Transcript Critical Thinking Chapter 11 Inductive Reasoning Lecture Notes © 2008 McGraw Hill Higher Education.

Critical Thinking
Chapter 11
Inductive Reasoning
Lecture Notes © 2008 McGraw Hill
Higher Education
1
Introduction
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Inductive Argument: an argument in which the
premises are intended to provide support, but not
conclusive evidence, for the conclusion.
Strong Inductive Argument: an inductive
argument in which the premises actually do make
the conclusion more likely to be true (rather than
false).
 Remember, strength comes in degrees.
Cogent Inductive Argument: a strong inductive
argument with true premises.
How can you know if the argument is inductive?
 If the argument is invalid, the charitable thing
to do is treat it as inductive.
 Indicator words: likely, probably, it’s plausible
to suppose that, etc.
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Inductive Generalizations

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Generalization: statement made about all or
most members of a group.
Inductive generalization: inductive argument
that relies on characteristics of a sample
population (i.e., a portion of the population)
to make a claim about the population as a
whole.


i.e., an inductive argument with a generalization
as a conclusion.
Example: All the bass Hank caught in the
Susquehanna have been less than 1lb. So,
most of the bass in the Susquehanna are
less than 1lb.
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Making Inductive Generalizations
stronger by making conclusions weaker.

Notice…

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All the bass Hank caught in the Susquehanna have
been less than 1lb. So, all of the bass in the
Susquehanna are less than 1lb.
..is a pretty weak argument. Even if Hank
fishes often, the Susquehanna is a big river
and his catches are not enough to justify
such a “sweeping conclusion.”
However, if we changed the conclusion to
“most of the bass are…” or, better yet,
“many of the bass are…” the argument
would be much stronger.
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Practice
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Page 288, Exercise 11.1
5
Evaluating Inductive Generalizations
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Three questions to ask:
Are the premises true? Use the skills you
learned in chapter 8 to determine whether
you are justified in accepting the premises.
Is the sample large enough? In general, the
larger the population you are generalizing
about, the larger your “sample population”
will need to be.
Is the sample representative? Only if the
sample shares all the relevant “percentages”
with the population as a whole.

Maybe Hank only fished with lures that were
attractive to smaller fish.
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Are the premises true?
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A deductive argument that is valid
and has all true premises leading to a
true conclusion is called a sound
argument
A deductive argument can have good
– that is, valid – argumentation and
still be unsound if the premises are
not all true.
So, likewise an inductive argument….
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Are the premises true?
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Premises of an inductive
generalization can provide strong
support for its conclusion
But if the premises are not all true, it
is not a cogent inductive argument.
A cogent argument has all true
premises and supplies strong support
for its conclusions.
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Are the premises true?

One or more false premises makes an
inductive argument uncogent, even if
its argumentation, its support for the
conclusion, is strong.

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Most CEOs of Fortune 500 companies are
women.
So, the CEOs of most big businesses are
probably women.
Weak
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Is the sample large enough?

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The size of the sample population
must be sufficient to justify the
conclusion about the population as
a whole.
If it is not, the argument is a hasty
generalization.
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Is the sample large enough?
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None of the thousands of rabbits Alan has
come across has tried to attack him.
So, most rabbits are not inclined to attack
human beings.
Strong and cogent
Brooke taught 3 students with purple hair
last year, and all of them were A students.
So, all students with purple hair must be
A students.
Weak
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Is the sample representative?
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A representative sample is like the
population as a whole in all relevant
ways.
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Two nuclear bombs were dropped on Japan,
and today Japan has one of the strongest
economies in the world.
So, all the concern about nuclear warfare
and the end of humankind is a bunch of
nonsense.
Weak
12
Practice
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Pages 291-92, Exercise 11.2
Lecture Notes © 2008 McGraw Hill
Higher Education
13
Opinion Polls and Inductive
Generalizations
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Terminology:
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Margin of error – indicated by ±; results of
poll will be within a range of + or – the
amount indicated
Random sample – a sample in which each
member of the population has an equal
chance of being selected as a member of
the sample population.
Self-selecting sample – a sample in which
the whole population invited to case a vote
Non-responses – those who don’t respond
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Opinion Polls and Inductive
Generalizations
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Take a population of 100 million and take a
representative poll of 4000 voters. There is a 95%
chance that the percentages of the 4000 will fall
within 2 percentage points of the whole population.

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Smaller polls have bigger margins of error:
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If 60% of the 4000 voted yes, there would be a 95%
chance that 58% to 62% of the 100 million, voted
yes. [i.e., 60% (2)]
1500 (3), 1000 (4), 600 (5), 100 (11)
But this only holds if the sample population is
representative.

To find the whole population’s opinion, you can’t poll
just the rich, or just those with houses, or just radio
listeners, just the happy, just those you are willing to
approach, just those who bother to answer emails,
or just those who don’t intimidate you, etc.
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Terminology Recap
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A strong argument has premises that
provide evidence that its conclusion is more
likely true than false.
A strong and reliable argument has
premises that provide evidence that its
conclusion is more likely true than false, and
it is an argument that a reasonable person
would act or bet on.
A strong but unreliable argument has
premises that provide evidence that its
conclusion is more likely true than false, but
it is an argument that a reasonable person
would not act or bet on.
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Practice
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Page 296, Exercise 11.3
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Statistical Arguments
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Statistical arguments go “the other way.” They take
generalizations and draw conclusions about smaller
samples of the population (usually individuals).
Example:
1.
2.
3.
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You’re a college student
90% of college students want no cumulative final.
So you probably don’t want a cumulative final.
The more “broad” the conclusion the better.
The higher the original percentage, the better.
Usually, if the percentage is around 50%, we will
call the argument unreliable, even if it is more likely
than not that the conclusion is true.

A “rule of thumb”: if it is would reasonable to bet on
it, then it is reliable.
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Reference Class
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The reference class is the group to which
statistics apply.
As a rule, the more specific the reference
class is, the better the argument is.
A statistical argument can be used to support
a conclusion about a group rather than an
individual.

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90% of college students are in favor of not having a
final exam.
So, 90% of Ling 21 students are in favor of not
having a final exam.
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Reference Class
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In a statistical argument, if you find out
more information about the person in
question, you “narrow” the group (class)
the person is in.
Example:
1.
2.
3.
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You are a college student who likes essays.
85% of college students who like essays want
cumulative fails.
Thus you probably want a cumulative final.
This additional information weakened our
justification for believing that you don’t
want a final.
Lecture Notes © 2008 McGraw Hill
Higher Education
20
Practice
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Pages301-302, Exercise 11.4
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Argument by Analogy
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Analogy: comparison of things based on
similarities.
Argument from analogy: an argument
that suggests that the presence of certain
similarities is evidence for further
similarities.
Common Form:
1.
2.
3.
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A and B have characteristic X
A has characteristic Y
So B probably has characteristic Y too.
Example:
1.
2.
3.
Tiffany and Heather are both tall and play
basketball.
Tiffany also plays volleyball.
So, Heather
probably
plays
volleyball too.
Lecture Notes
© 2008 McGraw
Hill
Higher Education
22
Evaluating Arguments from Analogy
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Most arguments from analogy are
inductive arguments, so they are
neither valid nor invalid.
Unlike deductive arguments, there are
no clear-cut ways to tell if inductive
arguments are strong or weak.
But there are good questions to ask to
help determine if an argument from
analogy is strong or weak.
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Evaluating Arguments from Analogy
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Squirrels and rats are rodents of similar
size and appearance.
Rats cause problems in the city, and
squirrels cause problems in the
suburbs.
Rats should be exterminated.
So, squirrels should be exterminated.
Is this a good argument?
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Evaluating Arguments from Analogy
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Tiffany, Heather, Amber and Krissy
are all tall and play basketball.
Tiffany, Amber and Krissy also play
volleyball.
So, Heather must also play
volleyball.
Sample size strengthens an
argument
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Evaluating Arguments from Analogy
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Jason’s German car was a lemon and so was
Fred’s, Joe’s, Roy’s and Bob’s.
So Dirk’s German car is probably a lemon,
too.
If they all owned Volkswagen the argument is
very weak.
If one owned a Volkswagen, one a BMW, one
a Benz and one a Porsche, the argument
would be stronger.
With increased sample size, diversity becomes
a mark of strength.
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Evaluating Arguments from Analogy
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Finally, consider the specificity of the
conclusion relative to the premises
The broader and less specific the conclusion
is, the stronger the argument is.
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Tiffany, Heather, Amber and Krissy are all tall and
play basketball.
Tiffany, Amber and Krissy also play volleyball.
So, Heather may have played a game of volleyball
at some time.
This argument is stronger than the previous
one about Heather.
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Evaluating Arguments from Analogy
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Are the premises true?
Are the similarities relevant?
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The more relevant similarities there are, the better.
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If Amber and Krissy are also tall and play both basketball and volleyball our
conclusion is even further supported.
The more diversity in the examples, the better.
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Irrelevant dis-similarities: hair color
Relevant dis-similarity: job status
The more examples which are also similar, the better.
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If we also learn that they both get scholarships if they play more than one
sport, our conclusion is more supported.
Are there relevant dis-similarities?
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Since being tall is helpful in volleyball, the fact that both Tiffany and
Heather are tall is relevant to the previous conclusion.
If Tiffany, Amber and Krissy are different in many ways, except for the fact
that they are all tall and play basketball and volleyball, it seems more likely
that their being tall and playing basketball is relevant to their playing
volleyball. Thus, Heather’s being tall and playing basketball is better
evidence that she also plays volleyball.
Is the conclusion too specific?

“Heather probably plays volleyball” is better supported than “Heather must
play volleyball.”
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Arguing by Analogy
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Employ the same questions and
evaluation as you construct your
own arguments from analogy.
Don’t be too specific.
Use relevant similarities.
Use many similarities.
Use a diverse and large group.
Use true premises.
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Practice
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Page 308-09, Exercise 11.6
Page 309-11, Exercise 11.8
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I
So far in this Chapter, we have looked
at Inductive Reasoning and …
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Opinion polls
Statistical arguments
Reference class
Arguments from analogy
Today we will look at
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Causal arguments
Lecture Notes © 2008 McGraw Hill
Higher Education
31
Induction and Causal Arguments
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Causes precede and are constantly conjoined with their
effects.
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Cause and effect on a billiard table
To argue that they are causally connected, we would cite
the fact that the cue ball’s striking of the other balls
always precedes and is constantly conjoined with the
movement of the other balls.
But (chapter 6) two things being constantly conjoined
isn’t enough to conclude a causal connection.
Additionally, one thing preceding another is not enough
to conclude a causal connection.
So how can we argue and conclude that two things are
causally connected?
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Two kinds of causal arguments
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Arguments about a single instance:
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Arguments about a general relationship.
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Example: My car wouldn’t start but I haven’t replaced the
battery in six years. It must have been the battery.
This is just an argument about the cause of one event.
Example: There is a high correlation between smoking
and lung cancer. Smoking must cause lung cancer.
This is suggesting a causal relationship between smoking
and lung cancer (not about a specific person’s lung
cancer).
Notice that it isn’t saying that everyone who smokes
gets lung cancer. The generous interpretation reads it
as: smoking makes one more likely to get lung cancer.
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Two kinds of causal arguments
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These arguments are inductive.
The premises provide evidence
(strong evidence) for the
conclusion.
The conclusion does not follow with
strict necessity from the premises.
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The dangers of correlation
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We can never PROVE (beyond doubt) that there is a
causal relation between two things:
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But BEWARE!
We have a tendency to view things that correlate as
causally related.
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The best we can do is observe correlation.
Correlation is the best we can do when it comes to
arguments for causation.
E.g., superstitions, belief about bad luck, etc.
But assuming they are connected when trying to prove
they are, begs the question.
To really establish a “link,” we must be careful to
eliminate bias and expectation.

This is why, to eliminate the “placebo effect” medical
scientists do “double blind” studies.
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Positive, negative & neutral
correlation
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Positive Correlation: if A is found together
with B more than 50%of the time B is
found, then A is positively correlated with
B.
 The higher the percentage, the more
likely A has a causal connection to B.
Negative Correlation: if A is found together
with B less than 50% of the time, then A is
negatively correlated with B.
 The lower the percentage, the more
likely that A prevents B.
Neutral Correlation: If two things are found
together to exactly the same extent that
they are found apart, there is no
Lecture Notes © 2008 McGraw Hill
correlation.
Higher Education
36
Correlation and Cause
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Even large amounts of correlation are not enough to establish
a causal connection.
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When arguing from correlation, you need to make sure that
there aren’t any other factors that might account for the
correlation.
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Example: Vitamin C study & proper rest (p.329)
But all in all, correlation is most often due to coincidence…
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Example: big-feet and competence in math (p.318-19)
News reporters have this problem all the time.
Even if x is correlated with y, it could be due to the fact that they
are both the causal result of some other thing z
…so it is wise to always be suspicious.
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Practice

Pages 319-20, Exercise 11.10
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Probability

Epistemic Probability expresses how likely we think
something is, given what we believe.

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Relative Frequency Probability takes information
about a group as a whole and applies it to individual
cases.
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“There is a pretty high probability that I’ll go to the
beach sometime this summer.”
“There is a 90% chance that the operation will be
successful.”
This is derived from the fact that, for 90% of the people
on which the operation was performed, the operation
was successful.
A priori probability are statements that can be
calculated prior to sensory observations.
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“There is a 50% chance of getting tails on this coin
toss.”
The nature of the coin determines the “objective”
probability of getting tails.
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Some Clues:
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If you need to know about the structure
of the bet itself to know the odds, then it
is probably a priori.
If you need to know about the specific
person/thing involved to know how likely
it is, then it is epistemic.
If you need to know about the
percentages of the population in question,
then it is frequency.

If it says, “a random person/thing” then it
probably is frequency.
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Practice

Pages 321-22, Exercise 11.11
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More on a priori probability
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How to figure a priori probability
The probably of either A or B is “Pr(A) +
Pr(B)”
 Probability of drawing either a King or a 7 is
a standard deck: 1/13+1/13 [i.e, 2/13
(15.4%)]
The probably of getting both A and B is “Pr(A)
x Pr(B)”
 Probably of drawing a K and then a 7 (or a
7 and then a K) is 1/13x1/13 [i.e., 1/169
(.59%)]
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Gambler’s fallacy:

Thinking that previous chance occurrences
affect future ones.
 The probably of a roulette wheel coming up
black is always 47.37%, even if it just came
up black 28 times in a row.
 Granted, if you haven’t started spinning the
wheel yet, the probably of it hitting black 29
times in a row is low.
 But, if you have already hit black 28 times,
the probably of getting 29 in a row now is
the same as the probably of hitting it once
(because one more is all you need for 29):
47.37%.
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Bet values
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Expected Value: The payoff or loss you can expect
from a bet.
How to figure expected value: take the payoff and
multiple it by your odds.

The expected value of a 1/100 chance at $100 is $1.
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If there are multiple payoff, you average them:
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1/100x$100=$1
1/3rd chance at 0, 1/3 chance at $50, 1/3 chance at
$100. Expected value $50.
(0+50+100)/3=$50
Deal or no deal?:

The “banker” always offers less than expected value
(the average of the amounts left), until the end when
he wants them to take the deal.
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Relative Value
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Of course, there are other reasons to take bets other than
payoffs.
Your own needs, preferences and resources can affect the
“value” of a bet as well.
The value a bet has, given such considerations, is the
“relative value.”

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Example: The relative value of betting $100 for a long shot at
a billion is high for a millionaire (he can afford it), but low for
a homeless person (who wouldn’t want to risk money he can
use to eat on a long shot at a billion).
Diminishing marginal value: as quantity of bets increase,
the relative value of the bets tends to decrease.

If you are really hungry, you are willing to buy a piece of pizza
for $10 (if that is all that is available). It’s relative value is
really high. But after you eat it, and your hunger subsides
somewhat, its relative value drops and you are less willing to
pay so much. Buy enough pizza, and you won’t be willing to
pay much at all. Buy too much (and eat it) and you won’t take
it for free.
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Practice

Page 326, Exercise 11.12
46