formal probability theory when we reason about probabilities formally, symbolically, or...

Formal Probability Theory

When we reason about probabilities formally, symbolically, or mathematically (same thing), we introduce some convenient shorthand.

If I write “P(…)”, I mean “the probability that…”.

P(“it’s raining”) = the probability that it’s raining.

Conditional Probability

There’s one more special symbol in formal probability theory.

“P(…/__)” means the probability that … given that we are assuming __.

Example

So “P(HIV/ test = positive)” represents the probability that someone has HIV given that they’ve tested positive for HIV.

And “P(test = positive/ HIV)” is the probability that someone will test positive, given that they actually have HIV.

Comprehension Question

Which of the following two probabilities is 100%?

a. P(dog/ animal)b. P(animal/ dog)

Definition of Conditional Probability

Conditional probabilities are defined in terms of unconditional probabilities:

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

P(philosopher/ beard) = percentage of philosophers with beards ÷ percentage of people with beards.

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

Last class we started with information about P(test = positive/ terrorist) and P(test = positive/ not terrorist)

And we wanted to find out information like:P(terrorist/ test = positive)P(terrorist/ test = negative)

Bayes’ Theorem

Bayes’ Theorem is an important (but simple) statement about the relation between P(A/ B) and P(B/ A).

P(A/ B) = [P(A) x P(B/ A)] ÷ P(B)

Bayes’ Theorem

What we did last time in class (and on the homework) was to compute P(A/ B) from P(B/ A), P(A), and P(B) using Bayes’ Theorem.

Homework 3

Base rate neglect continued.

You’ve gone to war to fight al Qaeda in Afghanistan. Your military has offered to pay a bounty of $1,000 USD to anyone who turns in an Al Qaeda member.

P(turned in/ al Qaeda)

Many of the locals are sympathetic with the fight against al Qaeda, though many are also sympathetic with your enemies.

You estimate: If someone is al Qaeda, then there’s a 5% chance someone will turn them in.

P(al Qaeda/ turned in)

Importantly, this is not the same thing as saying:

“If someone gets turned in, there is a 5% chance that they are al Qaeda.”

P(turned in/ not al Qaeda)

It’s not too likely that you’ll get non al Qaeda members turned in. However, $1,000 USD is a lot of money, and sometimes people will turn in non al Qaeda member, just to receive the bounty. $1,000 USD will feed, clothe, and house an Afghani for months. Also, sometimes people will turn in someone who is non al Qaeda, just because they don’t like that person, and want you to take them away.

You estimate: If someone is not al Qaeda, there’s a 1 in one million chance that they get turned in for the bounty. (Probability = 0.0001%).This is not the same thing as saying: “If someone gets turned in, then there is a 1 in one million chance that they are al Qaeda.”P(turned in/ not alQ) ≠ P(not alQ/ turned in)

Demographics

You also know that there are only 100 al Qaeda in all of Afghanistan ), and that the population of Afghanistan is around 35 million.

P(al Qaeda) = 100 ÷ 35,000,000P(not al Qaeda) = (35,000,000 – 100) ÷ 35,000,000

Question #1

How many people who are members of al Qaeda will get turned in if your estimates are correct?

There are 100 members of al Qaeda. There is a 5% chance that if someone is al Qaeda, they will get turned in. So 100 x 0.05 = 5.

Question #2

How many people who are NOT members of al Qaeda will get turned in if your estimates are correct?

There are 35 million people in Afghanistan. 100 of them are al Qaeda, so there are 35m – 100 people who are not al Qaeda in Afghanistan.

Question #2

How many people who are NOT members of al Qaeda will get turned in if your estimates are correct?

There is a 1 in one million chance that non-al Qaeda will get turned in. So 0.000001 x (35,000,000 – 100) = 35 – 0.0001 people will get turned in. Let’s round up to 35.

Question #3

How many people total will get turned in if your estimates are correct?

This one is easy.

5 true positives + 35 false positives = 40.

Question #4

What is the percentage of people who are members of al Qaeda out of the total number of people who get turned in? What is the chance that someone who is turned in is al Qaeda?

This one is also easy!

P(al Qaeda/ turned in) = 5 ÷ 40 = 12.5%.

Question #4

And obviously, the chance that someone turned in is not al Qaeda is:

P(not al Qaeda/ turned in) =100% – 12.5% = 87.5%.

Question #4

So even though the chance that someone who is not al Qaeda will get turned in is 1 in one million, the chance that someone who got turned in is not al Qaeda is 87.5%.

0.0001% = P(turned in/ not al Qaeda) ≠ P(not al Qaeda/ turned in) = 87.5%

Question #4

To think that someone who got turned in was obviously al Qaeda is to commit the base rate neglect fallacy.

The base rate of al Qaeda in Afghanistan, P(al Qaeda) = 100/ 35,000,000 = 1/350,000 = 0.0003%. That’s a very small number!

Quick Calculation Using Bayes’ Theorem

P(al Qaeda/ turned in) = [P(turned in/ al Qaeda) x P(al Qaeda)] / P(turned in)= [5% x (100/35,000,000)]/ (40/35,000,000)= (5% x 100)/ 40= 5/ 40= 12.5%

Question #5

Should we imprison the people who get turned in? Should we let them go? Should we investigate them more? What should we do?

Discuss.

Logic and Fallacies

One of our main critical thinking questions was:

Does the evidence support the conclusion?

How do we evaluate whether specific evidence supports a specific conclusion? How do we answer this question?

Arguments

The word ‘argument’ as it is used normally in English, means something like this:

“An exchange of diverging or opposite views, typically a heated or angry one: ‘I've had an argument with my father’.”

Arguments

In philosophy, we use the word ‘argument’ differently. A philosophical argument:

• Is not an exchange of views • Doesn’t need to present opposing or contrary

views• Is not typically heated or angry.

Arguments

Instead, a philosophical argument consists of two parts: the premises and the conclusion.

The premises are the ‘evidence’ that are given in support of the conclusion.

The conclusion is the ‘claim’ that the premises are supposed to support.

Example

Premise 1: Either the butler is the murderer, or the gardener is the murderer.Premise 2: The butler is not the murderer.Therefore,Conclusion: The gardener is the murderer.

Relevance

There is no requirement that the premises of an argument have anything to do with the consequent. For example, this is an argument:

Premise: There are exactly 117 hairs on my hand.Conclusion: It’s half past three o’clock.

Deductive Validity

We say that an argument is deductively valid when it has the following property:

If the premises of the argument are true, then the conclusion of the argument must be true.

A valid argument is “truth-preserving”: the truth of the premises gets passed on to the conclusion.

Note

In ordinary English, the meaning of ‘valid’ is slightly different.

Deductive validity is a relation between premises and conclusion. ‘Validity’ ordinarily means something like “true or relevant.” A ‘valid criticism’ is a criticism that is true or relevant to some issue being discussed.

Inductive Validity

We say that an argument is inductively valid when it has the following property:

If the premises are true, then the conclusion is likely to be true.

An inductive argument probably preserves truth.

Example

Suppose I eat at McDonald’s. I eat there 100 times. The first time I eat there, I get sick. The second time I eat there I get sick. In fact, on every occasion, each of the 100 times I ate there, I got sick every time.

Example

Premise 1: The first time I ate at McDonald’s I got sick.Premise 2: The second time I ate at McDonald’s I got sick.Premises 3-99:…Premise 100: The 100th time I ate at McDonald’s I got sick.Conclusion: Next time I will get sick again!

Example

This is an inductively valid argument. If the premises are true, the conclusion is likely to be true too.

But it is not a deductively valid argument. (Why?)

Soundness

A sound argument is one that (i) is valid and (ii) has true premises. [And we can distinguish between deductively and inductively sound arguments.]

Every sound argument is valid (by definition), but the reverse is not true. Some valid arguments are not sound.

Example

Consider the following argument:

Premise 1: All dogs have eight legs.Premise 2: I am a dog.Therefore,Conclusion: I have eight legs.

Example

This argument is valid. If the premises are true, then the conclusion must also be true. If all dogs truly have eight legs, and I am truly a dog, then it is true that I have eight legs!

However, the argument is not sound. The premises are false (and so is the conclusion).

Comprehension Questions

• Can arguments that are deductively valid have false conclusions? False premises?

• Can arguments that are not deductively valid have true conclusions? True premises?

• Can deductively sound arguments have false conclusions?

• Can arguments that are not deductively sound still be deductively valid?

Deductive Logic

Deductive logic (often just called ‘logic’) is the study of deductively valid argument forms.

Argument Forms Example

Premise 1: If horses had wings, they could fly.Premise 2: Horses cannot fly.Therefore,Conclusion: Horses don’t have wings.


Premise 1: If the butler committed the murder, then the murder weapon is the candlestick.Premise 2: The murder weapon is not the candlestick.Therefore,Conclusion: The butler did not commit the murder.


Premise 1: If Sally is free this evening, George will take her to dinner.Premise 2: George did not take Sally to dinner.Therefore,Conclusion: Sally was not free this evening.


All of these arguments share a deductively valid argument form:

Premise 1: if A, then B.Premise 2: not B.Conclusion: not A.

Any argument with this form is valid, no matter what ‘A’ and ‘B’ are.


Premise: Everyone is happy.Conclusion: There is not someone who is not happy.

Premise: Everyone is F.Conclusion: There is not someone who is not F.


Premise: You can’t be happy and succesful.Conclusion: If you’re happy, you’re not successful.

Premise: not (A and B).Conclusion: if A then not B.


Premise: Either Fred took the train or he took the ferry.Premise 2: Fred did not take the ferry.Conclusion: Fred took the train.

Premise: Either A or B.Premise: not BConclusion: A

Deductive Logic

The goal of deductive logic is to identify deductively valid argument forms.

We can use these as a formal test for validity: if an argument has a certain form, then that argument is deductively valid.

Invalidity

An argument that is not valid is called invalid.

Valid: If the premises are true, then the conclusion must be true.

Invalid: The premises can be true while the conclusion is false.

Invalidity

Unfortunately, there is no formal test for (deductive) invalidity.

There is no way of looking at the form of an argument and telling that the premises do not guarantee the conclusion.

Fallacies

A fallacy is an invalid argument, usually one that might mislead someone into thinking it’s valid.

We’ve already encountered a number of fallacies in this course: the fallacy of quoting out of context, the regression fallacy, the conjunction fallacy, the base rate neglect fallacy, etc.

No Formal Fallacies

We must remember, however, that there are no formal fallacies (Wikipedia, for instance, is wrong about this very fact).

Let me give an example.

Affirming the Consequent

According to the Wikipedia article “List of Fallacies” the argument “affirming the consequent” is a “formal fallacy” meaning “an error in logic that can be seen in the argument's form without requiring an understanding of the argument's content.”

I claim that there is no such thing.


Here is the offending form:

Premise 1: If A, then B.Premise 2: B.Therefore,Conclusion: A.


There are certainly invalid instances of this form:

Premise 1: If Rex is a dog, then Rex is an animal.Premise 2: Rex is an animal.Conclusion: Rex is a dog.

INVALID! Even if the premises are all true, the conclusion might be false– Rex could be a cat.

But there are also VALID instances of the form.

Like when A = B…

Premise 1: If Rex is a dog, then Rex is a dog.Premise 2: Rex is a dog.Therefore,Conclusion: Rex is a dog.VALID!

Or when B entails A…

Premise 1: If Rex is a dog, then Rex is a big dog.Premise 2: Rex is a big dog.Therefore,Conclusion: Rex is a dog.VALID!

Or when A is a logical truth…

Premise 1: If 2 + 2 = 4, then Rex is a dog.Premise 2: Rex is a dog.Therefore,Conclusion: 2 + 2 = 4.VALID!

Or when B is a contradiction

Premise 1: If Rex is a dog, then there is a greatest prime number.Premise 2: There is a greatest prime number.Therefore,Conclusion: Rex is a dog.VALID!

And AtC is often inductively valid

Premise 1: If the cat were angry at us, then he’d scratch up the furniture.Premise 2: The cat scratched up the furniture.ThereforeConclusion: The cat is probably angry at us.

(This is also called “inference to the best explanation.”)

No Formal Fallacies

I could go on, but I won’t.

The point here is not about “affirming the consequent”. There are no formal fallacies. There is no formal test for an invalid argument. There are no “logical fallacies.” You cannot (legitimately) criticize an argument on the basis of its form.

(small exception)

There’s one small exception to this rule.

If an argument has NO premises AND the conclusion is a formal contradiction then AND ONLY THEN can you conclude that it’s invalid on the basis of its form.

Fallacies

Just because there are no informal fallacies does not mean that there are no fallacies.

There are lots of fallacies, because fallacies are just invalid arguments, arguments where the evidence does not support the conclusion.

To find such fallacies we need our smarts, not our logic.

Next time!

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