DEDUCTIVE ARGUMENT

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DEDUCTIVE ARGUMENT

• Recall that a deductive argument is an argument
  which is either valid or which is intended to be
  valid by the person making the argument.
• Also recall that an argument is valid when, on
  the assumption that the premise or premises of
  the argument is true, then the conclusion cannot
  be false.
• Further recall that an argument is invalid when
  its conclusion does not follow necessarily from
  its premises, or the conclusion of an invalid
  argument might be false even if its premise or
  premises is true.

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INDUCTIVE ARGUMENT I

• The premise or premises of an inductive argument
  provide support for the conclusion of the
  argument, but the support is not conclusive.
• Thus in an inductive argument the premises could
  be true and the conclusion false.
• Hence an inductive argument is an invalid rather
  than a valid form of argument.

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INDUCTIVE ARGUMENT II

• However, that an inductive argument is not valid
  (not an argument whose conclusion cannot be false
  if its premises are true) does not mean that it
  is not a good argument.
• Remember that a good argument is one which gives
  us grounds for accepting its conclusion.
• These grounds need not be conclusive, but could
  be strong in that, if the premises of the
  argument are true then it is unlikely that the
  conclusion is false.
• Inductive arguments can be good, and they are
  said to fall anywhere on the scale from very
  strong to very weak.

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INDUCTIVE ARGUMENT III

• Recall that an argument is strong when its
  conclusion is unlikely to be false on the
  assumption that its premises are true.
• Remember too that a weak argument is an argument
  which is not strong, or a weak argument is an
  argument whose conclusion is not unlikely to be
  false, even on the assumption that the premises
  are true.
• MP An inductive arguments premises can give
  powerful support for its conclusion, no support
  at all, or anything in between, so, again, it
  can range from very strong to very weak.

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INDUCTIVE VS. DEDUCTIVE

• I. M. Copi A deductive argument is one whose
  conclusion is claimed to follow from its premises
  with absolute necessity, this necessity not being
  a matter of degree and not depending in any way
  on whatever else may be the case.
• I. M. Copi An inductive argument is one whose
  conclusion is claimed to follow from its premises
  only with probability, this probability being a
  matter of degree and dependent upon what else may
  be the case.
• Thus Copi says that probability is the essence
  of the relation between premises and conclusion
  in inductive arguments.

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INDUCTIVE GENERALIZATIONS I

• A generalization is an argument offered in
  support of a general claim, and in an inductive
  generalization we generalize form a sample to an
  entire class.
• MP We reason that, because many (or most or
  all or some percentage) of a sample of the
  members of a class or population have a certain
  property or characteristic, many (or most or all
  or some percentage) of the members of the class
  or population also have that property or
  characteristic.
• For instance, Most students in my classes are
  well-mannered, (premise) therefore most IPFW
  students are well-mannered. (conclusion)

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INDUCTIVE GENERALIZATIONS II

• In the premise of an inductive generalization,
  the members of a sample are said to have a
  certain property.
• This is the property in question. (In the
  previous example the property in question is
  well-mannered.)
• In the conclusion of the inductive
  generalization, the property in question is
  attributed to many (or most or all or some
  percentage) of the entire class or population.
• This class is called the target or target class
  or target population.

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EXAMPLES

• Premise Many of the trees in this part of the
  woods are maples. The sample is the trees in
  this part of the woods, and the property in
  question is being a maple.
• Conclusion Many of the trees in the woods are
  maples. The target class is the trees in this
  woods.
• Premise Every dish of Spanish cuisine which I
  have tasted has been delicious. The sample is
  dishes of Spanish cuisine which I have tasted,
  and the property in question is being delicious.
• Conclusion All dishes of Spanish cuisine are
  delicious. The target class is the dishes of
  Spanish cuisine.

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REPRESENTATIVENESS AND BIAS I

• MP In an inductive generalization we use a
  sample to reach a conclusion about a target
  class. Therefore, the sample must represent the
  target class.
• For instance, Ninety percent of students polled
  at IPFW say that they are in favor of
  euthanasia. (premise) Therefore, ninety percent
  of all IPFW students are in favor of euthanasia.
  (Here the sample is the random sample of students
  polled, the property in question is the property
  of being in favor of euthanasia, and the target
  class is all IPFW students.)
• A sample is a representative sample to the extent
  to which it possesses all features of the target
  relevant to the property in question, and
  possesses them in the same proportions as the
  target class.

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REPRESENTATIVENESS AND BIAS II

• In the previous example, the sample is
  representative if all the people polled were in
  fact IPFW students, since being an IPFW student
  is the feature of the target class which must be
  possessed by the sample class, and the sample is
  representative if it possesses the property in
  question of being in favor of euthanasia in the
  same proportion as the target class of all IPFW
  students, namely 90. Both of these things are
  true here, and so the sample does represent the
  target class.

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REPRESENTATIVENESS AND BIAS III

• MP The less confidence we have that the sample
  of a class or population accurately represents
  the entire class or population, the less
  confidence we should have in the inductive
  generalization based on that example.
• For instance, Most people interviewed said that
  they are opposed to abortion. (premise)
  Therefore, most people are opposed to abortion.
  (Conclusion)
• This would not be a representative sample of the
  target class all people if the poll was
  conducted only amongst members of fundamentalist
  churches.

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REPRESENTATIVENESS AND BIAS IV

• MP Whether an inductive generalization is
  strong or weak depends on whether or not the
  sample accurately represents the target class.
• MP A sample that doesnt accurately represent
  its class is a biased sample. (The sample in the
  previous example regarding abortion is a biased
  sample.)
• MP A sample accurately represents a target
  class to the extent that it has all relevant
  features of the target class in the same
  proportion as the target.

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REPRESENTATIVENESS AND BIAS V
sample
Premise Many of the students in this class are
over thirty. Conclusion Many of the students in
this university are over thirty.
target
property in question
relevant feature shared by sample and target
classes
same proportions
MP A sample accurately represents a target
class to the extent that it has all relevant
features of the target class in the same
proportion as the target.
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REPRESENTATIVENESS AND BIAS VI

• How do we know when a feature is relevant?
• MP A feature or property P is relevant to
  another feature or property Q if it is reasonable
  to suppose that the presence or absence of P
  could affect the presence or absence of Q. (See
  the example on page 383.)
• For instance, a persons economic status is
  likely to affect his view on whether or not there
  should be tax cuts for the wealthy, while the
  color of his hair is not.
• A problem here is that our knowledge of what
  features affect other features is limited.

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REPRESENTATIVENESS AND BIAS VII

• If a class of things is homogeneous (composed of
  parts all of the same kind essentially alike of
  the same kind or nature) then we can be
  confident in the representativeness of our
  sample from that class. (An apple from a barrel
  of apples is likely representative of the whole.)
• This is not true of heterogeneous populations
  (differing in kind unlike composed of parts of
  different kinds). (If the class or population is
  the inventory of a grocery store, then an apple
  or apples from that store is unlikely to be
  representative of the whole.)

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RANDOM SAMPLES

• MP The most widely know method for achieving a
  representative sample in a heterogeneous
  population is to select the sample at random
  from the target population.
• A random sample of a population df. One in which
  every individual in the population has an equal
  chance of being selected.
• In addition, a sample can be biased even though
  it is randomly selected, if it is selected from a
  subgroup of the population which itself is not
  representative of the target population. (See the
  examples on page 384.)

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RANDOM VARIATION

• In an inductive generalization we look at a
  sample of a population or class, and draw a
  conclusion about the whole population or class
  the target.
• MP When you generalize from a sample of a
  population to the entire population, you must
  allow room for the random variation that can
  occur from sample to sample.
• For instance, if we take a sample of marbles from
  a bin containing both white and black marbles,
  and find that 50 of the marbles in that sample
  are black, we would not conclude, based on that
  single example, that exactly 50 of the marbles
  in the bin are black. The percentage of black
  marbles in the next sample we take might be 25
  or 75.

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ERROR MARGIN I

• The preceding example shows that we expect a
  random variation from sample to sample in a
  population or class of things (the target class).
• And because we expect random variation from
  sample to sample, we make a mistake in inductive
  generalization if we do not allow room for the
  random variation which can occur from sample to
  sample.
• The room allowed in the random variation that can
  occur from sample to sample in a class of things
  is called the error margin.

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ERROR MARGIN II

• If in our sample of marbles 50 of the marbles
  are black, we can only conclude that 50 of the
  marbles in the entire bin (the target population)
  are black plus or minus () a few points.
• The range of these points () indicates the error
  margin.
• The larger the sample the smaller the error
  margin. If we have a large sample, we need not
  allow as much room for error due to random
  variation.
• Thus if our sample of marbles was 500, the error
  margin is less than it would be if our sample was
  50 marbles.

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ERROR MARGIN III

• The error margin concerns the random variation
  that can occur from sample to sample in a
  population, and, from any sample taken from that
  population, we can at best conclude that the
  target population has the same percentage as the
  sample class plus or minus some points.
• Thus if 50 of the marbles in our sample are
  black, we can only conclude that 50 of the
  marbles in the target population are black some
  points.
• The larger the sample the smaller the error
  margin.
• Also, the wider the error margin the more
  confident we can be in our inductive
  generalization.
• Again, suppose that 50 of the marbles in a
  sample are black, we can be more confident if we
  conclude that 50 of the marbles in the entire
  bin are black 10 points than 2 points.

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CONFIDENCE LEVEL I

• MP The level of confidence that we have in the
  conclusion of our inductive generalization
  depends on the size of the random sample from
  the target population and on the error margin we
  allow for random variation.
• MP The larger the sample or the more room we
  allow for random variation (i.e. the larger the
  error margin we allow), the more confidence we
  have in the conclusion.

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CONFIDENCE LEVEL II

• Thus the confidence level df. The probability
  that the percentage of things in any given random
  sample will fall within the error margin.
• For instance, imagine that we say that 50 of
  marbles in a bin are black, with an error margin
  of 10 points. Then a confidence level of 80
  means that 80 of random samples of a certain
  size such as 50 marbles taken from the bin
  will fall within the error margin for that size
  sample. If the error margin is 10 points, then we
  would say that 80 of random samples consisting
  of 50 marbles will be 50 black 10 points. That
  is, 80 of random samples selected from the
  population would have 40-60 black marbles, or
  20-30 of the marbles in any random sample are
  likely to be black.
• MP If the random sample size were increased
  say from 50 to 100 marbles the confidence level
  would go up say from 80 to 90 or the error
  margin would shrink say from 10 points to 5
  points or both.

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SUMMARY

• Suppose that 50 of marbles in a bin are black.
• The error margin is the range of random variation
  in this percentage (50) that can occur from
  random sample to random sample of marbles taken
  from the bin.
• Thus if the error margin is 10 points, then we
  would expect 40 to 60 of the marbles in any
  given random sample from the population to be
  black. This is the range of random variation.
• The larger the random samples say 100 rather
  than 50 marbles the smaller this range.
• Thus if we take samples of 100 marbles rather
  than 50 we the error margin would now be less
  than 10 points.
• The larger the range of random variation
  perhaps 20 points rather than 10 the more
  probable it is that the percentage of marbles in
  any random sample from the bin will fall within
  that range.

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MAKING AN INDUCTIVE GENERALIZATION I

• MP An inductive generalization occurs when we
  dont know what percentage of a class or
  population has a given feature and we want to
  find out.
• MP So we make an inference from a sample.
• MP But our inference must make an allowance
  for the random variation that will occur from
  sample to sample.
• Thus we must allow for error margin.

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MAKING AN INDUCTIVE GENERALIZATION II

• MP If x percent of the sample have some
  feature e.g. 50 of 50 marbles chosen from the
  bin are black, we can only conclude that
  somewhere around x percent of the total
  population will have the feature in question
  e.g. somewhere around 50 of all the marbles in
  the bin will be black.
• MP The greater the margin of error we allow
  e.g. 10 points rather than 5 points, the
  higher our confidence level is.

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MAKING AN INDUCTIVE GENERALIZATION III

• Also, the larger the sample size the higher our
  confidence level is.
• Thus we can be more confident that 50 of all of
  the marbles in our bin are black if our sample
  size is 500 rather than 50 marbles.
• MP Except in populations known to be
  homogeneous e.g. we know that all the marbles in
  the bin are black, the smaller the sample in an
  inductive generalization, the more guarded the
  conclusion should be.

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GUARDED CONCLUSIONS I

• A conclusion of an inductive generalization can
  be made more guarded by decreasing the precision
  of the conclusion.
• The precision of the conclusion of an inductive
  generalization is decreased when, rather than
  saying 90 of the marbles in the bin are black,
  we say Most of the marbles are black, and
  decrease the precision even further if we say
  Many of the marbles are black.
• MP Decreasing the precision of the conclusion
  of an inductive generalization is an informal way
  of increasing the error margin.

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GUARDED CONCLUSIONS II

• A conclusion of an inductive generalization can
  also be made more guarded by expressing a lower
  degree of probability that it is true.
• The precision of the conclusion of an inductive
  generalization is decreased when, rather than
  saying It is certain that most of the marbles in
  the bin are black, we say It is very likely
  that most of the marbles are black, and decrease
  the precision even further if we say It is
  likely that most of the marbles are black.
• MP Informally expressing a lower degree of
  probability for the conclusion is an informal way
  of lowering the confidence level of the
  conclusion.

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GUARDED CONCLUSIONS III

• MP The less guarded the conclusion of an
  inductive generalization, the larger the sample
  should be, unless the population is known to be
  homogeneous.
• Lets say that we ask a number of IPFW students
  how many hours they study each week for each hour
  that they spend in class, and we find of those
  surveyed that 60 of them say that they spend two
  or more hours studying each week for each hour
  that they spend in class. If we conclude, based
  on our survey, that 60 of all students at IPFW
  spend two hours studying for each hour of class
  time, we would need a larger student sample than
  if we concluded that A majority of IPFW students
  spend two hours studying for each hour of class
  time. Also, if we conclude that it is very
  certain that 60 of all students have these study
  habits, we would need a larger sample than if we
  concluded that it is very certain that a number
  of students have these study habits.

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INDUCTIVE GENERALIZATIONS

• We should ask of any inductive generalization
  (generalizing from a sample to an entire class
  the target, and saying that many, most, or some
  percentage of the members of that class have a
  certain property x which many, most, or some
  percentage of the members of the sample have)
• 1. How well does the sample represent the target
  class or population? (Remember that the sample
  should be taken at random if the population is
  heterogeneous, so that each member of the
  population has an equal chance of being
  selected.)
• 2. Are the size and representativeness of the
  sample appropriate for how guarded the conclusion
  is? (See the examples on page 391.)

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ANALOGICAL ARGUMENTS I

• An analogy is a comparison of two or more
  objects, events, or other phenomena.
• In an analogical argument we reason that, because
  two or more things are alike in one or more ways,
  they are probably not necessarily alike in
  another way or ways as well.
• Analogical arguments are also called inductive
  analogical arguments and arguments from
  analogy, and they say that the more ways two or
  more things are alike, the more likely not
  necessarily it is that theyll be alike in some
  further way.

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AN EXAMPLE OF ANALOGICAL REASONING

• If two or more apples are alike in color and
  shape, and you have tasted one and it tastes
  good, you reason analogically in thinking that
  the other apples like it would also taste good.

Still Life with Basket of Apples, Paul Cézanne,
1890-1894
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THE PATTERN OF ANALOGICAL ARGUMENTS

• Things A, B, C . . . have properties a, b, c . .
  . Further, A and B have an additional property x.
  Therefore, C has property x.
• In our previous example, apples A, B, and C have
  the same color, shape, and size (properties a, b,
  and c). And apples A and B both taste sweet
  (property x). Therefore we conclude that apple C
  would taste the same as A and B if we were to
  bite into it.
• Analogical reasoning is probable only, not
  certain. It may be that apple C is sour rather
  than sweet like A and B.

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ANALOGICAL ARGUMENTS II

• The feature in question in an analogical argument
  is the feature (property or characteristic)
  mentioned in the conclusion of the argument.
• Thus in the previous example involving apples,
  the feature in question is the property x or the
  property of tasting sweet.
• The things that have the feature in question are
  the sample apples A and B in the previous
  example.
• The thing which we conclude has the feature in
  question is called the target item. This is
  apple C in the previous example.

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AN ANALOGICAL ARGUMENT
The sample
Premise 1 Lars and Mavis each own new BMWs, and
each BMW runs extremely well. Premise 2 Boris
just bought a new BMW. Conclusion Boriss BMW
will run extremely well.
Target item
The feature in question
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ANALOGICAL ARGUMENTS III

• As with any analogical argument, the conclusion
  of the the preceding argument is probable only,
  not certain. This means that the premises could
  be true even though the conclusion turns out to
  be false.
• Accordingly, analogical arguments are not valid
  arguments. (Remember that an argument is valid
  when, on the assumption that its premises are
  true, the conclusion which follows from the
  premises cannot be false.)
• However, analogical arguments are both good and
  strong arguments. (Recall that a good argument
  provides grounds for accepting its conclusion,
  and an argument is strong when it is unlikely
  that its conclusion is false on the assumption
  that its premises are true.)

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INDUCTION AND ANALOGY I

• MP In an inductive generalization, we
  generalize from a sample of a class or population
  to the entire class or population.
• For instance, Apples A, B, and C taste sweet,
  therefore, all the apples of the bunch which
  include A, B, and C are probably sweet is an
  inductive generalization. The sample class is A,
  B, and C, and the target class is the entire
  bunch.
• MP In an analogical argument, we generalize
  from a sample of a class or population to another
  member of the class or population.
• For instance, Apples A, B, and C taste sweet,
  therefore, apple D is probably also sweet is an
  analogical argument. The sample class is A, B,
  and C, and the other member of the class is D.

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INDUCTION AND ANALOGY II

• Inductive generalization and argument from
  analogy are similar in that, in reasoning from a
  sample class, we draw a conclusion about either
  the entire class from which the sample is taken
  (inductive generalization) or about another
  member of the same class as the sample class
  (analogical argument).
• Because they are similar in this regard
  reaching a probable conclusion from a sample
  class which concerns a number of other members of
  the same class the same evaluation questions
  apply to both kinds of argument.

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ANALYZING AN ANALOGICAL ARGUMENT I

• An implied target class of an analogical argument
  is the class to which the sample items and
  target item belong.
• Recall the argument Lars and Mavis each own new
  BMWs, and each BMW runs extremely well.
  Therefore Boriss new BMW will run extremely
  well. Here the sample items are the new BMWs of
  Lars and Mavis, and the target item is the new
  BMW of Boris. Accordingly, the implied target
  class is the class of new BMWs.
• MP Having spotted the implied target class, we
  can treat the analogical argument as an inductive
  generalization from a sample to the implied
  target class or population.

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ANALYZING AN ANALOGICAL ARGUMENT II

• So treating an analogical argument as an
  inductive generalization means that we have to
  ask
• 1. How well does the sample class represent the
  implied target class or population?
• 2. Are the size and representativeness of the
  sample appropriate for how guarded the conclusion
  is?

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ANALYZING AN ANALOGICAL ARGUMENT III

• 1. How well does the sample class represent the
  implied target class or population?
• We would expect all new BMWs to be made pretty
  much the same (homogeneous), and so the BMWs of
  Lars and Mavis can be taken to be representative
  of all new BMWs if they are the same model, with
  the same engine, etc. However, unless we had such
  information we should not infer that two new BMWs
  are necessarily representative of all new BMWs.

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ANALYZING AN ANALOGICAL ARGUMENT IV

• 2. Are the size and representativeness of the
  sample appropriate for how guarded the conclusion
  is?
• The conclusion that Boriss new BMW will run
  extremely well is not guarded at all, since it is
  inferred that it will run well simply because
  those of Lars and Mavis do. But are two BMWs
  enough to conclude that all BMWs run extremely
  well? Here the size is not appropriate for how
  unguarded the conclusion is.
• However, if the BMWs of Lars, Mavis, and Boris
  are the same model with the same features made at
  the same plant, then we could reasonably expect
  Boriss new BMW to run extremely well.

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THE FALLACY OF THE HASTY GENERALIZATION I

• The fallacy of the hasty generalization df.
  Basing an inductive generalization on a sample
  that is too small.
• And if the sample is too small, then the argument
  cannot be strong, but rather is weak.
• For instance, thinking that all IPFW students are
  in favor of decriminalizing marijuana (the
  generalization) because all students polled on
  the issue were so in favor (the sample). However,
  only three students were asked their opinion (the
  sample was too small).

44
THE FALLACY OF THE HASTY GENERALIZATION II

• The fallacy of the hasty generalization can also
  apply to the conclusion of an analogical
  argument.
• For instance, Bob and Ted and Jane and Alice are
  IPFW students. We find that Bob and Ted and Jane
  are in favor of decriminalizing marijuana. We
  commit the fallacy of the hasty generalization if
  we conclude that Alice is also in favor of
  decriminalizing marijuana.
• Here the sample is also too small for us to say
  that the argument is strong, and so it too is
  weak.

45
APPEAL TO ANECDOTAL EVIDENCE

• Appeal to anecdotal evidence df. A form of the
  fallacy of hasty generalization presented in the
  form of a story.
• My brother and I once took a cab in Vienna where
  the cab driver, without our knowledge since we
  did not know the city, took a wrong route to our
  destination. As a result, our fare was higher
  than it should have been although we did not know
  this. However the driver knew it and promptly
  refunded part of our money.
• To conclude from this anecdote that all, most, or
  even many cab drivers in Vienna are kind and
  honest would be a fallacy of hasty generalization
  in the form of an appeal to anecdotal evidence.
• In addition, if my brother and I had concluded
  that the next cab driver we met in Vienna would
  be similarly kind and honest would be the same
  kind of fallacy.

46
REFUTATION VIA HASTY GENERALIZATION

• Refutation via hasty generalization df. The
  fallacy of rejecting a general claim on the basis
  of an example or two which run counter to the
  claim (the sample, once again, is too small).
• For example, a general claim, based on a suitable
  sample size, might be that IPFW students are
  polite. Rejecting this claim after meeting one or
  two rude IPFW students would be an example of a
  refutation via hasty generalization.
• However, a universal general claim one which
  makes a claim about every member of a class, or
  attributes a particular property of every member
  of the class can be refuted by a single
  counterexample.
• Thus saying that all IPFW students are polite is
  a universal general claim which would be refuted
  by a single rude IPFW student.

47
BIASED GENERALIZATION

• Biased generalization df. A fallacy in which an
  entire class is generalized about which is based
  on a biased example. The example is biased in
  that it is a sample which does not represent the
  target class very well. (See the example on page
  398.)
• Biased analogy df. A fallacy in which something
  is concluded about some member of a class which
  is based on a biased example from that class.
  Again, what makes the example biased is that it
  does not represent the target class very well.
  (See the example on page 398.)

48
UNTRUSTWORTHY POLLS I

• Polls based on self-selected examples. The
  members of a self-selecting example put
  themselves in the example.
• Thus if a radio station has a call-in poll on
  some topic, then people who call the station to
  give their opinion about the poll put themselves
  in the example (those calling in to report their
  views) by calling in.
• Such a poll is untrustworthy because people who
  call in will have strong enough views about the
  topic to take the time to call in, and so will be
  oversampled or overrepresented, while those who
  do not call in will be undersampled or
  underrepresented. However, the view of those who
  call in may not in fact represent the majority
  view. And if not, then the poll is not
  trustworthy.

49
UNTRUSTWORTHY POLLS II

• Person-on-the-street interviews oversample people
  who walk and undersample people who drive. They
  also oversample people who frequent the area
  where the interviews are conducted, and
  undersample people who do not frequent that area,
  and oversample people who look friendly and
  willing to talk and undersample people who not.
• Telephone surveys oversample people who have
  phones and who answer them and undersample people
  who do not have phones, or who dont answer them,
  or who have unlisted numbers, or who are
  unwilling to take the time to be interviewed.

50
UNTRUSTWORTHY POLLS III

• Questionnaires oversample people who have the
  time and willingness to answer them and
  undersample people who do not have either or
  both.
• MP If the nonrespondents are atypical with
  respect to their views on the question(s) asked,
  the survey results are unreliable.
• Polls commissioned by advocacy groups. MP
  Polls of this sort can be legitimate, but
  questions might be worded in such a way as to
  elicit responses favorable to the group in
  question. (See the example on page 401.)
• MP Also, the sequence in which questions are
  asked can affect results. (See the example on
  page 401.)
• Questions can also be loaded. (See the example on
  page 402.)

51
UNTRUSTWORTHY POLLS IV

• Push-polling is not really polling, but marketing
  since, in ostensibly asking a person her opinion,
  the question is asked in such a way that she is
  pushed in the direction desired by the marketer.
  (See the example on page 402 where the respondent
  is pushed in a certain direction by the framing
  of the question hence the name push-polling.)

52
THE LAW OF LARGE NUMBERS I

• The large of large numbers df. The larger the
  number of chance-determined repetitious events
  considered, the closer the alternatives will
  approach predictable ratios.
• The chance of getting heads on any single flip of
  a coin is 1 out of 2 or 50. That is the
  predictable ratio. The law of large numbers says
  that, the more times you flip a coin, the closer
  it will come to 50. This is the case even though
  you may get heads 10 times in a row, or 75 times
  out of a hundred. But the more you throw, the
  greater the likelihood that the percentage of
  heads will near 50 the predictable ratio.

53
THE LAW OF LARGE NUMBERS II

• MP The reason smaller numbers dont fit the
  percentages as well as bigger ones is that any
  given flip or short series of flips can produce
  nearly any kind of result 10/10 heads, 8/10
  heads, 3/10 heads, or all tails.
• MP The law of large numbers is the reason that
  we need a minimum sample size even when our
  method of choosing a sample is entirely random.
• Because smaller sample sizes increase the
  likelihood of random sampling error, to infer a
  generalization with any confidence we need a
  sample of a certain size before we can trust the
  numbers to behave as they should.
• Thus we need to interview more than a few IPFW
  students before we conclude that most students
  are in favor of invading Iraq.

54
THE GAMBLERS FALLACY I

• The gamblers fallacy df. The belief that recent
  past events in a series of events can influence
  the outcome of the next event in the series.
• MP This reasoning is fallacious when the
  events have a predictable ratio of results.
• For instance, a person commits the gamblers
  fallacy when he thinks that, because he has
  flipped a coin four times in a row (a series of
  past events), and it has been heads each time,
  that that changes the odds of the next flip being
  heads to anything other than 50 - the
  predictable ratio.

55
THE GAMBLERS FALLACY II

• It is a fallacy because the past history of heads
  in the series does not change or effect the
  predictable ratio of 50 heads for the next flip.
• MP Its true that the odds of a coin coming up
  heads five times in a row are small only a
  little over 3 in 100 but once it has come up
  heads four times in a row, the odds are still
  50-50 that it will come up heads next time.