Transcript Inductive Thinking - Where can my students do assignments that

Inductive Thinking
• 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.”
Inductive Thinking II
• 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.
Inductive Thinking III
• 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.
Inductive Thinking IV
• 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.
Causation
• 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.
Causation II
• Our text points out that 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.
Causation III
• 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.
Causation IV
• In Lewis Carroll’s Through the Looking Glass the
following scene occurs in which causal connections
and temporal sequences are deliberately confused.
• Please turn to page 181-182 of the text and read the
conversation between Alice and the Queen.
• Obviously, the order in which things happen make a
difference here because the events are causally
related. It may not matter whether the Queen speaks
and then smiles or smiles and then speaks, but when
it comes to a wound followed by a pain, the second
must occur after the first because it is a consequence
of it. The Queen’s mistake is to see just a series of
events where there are actually causal relations, and
causes and effects cannot be reversed.
Causation V
• To take another 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-and-effect
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.”
Causation VI
• 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-andeffect relationship between two things.
Mill’s Methods
• 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.
Mill’s Method of Agreement
• 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.
Mill’s Method of Agreement II
• 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
Mill’s Method of Agreement III
• Please turn to page 183 of your textbook.
• 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.
Method of Difference
• 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.
Method of Difference II
• 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
Effect
a, c, e, f, h
no illness occurred
a, d, e, g, i
no illness occurred
b, d, e, f, h
no illness occurred
b, c, e, g, j
illness occurred
Therefore j is the cause
Method of Difference III
• 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.
Joint Method of Agreement and Difference
• 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.
Joint Method of Agreement and Difference
II
• Prior factors
a, c, e, f, h
a, d, e, g, h
b, d, e, f, h
b, c, e, g, i
a, d, e, g, 1
a, d, e, f, 1
Therefore h is the cause
Effect
illness occurred
illness occurred
illness occurred
no illness occurred
no illness occurred
no illness occurred
Joint Method of Agreement and Difference
III
• 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.
Method of concomitant variations
• 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
Method of concomitant variations II
• 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 halfmoon 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 causeeffect link: the larger the moon, the higher the tide.
Method of concomitant variations III
• 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.
Necessary and Sufficient Conditions
• 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.
Necessary and Sufficient Conditions
• 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.”
• 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.
• 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.
Proximate and Remote Causes
• 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.
Proximate and Remote Causes II
• 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.
Proximate and Remote Causes III
• 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.
Proximate and Remote Causes IV
• 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.
Proximate and Remote Causes V
• 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.
Proximate and Remote Causes VI
• 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.
Problems in Determining Causation
1. 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?
Problems in Determining Causation II
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.
Problems in Determining Causation
III
• 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.
Problems in Determining Causation
IV
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.
Problems in Determining Causation V
• 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.
Summary
•
Steps for identifying genuine causal relationships
from mere temporal sequences. First we must
apply Mill’s four methods:
1.
2.
3.
4.
•
Agreement
Difference
Agreement and difference
Concomitant variations
Then we should differentiate between
1. Necessary and sufficient conditions
2. Proximate and remote causes
•
Finally, we should be careful to distinguish:
1. Cause from effect
2. Causation from correlation
3. The logical from the psychological
Similes and Metaphors
• 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.
Analogies
• 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.
Analogies II
• 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.
Analogies III
• 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.
Analogies IV
• 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.
Analogies V
• 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.
Analogies VI
• 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.
Analogy and Induction
• 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.
Criteria for determining the strength
of analogical arguments
1. The two cases must be alike in all essential
respects, and the greater the relevant similarities
the more probable the argument.
• For example:
–
–
–
•
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.
Criteria for determining the strength
of analogical arguments II
2.
•
•
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.
Criteria for determining the strength
of analogical arguments III
3.
•
•
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.
Criteria for determining the strength
of analogical arguments IV
• 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.
Legal Reasoning
• 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.
Legal Reasoning II
• 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.
Legal Reasoning III
• 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.
Moral Reasoning
• As in law, arguments from analogy are also useful in
deciding moral questions. Find examples of arguments
from analogy in the Moral Reasoning handout.
Common Areas of Argument from Analogy
• 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.
Common Areas of Argument from Analogy
II
• 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?
Common Areas of Argument from Analogy
II
• 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? Please turn to
page 319 in the textbook.
Reductio ad absurdum
• 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.
Reductio ad absurdum II
• 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.
Reductio ad absurdum III
• 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.
Reductio ad absurdum III
• 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.