Review Part 2

LANGUAGE AND REASONING

L&R on the Final

There are about 5-6 (out of 30) multiple choice questions involving “language and reasoning” on the final. These questions test the concepts of ambiguity, vagueness, analogy, different kinds of definitions, and necessary and sufficient conditions.

What to Study

Joe Lau’s module on Meaning Analysis (recommended reading for 25/4) contains everything you need to know about “language and reasoning” on the final.

The slides on 25/4 “Definitions and Arguments” are also a good source of information.

Ambiguity

An ambiguous word is a word with more than one meaning.

‘Bank’ in English can mean a financial institution or the side of a river.

Sometimes sentences can be ambiguous even if none of their words are. “Flying planes can be dangerous.”

False Equivocation

Equivocation (or “false equivocation”) is when one word is used with two meanings in the same argument, rendering it invalid.

Note that not every argument containing two words with different meanings is fallacious or invalid. It’s only when the argumenter acts as though they have the same meaning.

Vagueness

A vague word is not ambiguous– it has only one meaning. But it can be unclear whether it applies in any particular circumstance:

• Tall• Rich• Bald• Beautiful

Definitions

You should know, and be able to recognize:• Reportative definitions: gives the meaning of the

word as actually used by speakers• Stipulative definition: provides an entirely new

meaning• Precising definition: uses the normal meaning,

but with added (stipulated) conditions• Persuasive definition: Sneaks in emotional

content where it does not belong.

Wide/ Narrow

A definition is too wide when its extension includes things that are not in the extension of the word being defined.

It’s too narrow when the extension of the word being defined includes things that are not in the extension of the definition.

Necessary and Sufficient Conditions

In class we talked about necessary and sufficient conditions in the following way:

A is sufficient for B = if A then BA is necessary for B = if B then A

Use on the Final

On the final these terms are used slightly differently. Examples:

• “Being a dog is sufficient for being an animal.”• “Being a woman is necessary for being a

mother.”

How It Relates

It doesn’t make sense to say:

“If being a dog, then being an animal.”

But we can say:

“If something is a dog, then it is an animal.”

Analogies

An analogy is when you take two different things and compare them in some respect.

“Life is like a box of chocolates: you never know what you’re going to get.”

Analogical Arguments

An analogy is often used as the premise in an argument. “X is like Y; Y is Z; therefore X is Z.”

Analogical Arguments

The universe is a complex system of interacting parts that serve a purpose.A watch is a complex system of interacting parts that serve a purpose.If we found a watch by itself in the wilderness we’d assume that some intelligent designer had made it.Therefore we should assume that an intelligent designer made the universe.

FALLACIES

Fallacies on the Final

There are about 10/30 multiple choice questions about fallacies on the final, and 1/10 Part II questions on fallacies, for between 25% and 30% of the total marks available.

You should obviously study fallacies.

Names of Fallacies

The final will not require that you memorize the names of the fallacies (straw man, ecological, regression, etc.).

What you need to know is why certain arguments are bad arguments. You need to be able to identify fallacies, just not identify them by name.

What to Study

I obviously can’t go over all the fallacies we talked about throughout class here. So I will talk about some of the more difficult ones. This does not mean that others won’t be on the exam. Relevant lectures include:• Cognitive Biases 1 (28/1)• Cognitive Biases 4 (7/2) + optional readings• Fallacies (25/3)• Probability (23/4)

Appeals to Irrelevant Things

• Appeal to pity: “You have graded my exam wrong because I won’t graduate if I fail.”

• Appeal to popularity (ad populum): “Surely God exists; billions of people believe in him.”

• Appeal to motive: “The drugs produced by big drug companies can’t work like they say; those companies make lots of profits selling them.”

• Appeal to origin (genetic fallacy): Pope Benedict was in the Hitler Youth. He therefore supports the Nazis.

Straw Man Fallacy

The Straw Man Fallacy (sometimes in the UK called “Aunt Sally Fallacy”) is when you misrepresent your opponent, and argue against the misrepresentation, rather than against your opponents claim.

Assuming the Original Conclusion

Assuming the original conclusion* involves trying to show that a claim is true by assuming that it is true in the premises. It has the form:

X is true. Why? Because X.

*This is Aristotle’s name for the fallacy.

Circular Reasoning

In the final, assuming the original conclusion is described as “circular reasoning.” The metaphor is that this argument leads you in a circle:

Why should I believe X? Because X. Why should I believe X? Because X…

Technically Valid

One thing to point out is that unlike most fallacies, Assuming the Original Conclusion/ Circular Reasoning/ Begging the Question is technically valid.

This is because obviously X ├ X

Regression to the Mean

Whenever two variables are imperfectly correlated, extreme values of one variable on average are paired with less extreme values of the other.

Tall parents have tall children, but the children tend to be less tall than the parents. Students who do very well on Exam 1 tend to do well on Exam 2, but not as well as they did on Exam 1.

Regression to the Mean

This is true of any two imperfectly correlated variables.

Companies that do very well one year on average do well the next year, but not quite as well as the previous. Students who do well in high school on average do well in college, but not as well as in high school.

Regression to the Mean

“Regression” just means going back, and “mean” means average. “Regression to the mean” is just a fancy way of saying going back to average.

On Average

It’s important to note that sometimes, regression to the mean doesn’t happen (or doesn’t happen immediately).Tall parents can have children who are even taller than they are. But the average height of children born to tall parents is lower than the average height of the parents. On average, things regress to the mean.

Regression Fallacy

The regression fallacy involves attributing a causal explanation to what is nothing more than regression to the mean.

If you feel very bad when you wake up hung over, you will likely feel better in an hour. If you eat a greasy meal when you wake up, and feel better in an hour, you might commit the regression fallacy and assume your meal made you feel better.

Base Rate

The base rate of X is the proportion of X in the population. The base rate of terrorists is the # of terrorists out of the total # of people; the base rate of women is the # of women out of the total # of people, etc.

It is the probability that a randomly selected person will be X (a terrorist, a woman…)

Base Rate Neglect

When the base rate of some X is very low, even very accurate tests for X can be highly unreliable.

The base rate neglect fallacy is when you assume that a test is reliable without looking at the base rate– or even when knowing that the base rate is very low.

Problem #1

1. Suppose I notice that two students have turned in midterms with exactly the same answers on the multiple choice section. I know this: • If the two students cheated, the probability that

their answers would be the same is 100%.• If the two students did not cheat, then the

probability that their answers would be the same is (1/4) x 10 = 1/40 = 2.5%.

What is the probability that the two students cheated? a. 100%.b. 97.5%c. 2.5%d. It is impossible to tell, given the information in the question.

What is the probability that the two students cheated? a. 100%.b. 97.5%c. 2.5%d. It is impossible to tell, given the information in the question.

Base Rate Neglect

The reason it is impossible to tell the answer to this question is that we are not told the base rate of cheaters (how likely is it that a student cheats?). If the base rate is low, then even very good cheater-detection tests will be highly unreliable.

Example

Suppose 2 in 1000 Lingnan students cheat. Then For every 1000 students I will receive identical multiple choice answers from 2 real cheaters and 2.5% of the 998 non-cheaters (around 25).

So the probability that a student cheated, given that his answer set matched another student is

2/ (25 + 2) = 7.5%

Ecological Fallacy

The ecological fallacy is when you assume that a correlation between groups of people corresponds to a correlation between individuals.

Example: Countries with a higher than average consumption of fish also have higher average wealth. Therefore, individuals who eat more fish are more wealthy.

Conjunction Fallacy

(A & B) is always less probable or equally probable than A, and always less probable or equally probable than B.

The conjunction fallacy is when someone is misled into thinking that (A & B) is more probable than A (or more probable than B).

Example Question

Tom is a wealthy investment banker. He drives a Ferrari, wears expensive suits, drinks expensive French wine daily, and thinks poor people are “dirty” and “disgusting.”

Example Question

Which is more probable?a. Tom eats at Café de Coral.b. Tom eats at Café de Coral because his

girlfriend works there.c. Tom eats at Café de Coral because he owns

stock in the companyd. Tom eats at Café de Coral because their pork

bone soup is delicious.

THE SCIENTIFIC METHOD

Scientific Method on the Final

There are 4-5 questions on scientific method on the final. These include questions about things like biased samples, causation vs. correlation, observational studies vs. controlled experiments, etc.

What to Study

Lectures on 28/3 “Experimental Design,” 8/4 “RCTs” and 23/4 “Probability.”

Scientific Method

Science proceeds by the hypothetico-deductive method, which consists of four steps:1. Formulate a hypothesis2. Generate testable predictions3. Gather data4. Check predictions against observations

Causation

Much of science is concerned with discovering the causal structure of the world.

We want to understand what causes what so we can predict, explain, and control the events around us.

Independence

In statistics, we say that two variables are independent when the value of one variable is completely unrelated to the other:

P(A/ B) = P(A) and P(B/ A) = P(B)

B happening does not make A any more likely to happen. (If that’s true, so is the reverse.)

Correlation

Two variables A, B that are not independent are said to be correlated.

A and B are positively correlated when P(A/ B) > P(A). If B happens, A is more likely to happen.

A and B are negatively correlated when P(A/ B) < P(A). If B happens, A is less likely to happen.

Causation and Correlation

One thing that can lead two variables A and B to be correlated is when A causes B.

For example, if having a cold causes a runny nose, then having a cold is correlated with having a runny nose:

P(cold/ runny nose) > P(cold)

Causation ≠ Correlation

But correlation does not imply causation. If A and B are correlated there are several possibilities:

• A causes B• B causes A• C causes A and C causes B• A and B are only accidentally correlated

Observational Studies

An observational study looks at data in order to determine whether two variables are correlated.

Controlled Experiments

In a controlled experiment there are two groups who get separate treatments.

One group, the “control group” gets the standard treatment. For example, all of the king’s servants ate meat and wine before Daniel suggested a different diet might be better.

Controlled Experiments

The other group, the “experimental group”, gets the treatment we plan to test.

If the test group has better results than the control group, we have good evidence that our new treatment should be adopted.

Randomization

Ideally, an experiment controls for as many variables as possible. To a large extent, this is done by randomly assigning individuals in the study to either the control group or the experimental group. This way, the members of the group are less likely to share features other than the variable we’re studying.

“Blinds”

In experimental studies, we say that the participants are blind if they do not know which group they are in: the control group or the experimental group.

Again, it’s not always possible to have blind participants, but this is considered best practice.

Blind Experimenters

Ideally, in experiments the researchers are blind to which group subjects are in.

This prevents the experimenter from accidentally indicating to the subjects which group they are in.

It also prevents experimenter bias.

Sample

In statistics, the people who we are studying are called the sample. (Or if I’m studying the outcomes of coin flips, my sample is the coin flips that I’ve looked at. Or if I’m studying penguins, it’s the penguins I’ve studied.)

Representative Samples

A perfectly representative sample is one where if n% of the population is X, then n% of the sample is X, for every X.

For example, if 10% of the population smokes, 10% of the sample smokes.

Selection Bias

The opposite of a representative sample is a biased sample.

One main cause of bias in a sample is selection bias. For example, in polling to see if candidate X will win the election, you might set up your poll in a rich part of town. Your sample will be biased by having more rich people in it than the general population.

Sample Size

An important lesson that we learned regarding sampling is that it is not necessary to have millions of people in your sample, even if there are tens of millions of people in the population you are generalizing about.

Be careful! Don’t assume that just because a sample contains only a couple hundred people that it is too small to be trustworthy.