Inductive Thinking

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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.

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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.

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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.

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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.

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Causation

• One of the most basic, most common, and most
  important kinds of knowledge we seek is knowledge
  of cause and effect. Why didnt 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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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.

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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.

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Causation IV

• In Lewis Carrolls 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 Queens mistake is
  to see just a series of events where there are
  actually causal relations, and causes and effects
  cannot be reversed.

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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 cans say, I see a
  cause-effect connection, but only by claiming,
  I can prove it.

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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 wifes 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 farmers
  wife approaches you and doesnt 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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Mills 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 womens 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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Mills 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.

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Mills 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.

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Mills 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,
  Mills 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 Mills.

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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.

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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

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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.

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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.

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Joint Method of Agreement and Difference II

• Prior factors Effect
• a, c, e, f, h illness occurred
• a, d, e, g, h illness occurred
• b, d, e, f, h illness occurred
• b, c, e, g, i no illness occurred
• a, d, e, g, 1 no illness occurred
• a, d, e, f, 1 no illness occurred
• Therefore h is the cause

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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.

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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.

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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
  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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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 peoples freedom varies inversely with their
  neuroses the more neurotic they are, the less
  they are in charge of their lives.

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Necessary and Sufficient Conditions

• Aside from Mills 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 cant 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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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 thats 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 cant 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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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 persons 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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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
  didnt 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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Proximate and Remote Causes III

• Each attorneys 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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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 mirrors breaking. He is looking
  for something else.

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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 mirrors 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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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 sisters diary, which she had always kept
  hidden. Finally, your father has the answer he
  has been looking for.

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Problems in Determining Causation

• 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 Georges insomnia (the feature about
  George in question) was caused by his jitters
  (the only relevant difference). She may fail to
  consider the possibility that Georges 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?

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Problems in Determining Causation II

• 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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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.

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Problems in Determining Causation IV

• 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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Problems in Determining Causation V

• From a logical perspective, the fact that the
  gambler usually loses is proof against the
  gamblers idea that her system works, but from a
  psychological viewpoint the occasional win
  confirms the gamblers belief. Obviously, it is
  more realistic to look at this situation from a
  logical perspective.

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Summary

• Steps for identifying genuine causal
  relationships from mere temporal sequences.
  First we must apply Mills four methods
• 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
• Cause from effect
• Causation from correlation
• The logical from the psychological

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Criteria for determining the strength of
analogical arguments

• 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.

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Criteria for determining the strength of
analogical arguments II

• The greater the number of cases compared, the
  stronger the probability of the conclusion.
• For example Jims Buick leaks oil. Therefore,
  Tims 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.

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Criteria for determining the strength of
analogical arguments III

• 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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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.

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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.

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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.

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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.

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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.

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Common Areas of Argument from Analogy

• Arguments from analogy are found in many areas of
  study and have many practical applications. Once
  again, lets 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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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?

59
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.

60
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 Im 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.

61
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.

62
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.

63
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.