Transcript Inductive Thinking - Where can my students do assignments that

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Inductive arguments are those in which the
premises are intended to provide support,
but not conclusive evidence, for the
conclusion.
To use the example we have been using in
the book, in deduction we argue that “All
fish have gills, tuna are fish, therefore tuna
have gills.” In induction we argue that
“Tuna, salmon, cod, sharks, perch, trout,
and other fish have gills, therefore all fish
have gills.”
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To be even more precise, in using deductive
arguments we make explicit in the conclusion
what is implicit in the premises. In inductive
arguments, we extend the premises and make
a claim beyond the cases that are given.
Induction hazards an educated guess based
on strong but not on absolute proof about
some general conclusion that can be drawn
from the evidence.
However we characterize induction, we can
see that it is not nearly as reliable as
deduction because the conclusion is never
certain.
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In the previous example, it is probably true
that all fish have gills, but we have not
examined all species of fish, so we never
know that our claim is true. The same can
be said for the statement that the sun will
rise every day, which is based on all
recorded instances in the past but not on
all possible instances.
Because inductive arguments do not
guarantee that their conclusions are true,
we evaluate them according to the
strength of the support they provide for
their conclusion.
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An inductive argument is strong when its
premises provide evidence that its conclusion
is more likely true than false. An inductive
argument is weak when its premises do not
provide evidence that its conclusion is more
likely true than false.
Instead of striving for certainty, we have to
settle for a high degree of probability. Used
properly, induction can lead to extremely
reliable generalizations, as science has
repeatedly shown. For example, Charles
Darwin established the theory of evolution
using inductive reasoning.
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One of the most basic, most common, and
most important kinds of knowledge we
seek is knowledge of cause and effect.
Why didn’t my alarm clock go off when it
was supposed to? Why did I get a “D” on
my critical thinking exam? We want to
know the cause of what happened. In the
absence of a good account, we will often
accept a bad one - as in the case of
superstition and mythology. Some people
have believed that they can appease the
gods by sacrificing a virgin. Some people
believe that if a black cat crosses their
path, bad luck will follow, and so forth.
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To bring rain we may not do a rain dance,
but we are only half-joking when we say,
“Of course it rained; I just washed my car.”
In all of these cases, a false connection has
been established between two events such
that we assume that one event is
responsible for the other when they are
actually unrelated.
It can be difficult to recognize genuine
causal connections and distinguishing
them from mere temporal succession.
In our reasoning we need to separate a
necessary train of happenings from an
accidental one.
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We can say that some events are subsequent,
meaning that they just happen to follow, while
others are consequent; they occur because of
the earlier event. The trick is to differentiate
between the two, and to identify a causal
connection only when one event compels the
another to occur.
We can, for example, justifiably assert that the
following causal sequences took place: the
water boiled because the temperature was
raised to 212° F; every time I let go of the
chalk, the chalk falls to the floor.In these
cases the sequence was necessary, not
accidental; given one event, the other had to
happen.
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To take an example, one that the philosopher
David Hume liked, every time you have seen one
billiard ball strike another, it has caused the other
to move. So, you assume there is a cause-andeffect relationship there. You have witnesses the
same pairing of events over and over again – it is
no mere coincidence. But, Hume asks us, when
you think about it, what have you really seen?
Just the pairing of two events, one billiard ball
striking the other and then the other billiard ball
moving. You have witnessed what Hume called
“constant conjunction.” The two events always
happen one before the other – they are
“constantly conjoined.” You never see “necessary
connection” or “causal power.” Because of Hume,
we can’s say, “I see a cause-effect connection”,
but only by claiming, “I can prove it.”
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To make the same point, the philosopher
Bertrand Russell asks you to consider
yourself in the position of a chicken on a
farm. Every day that you can remember,
the farmer wife’s has approached you and
then fed you. You have come to associate
the two in terms of cause and effect. But
then comes the day when the farmer’s wife
approaches you and doesn’t feed you.
Instead, she wrings your neck. The moral
of the story is that we need to be careful in
assuming a cause-and-effect relationship
between two things.
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The nineteenth-century English
philosopher John Stuart Mill (11806-1873)
considerably refined the process of
identifying causal connections. John Stuart
Mill began learning Greek at the age of
three. By eight, he was reading Plato. He
was extremely influential in the
development of utilitarian ethics, but also
crucial in the establishment of the first
women’s rights organization.
Mill specified four “methods” that can be
used to recognize cause-effect chains: that
of agreement, difference, agreement and
difference, and concomitant variations.
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The method of agreement is described by Mill as
follows:
If two or more instances of the phenomenon under
investigation have only one circumstance in
common, the circumstance in which alone all the
instances agree, is the cause (or effect) of the given
phenomenon.
For example, consider an individual doing research
on why some students are successful in an
especially difficult subject, say, mathematical logic.
In reviewing the data, the researcher finds many
circumstances in which students are successful in
mathematical logic, such as instructors using
particular approaches to teaching the subject or
assigning particular tests. However, the researcher
discovers that in all instances in which students are
successful they are highly motivated.
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High student motivation is the only condition that
is common to all instances of student success in
mathematical logic. From this observation, using
the method of agreement, the researcher concludes
that the necessary condition for student success in
mathematical logic is high motivation.
A
B
C
D
E
All instances exhibit P (the phenomenon)
C1
C1
C1
C1
C1
C2
C3
C6
C3
C2
C4
C4
C5
C7
C6
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Although this method can be useful, if
suffers from a major defect: that there is very
often more than one common factor. In the
example of the students, they may have
drank from the same water fountain, been to
the same party the night before, been
exposed to someone with a contagious
disease, and so forth. This having been said,
Mill’s methods are a form of inductive
reasoning. There was a recent out break of
E. coli at a county fair. Health officials were
able to determine that water was the source
of the deadly E. coli by using causal
reasoning like Mill’s.
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The method of difference is described by Mill as
follows:
If an instance in which the phenomenon under
investigation occurs, and an instance in which it
does not occur, have every circumstance in
common save for one, that one occurring only in
the former; the circumstance in which alone the
two instances differ, is the effect, or the cause,
or an indispensable part of the cause, of the
phenomenon.
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In our previous example about the dining
hall, suppose that none of the students
became ill except for the one who ate
pumpkin pie for dessert. She had eaten
the appetizer and the main course just as
the other students did who did not become
ill.
Prior factors
a, c, e, f, h
occurred
a, d, e, g, i
occurred
b, d, e, f, h
b, c, e, g, j
Effect
no illness
no illness
no illness occurred
illness occurred
Therefore j is the cause
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The problem with this approach is that,
just as the areas of agreement can be
numerous, so can the differences. Because
of the number of variables involved, we
can never be sure when we have found the
consequential difference. Even though
pumpkin pie may have been the cause, it
may not have been the cause. There
could have been additional variables. For
instance, she could have broken up with
her boyfriend that day, drank alcohol the
night before, and so forth. The
possibilities are numerous.
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To try and fill the gaps in both methods
Mill suggests a third approach called the
joint method of agreement and difference.
Here we judge as the cause that element
which all preceding events have in
common (agreement) after factoring out
any common elements that did not result
in the subsequent event (difference). We
are then left with the one common element
present only in positive instances, and that
is taken as the cause.
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Prior factors
a, c, e, f, h
occurred
a, d, e, g, h
b, d, e, f, h
b, c, e, g, i
occurred
a, d, e, g, 1
a, d, e, f, 1
Therefore h is the cause
Effect
illness
illness occurred
illness occurred
no illness
no illness occurred
no illness occurred
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Both e and h are present in cases where
illness occurred, but by extending the
number of cases further, e drops out as a
possible cause. e is present even when
there is no illness, so it cannot be the
cause. H, on the other hand, is present
only (and always) when illness occurred, so
it must be the cause.
So, as in the case of the method of
difference, when pumpkin pie appears to
be the cause then we can ask if there is
anyone who ate pumpkin pie that did not
get sick. If we find such persons then we
can eliminate pumpkin pie as the cause of
the illness.
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The last approach, the method of
concomitant variations, is usually
employed when a continuous flow of
events is involved and we cannot control
for the negative occurrences. Here we try
to establish causation by recognizing a
correlation in the way one set of event
varies in relation to another. That is, we
see a correlation in degree and regularity
between two events, such that we infer
that the first must be causally related to
the second.
blood pressure
Concomitant graph
300
200
Series2
100
Series1
0
1
2
3
Heart attacks
4
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For example, people have observed that
the height of the tide depends upon the
phases of the moon. When the moon is
full the tide is highest; a half-moon is
followed by a medium tide; and a low tide
seems to be related to a quarter or a
crescent moon. Because of the
consistency and predictability of the
relation, we can infer a cause-effect link:
the larger the moon, the higher the tide.
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Other examples are the age of a tree and its
thickness; and the darkness of our tan and the
length of time we were in the sun. Economists
will use this method in declaring that as
mortgage rates decline investment in homes
increases. Freudians psychologists will argue
that people’s freedom varies inversely with their
neuroses; the more neurotic they are, the less
they are in charge of their lives.
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Aside from Mill’s formal methods, one basic way of
proving causal connections is to ask whether the
second event could have occurred without the first.
If it could not, then the first event can be named as a
cause. In technical terms this means identifying the
first event as a necessary condition for the second.,
a sine qua non or indispensable prior factor.
Consider this example from a Moore and Parker
Critical Thinking text:
The presence of oxygen is a necessary condition for
combustion.
This tells us that we can’t have combustion without
oxygen, or “If we have combustion (C), then we must
have oxygen (O).” Notice that the necessary
condition becomes the consequent of a conditional:
If C then O.
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A sufficient condition guarantees whatever it
is a sufficient condition for. Being born in
the United States is a sufficient condition for
U.S. citizenship – that’s all one needs to be a
U.S. citizen. Sufficient claims are expressed
as the antecedents of conditional claims, so
we could say “If John was born in the United
States (B), then John is a U.S. citizen (C): If C
then P.
You should also notice the connection
between “if” and “only if” on the one hand
and necessary and sufficient conditions on
the other. The word “if,” by itself, introduces
a sufficient condition; the phrase “only if”
introduces a necessary condition. So, the
claim “X is a necessary condition for Y” could
be symbolized “if X then Y.”
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Some other examples would be:
In sports, having a positive attitude is a
necessary condition for winning; you can’t win
without it. However, it may not be sufficient.
You also need good training, strength, skill,
stamina, a mutually supportive team, and so
forth.
It is sometimes said that to be happy we need
good health. However, good health may be a
necessary condition but it is not a sufficient
condition for happiness. We would probably be
unhappy if we were not healthy, but just being
healthy is not enough to make us happy. As for
what the sufficient conditions are for happiness,
that has been a quest of philosophers and
humankind for centuries.
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Sometimes conditions are not the same as
causes. In the case of a fire, a spark is both a
(necessary) condition and a cause, but if I lend a
friend my car which he then drives into a tree,
injuring himself, my lending him the car did not
cause the accident even though it was a
necessary condition for it.
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A distinction often made among causal
connections is between a proximate and a
remote cause. A proximate cause is that
which immediately triggers an event. It
functions as the factor that precipitates
some happening. For example, the
proximate cause of a person’s death could
be heart failure.
A remote cause on the other had, is the
background cause that ultimately produces
a certain effect; these causes are usually
multiple. They stretch backward in time as
links in the cause-effect chain, and
contribute to the inevitable and final
outcome.
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For example, the proximate cause of a death
might have been heart failure but the remote
causes could have been a gunshot wound,
preceded by a jealous quarrel.
At a criminal trial the prosecuting attorney will
often stress the proximate cause while the
defense attorney will draw attention to the
remote ones. For example, a prosecutor
might emphasize that the accused was caught
stealing a toy. The defense attorney might
argue that it was Christmas, the person was
unemployed, she didn’t have any friends or
family, she was to far down on the waiting list
for some of the toys for tots type programs,
and so forth.
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Each attorney’s case seems convincing because
each is referring to a different type of cause.
Some causes are certainly main ones and others
are peripheral, but rarely do we find one event
that can be labeled as the cause.
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Imagine that you are a child and that your father
enters the living room and asks what caused the
large mirror over the fireplace to break. The
proximate cause was that the mirror, a very
fragile object, was struck with sufficient force by
another object of sufficient rigidity. But your
father is not interested in the proximate cause
of the mirror’s breaking. He is looking for
something else.
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The second type of cause that we can
identify is a remote cause. A remote cause
of a given event is part of the chain of
events that led to the occurrence of that
event. Typically, for any given event, there
are many remote causes. For example, the
remote cause of the broken mirror might
have been a shoe flying through the air.
This is an event within the chain of events
that led to the mirror’s breaking. But this
does not satisfy your father either. So you
tell him that if your sister had not let go of
the shoe, the mirror would not have
broken. You have identified another
remote cause, yet it, too, does not satisfy
your father.
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The nature of the information sought
determines how far back in the chain of events
one needs to go in seeking a remote cause. In
the case of the broken mirror, your father
continues to question you and eventually
discovers that you were sitting on the fireplace
mantel reading aloud your sister’s diary, which
she had always kept hidden. Finally, your father
has the answer he has been looking for.
1.
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Distinguishing cause and effect. In the
method of concomitant variations, as well
as in other methods, it is sometimes hard
to determine which factor is the cause
and which the effect.
For example, George seems unusually
jittery and remarks that he did not sleep
well. His wife thinks George’s insomnia
(the feature about George in question)
was caused by his jitters (the only
relevant difference). She may fail to
consider the possibility that George’s
being jittery was the effect of his poor
sleeping rather than the cause.
Do the times create great leaders, or do
great leaders create the times?
2.
Causation and correlation. Sometimes,
two things or events are clearly
associated or linked. Where you find X,
you will also find Y. A relationship such
as this, in which two things are
frequently, or even constantly, found
together is a correlation. In a correlation,
two things share a mutual relationship;
where one is found, the other is often, or
always, found. By contrast, in the
relationship of causation, one thing
produces or brings about the other.
Sometimes, a correlation is an indicator
of a cause-and-effect relationship.
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From the text,
◦ Chance correlations must be guarded against. For
example, Arizona has a high death rate from lung
disease. However, that does not mean the climate
is unhealthy, but only that many people with lung
disease move to Arizona (for the clean air). In the
same way, in Holland the more storks there are, the
greater the number of babies. Does that mean
storks bring babies, as mother told us? No, it is
rather that as the number of buildings grown with
the population, more nesting areas are available for
storks. Storks do not bring babies, but babies do
bring storks.
3.
The logical and the psychological. A third
problem has to do with our tendency to
attribute causation to events that are
connected only periodically, not constantly.
The prime example is that of gambling. The
steady gambler is the steady loser since the
odds are always with the house. However,
gamblers are rewarded sometimes and that
reinforces their belief that they have a
winning system (or good luck). A behavioral
psychologist tells us that intermittent
reinforcement is a very powerful tool.
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From a logical perspective, the fact that the
gambler usually loses is proof against the
gambler’s idea that her system works, but from
a psychological viewpoint the occasional win
confirms the gambler’s belief. Obviously, it is
more realistic to look at this situation from a
logical perspective.
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Steps for identifying genuine causal
relationships from mere temporal
sequences. First we must apply Mill’s four
methods:
1.
2.
3.
4.
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Agreement
Difference
Agreement and difference
Concomitant variations
Then we should differentiate between
1.Necessary and sufficient conditions
2. Proximate and remote causes
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Finally, we should be careful to distinguish:
1. Cause from effect
2. Causation from correlation
3. The logical from the psychological
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Similes and metaphors are figures of
speech that are basically poetic devices
that draw together events, objects, or
ideas, which are otherwise dissimilar, in a
striking comparison.
Similes, from the Latin, meaning “likeness,”
use the terms “as” or “like” to make the
comparison explicit, whereas metaphors,
from the Greek meaning “transfer,”
dispense with the indicator terms and
imply the connection by substituting the
language of the one for the other.
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Whereas similes and metaphors compare things
that are essentially different except for one
similarity, analogical arguments compare things
that are alike in all essential respects and then
claimed to be alike in some further respect.
From the Greek, ana logon, “according to a
ratio,” analogies declare a relationship between
two things, a parallel connection, usually
between ideas or a set of ideas.
In mathematics, for example: 5 is to 10 as 10 is
to X . X being 20.
Or, up is to down as right is to?
Left, because the relationship is one of
opposites.
These are analogy questions.
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An analogy is a comparison of things based on
similarities those things share.
Although analogies are interesting and
important for many reasons, including their use
in poetry, we shall focus on one: their
importance in constructing inductive arguments.
Arguments from analogy claim that certain
similarities are evidence that there is another
similarity.
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Extended beyond mathematics, analogical reasoning
has had an extremely wide application.
For instance, physical scientists have argued that the
atomic nucleus is like a miniature solar system, so
whatever physical forces disrupt the one will disrupt
the other.
Just prior to the Revolutionary War some royalists
argued that the colonies were like the children of the
mother country, and just as children should remain
loyal to their parents, the colonies should not revolt
against England. On the other hand, the
revolutionaries argued that the colonies were like
fruit in an arbor, and when the fruit is ripe it is
natural that it should drop from the tree.
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These examples illustrate the nature of analogical
argument, but the last example also shows one of its
basic weaknesses. That is, almost anything can be
proven by carefully selecting the comparison.
If we want to argue for the blessings of old age we
can compare it to the maturing of a fine wine or say
that one achieves senior status in the community
acquires patience and wisdom, free from the tyranny
of passions.
On the other hand, we could show the sadness of old
age by comparing it to a house that is decrepit and
crumbling, a pitiful ruin dimply reflecting its former
dignity.
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The English theologian William Paley (1743-1805)
presented one of the best known analogical
arguments. Paley tried to support the view of St.
Thomas Aquinas that the world exhibits evidence of
a purposeful design and therefore proves the
existence of an intelligent designer, that is, God.
Paley did this by comparing the world to the
mechanism of a watch. If we were on a deserted
island and found a watch ticking away in perfect
order, we would assume that a watchmaker had
produced the watch. The odds of all the random
parts coming together and forming a functioning
watch by pure dumb luck seems unlikely. In the
same way, it is unlikely that just dumb luck and a
big bang could create a world such as this that is
well-organized and functional.
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However, we could also compare the world to an
organism rather than a mechanism, one with
biological parts that can become diseased; with
systems, vital organs, and limbs that develop
and degenerate; and with energy and matter at
the core, not mind or spirit. The blind
watchmaker.
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In an inductive generalization, we generalize
from a sample of a class or population to the
entire class or population.
In an analogical argument, we “generalize”
from a sample of a class or population to
another member of the class or population.
1.
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The two cases must be alike in all
essential respects, and the greater the
relevant similarities the more probable
the argument.
For example:
◦
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◦
•
Jim and Tim are both burly and play football.
Jim also wrestles.
So, Tim must also wrestle.
This is obviously a weak analogy. It would be
made stronger if it was noted that they are
best friends, rarely do anything apart, attend
a college that gives scholarships only to
athletes who play more than one sport, and
so forth.
2.
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The greater the number of cases compared,
the stronger the probability of the conclusion.
For example: Jim’s Buick leaks oil. Therefore,
Tim’s Buick will leak oil, also.
This case is not enough to make a fair
statement. If we tested 5,000 Buick cars and
all of them leaked oil, then we would have a
stronger case.
3.
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The greater the dissimilarity of the cases used
as the base of the analogy, the higher the
probability of the conclusion.
Example in the book: If we say that a company
is like a football team in that they are both
organizations of individuals devoted to the
achievement of a common goal, and just as
teamwork is necessary in winning football so
teamwork is essential to business success.
If the characteristics applied to high school
teams, as well as college teams, professional
and amateur, and so forth, that is stronger
evidence than citing just one football team.
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That is to say, if all subsets exhibit the same
characteristics plus the factor of teamwork,
then the argument that business (which is
similar to them) should do likewise and
becomes more powerful.
If all three rules are followed, the likelihood
of the analogy being correct is increased
considerably, although we can never be
certain of our conclusion.
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Many of the arguments used by lawyers in the
United States and Canada to support a trial are
analogical arguments. The reason is that the
legal systems of these countries were derived
many years ago from the English system, and an
essential feature of the English system is its
dependence on precedent. According to the
requirement of precedent, similar cases must be
decided similarly.
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Consider a “law” that we are all familiar
with, the First Amendment to the U.S.
Constitution, which provides for freedom
of speech and religious expression.
Suppose that you decide, in reliance on the
First Amendment, to pass out religious
pamphlets on a downtown street corner.
Suppose further that most of the people
your hand your pamphlets to merely
glance at them and then throw them on
the street and that the gathering litter
makes the area look like a garbage dump.
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To prevent the litter, the police tell you
that you can hand out your pamphlets only
in the vicinity of trash cans. You object
that such a restriction violates your First
Amendment rights, and you take the issue
to court.
In presenting your case, your lawyer will
argue that the case is analogous to a
number of other cases where the state
attempted to limit not the content of
religious expression, but the time, place,
and manner of its expression. Your lawyer
will attempt to show that your case is
analogous to cases in which the
government failed to prove that the
restriction was so tailored.
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As in law, arguments from analogy are also useful
in deciding moral questions. Find examples of
arguments from analogy in the Moral Reasoning
handout.
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Arguments from analogy are found in
many areas of study and have many
practical applications. Once again, let’s
consider law:
American law has its roots in English
common law, so legal decisions are often
made on the basis of precedence. For
example, in deciding whether or not the
free speech guaranteed by the First
Amendment applies to cyberspace
communications, a judge would be
expected to appeal to earlier and
analogous free speech cases.
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In deciding whether another case is
analogous, we must apply our rules to test
the strength of analogous arguments:
The two cases must be alike in all respects,
and the greater the number of similarities,
the more probable the argument.
Are there a good number of relevant
similarities, and few, if any, relevant
dissimilarities? Is the conclusion of the
judicial ruling properly specific?
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Arguments from analogy are often effective in
matters of ethics. One strategy used in moral
reasoning is to argue that a controversial issue
is analogous to one that is not controversial. In
her article “A Defense of Abortion,” Judith Jarvis
Thompson argues in favor of the morality of
abortion. Using a creative scenario, Thomson
argues that a person would have no moral
obligation to stay connected to a famous
violinist who was linked to he kidneys without
her knowledge or consent. She then argues by
analogy that a woman similarly has no moral
duty to carry her pregnancy to term. There are
some similarities here. There are also
dissimilarities. The question is, how relevant
are they? Does the analogy work?
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From Moore and Parker: One common
strategy for establishing the truth of a
claim is showing that its contradictory
implies something false, absurd, or
contradictory. This strategy, called
indirect proof, is based on the same idea
as remarks like this: “If Phillips is
conservative, then I’m the King of
England.” Obviously, this is just a way of
saying that Phillips is not conservative,
because it is clear that I am not the King of
England.
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If we want to argue that a claim is true by
using indirect proof, we begin with its
contradictory. To argue either for P or for
not-P, we begin with the other one and try
to show that it implies a false claim.
For example, if we wanted to prove that
your critical thinking instructor is not
wealthy, we would start by assuming the
opposite, that is, your critical thinking
instructor is wealthy. This can be shown
to imply that she can buy Dodge Vipers,
mansions, designer clothes, and so forth.
Because this is all obviously ridiculous, we
have proven that, sadly, your critical
thinking instructor is not wealthy.


This pattern of reasoning is sometimes
called reductio ad absurdum (reducing to
an absurdity, or RAA, for short), because it
involves showing that a claim implies a
false, absurd, or contradictory result.
Once again, the strategy is this:
To prove P,
Assume not-P.
Show that a false, absurd, or contradictory
result follows from not-P.
Conclude that not-P must be false.
Conclude that P must be true.

In the case of reducing analogies to an
absurdity, we need to show that the analogy has
many dissimilarities, so that to assume
similarities between the two things might be
ridiculous.