Download Logic and Fallacies

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 ÷ 35,000,000
P(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.
Argument Forms Example
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.
Argument Forms Example
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.
Argument Forms Example
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.
Argument Forms Example
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.
Argument Forms Example
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.
Argument Forms Example
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 B
Conclusion: 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.
Affirming the Consequent
Here is the offending form:
Premise 1: If A, then B.
Premise 2: B.
Therefore,
Conclusion: A.
Affirming the Consequent
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.
Therefore
Conclusion: 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!