assuming that reduction rate (which seems reasonable)half of the families would have a boy first, but since every family accounted for eventually has a boy, we assume two halves each with 1 boy average. so ?:2

if 1*1/4+2*1/8+3*1/16 . . . _ 1.000 then that accounts for a 1:1 ratio in the half that has girls, so after you add back in the half that has no girls, you are still left with a ratio of girls to boys at 1:2

>>3259

I don't follow.

Every family has exactly one boy.

What is the average number of girls in a family? It is 1/4 + 2/8 + 3/16 + ... = 1.000

So the ratio of boys to girls is 1 : 1.000.

(Also, I have a hunch that the infinite sum is exactly 1, but I don't know how to go about proving it.)

I don't follow.

Every family has exactly one boy.

What is the average number of girls in a family? It is 1/4 + 2/8 + 3/16 + ... = 1.000

So the ratio of boys to girls is 1 : 1.000.

(Also, I have a hunch that the infinite sum is exactly 1, but I don't know how to go about proving it.)

>>3258

>Those are two separate, competing objectives. Parents want boys. The government wants population control. Those two concerns are in tension with one another.

fair point, I confused the objective with the proposal.

>It is a compromise between the parents wanting to have many male children and the government wanting to limit the number of children the parents can produce.

So they wanted *many* male children? That's different from wanting male children. If a family only wants *a* male child, they will make the attempt until they get one male child, which is different from continuing after getting a male child. The government's proposal is not a compromise in the first situation, since it matches exactly the situation that would already be happening, and therefore does not limit births at all. In the second situation, it does limit births, by capping the average at 2 children per family average, as opposed to the 3-4 which might have occurred otherwise. A good compromise, and an effective solution, if only the question had named its terms properly.

>Those are two separate, competing objectives. Parents want boys. The government wants population control. Those two concerns are in tension with one another.

fair point, I confused the objective with the proposal.

>It is a compromise between the parents wanting to have many male children and the government wanting to limit the number of children the parents can produce.

So they wanted *many* male children? That's different from wanting male children. If a family only wants *a* male child, they will make the attempt until they get one male child, which is different from continuing after getting a male child. The government's proposal is not a compromise in the first situation, since it matches exactly the situation that would already be happening, and therefore does not limit births at all. In the second situation, it does limit births, by capping the average at 2 children per family average, as opposed to the 3-4 which might have occurred otherwise. A good compromise, and an effective solution, if only the question had named its terms properly.

>>3263

you're right actually. I've just done the math, and while that limit approaches zero, its sum is 1, as demonstrated by finding the area compounded by x>0, y=>X/(2^x+1), which is 1. That's a calculus answer though.

I misunderstood what you were saying until I tried it myself, but you're right, the average ratio is 1:1. which also means that the average children per household total is 2.

you're right actually. I've just done the math, and while that limit approaches zero, its sum is 1, as demonstrated by finding the area compounded by x>0, y=>X/(2^x+1), which is 1. That's a calculus answer though.

I misunderstood what you were saying until I tried it myself, but you're right, the average ratio is 1:1. which also means that the average children per household total is 2.

>>3256

Seems correct. Although in practice there will only be so many children, so a parameter will only be approximated by a population, the expected value can be figured exactly.

>>3258 >>3254

Right, the government might (and mostly did) simply limit families to one child. In which case if they got a girl, better luck next lifetime, I guess. In letting parents have children until they get a boy, some control is placed on population -- parents might really want several boys -- but parents at least won't miss out on having a boy. To be honest, I'm not well versed in the history of China, but I think both versions happened, at least in some part, at least for some period of time.

>>3264

I thought not giving the sex ratio of a birth might be an issue, although people have assumed 1:1 which was fine. Solving for an arbitrary ratio would be fine as well. Noted I should have said "Parents want families consisting of more than one boy." I guess one can't change the past, so one will change the future.

Logic Exercise 2:

In China, parents want families consisting of many boys. The government wants to limit births, so they compromise: parents may have children until they get a boy. For simplicity, we'll say every family has one boy and however many girls preceded that boy. At birth, the sex ratio may be assumed to be 1:1 -- that is 50% chance of male, 50% chance of female -- but arbitrary ratios may be used as well. What is the sex ratio of Chinese children (and please specify if not using a 1:1 birth ratio)?

>>3272

>finding the area compounded by

I convinced myself I can show convergence of the series, but it would be a bit long to type out. You mention area -- did you compute a integral?

Seems correct. Although in practice there will only be so many children, so a parameter will only be approximated by a population, the expected value can be figured exactly.

>>3258 >>3254

Right, the government might (and mostly did) simply limit families to one child. In which case if they got a girl, better luck next lifetime, I guess. In letting parents have children until they get a boy, some control is placed on population -- parents might really want several boys -- but parents at least won't miss out on having a boy. To be honest, I'm not well versed in the history of China, but I think both versions happened, at least in some part, at least for some period of time.

>>3264

I thought not giving the sex ratio of a birth might be an issue, although people have assumed 1:1 which was fine. Solving for an arbitrary ratio would be fine as well. Noted I should have said "Parents want families consisting of more than one boy." I guess one can't change the past, so one will change the future.

Logic Exercise 2:

In China, parents want families consisting of many boys. The government wants to limit births, so they compromise: parents may have children until they get a boy. For simplicity, we'll say every family has one boy and however many girls preceded that boy. At birth, the sex ratio may be assumed to be 1:1 -- that is 50% chance of male, 50% chance of female -- but arbitrary ratios may be used as well. What is the sex ratio of Chinese children (and please specify if not using a 1:1 birth ratio)?

>>3272

>finding the area compounded by

I convinced myself I can show convergence of the series, but it would be a bit long to type out. You mention area -- did you compute a integral?

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https://ponyville.us/townhall/res/3340.html

https://ponyville.us/townhall/res/3340.html

>>3249

I was thinking someone might come to it, but a simplifying insight is that in broad terms, parents are having babies, each birth an independent, random event. There is no way to skew the expected birth ratio by choosing which family will have a child.

I was thinking someone might come to it, but a simplifying insight is that in broad terms, parents are having babies, each birth an independent, random event. There is no way to skew the expected birth ratio by choosing which family will have a child.

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