Inductive Reasoning

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Chapter 10

• Inductive Reasoning

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The Nature of Inductive Reasoning

• What is an inductive argument?
• Any argument which is not deductive!
• I.e., any argument which does not provide a
  guarantee of the truth of the conclusion if the
  premises are true.
• Inductive arguments are probabalistic.

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The Nature of Inductive Reasoning

• What else?
• Inductive arguments are ampliative, while
  deductive arguments are non-ampliative.
• An argument is ampliative (defn) iff there is
  information contained in the conclusion that is
  not already contained in the premises.
• Thats the trade-off! You lose the guarantee of
  the truth of the conclusion for amplification.

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The Nature of Inductive Reasoning

• The logical strength of inductive arguments is
  not dependent on the form of the argument, but
  rather the content of the premises. (Its the
  opposite for deductive arguments.)

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The Nature of Inductive Reasoning

• However, there are 4 forms of inductive arguments
  that usually regarded as logically strong so long
  as certain conditions are met.
• Inductive generalization
• Statistical syllogism
• Induction by confirmation
• Analogical reasoning

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Inductive Generalization

• Has the following form
• Z percent of observed Fs are G
• It is probable, therefore, that Z percent of all
  Fs are G.

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Inductive Generalization

• E.g.
• 60 of students at STFX who were questioned
  believe in God. It is probable, therefore, that
  60 of students at STFX believe in God.

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Inductive Generalization

• When assessing these arguments, ask
• Is the sample representative?
• Is the sample large enough?

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Statistical Syllogism

• Has the following form
• Z percent of all Fs are G
• x is an F
• Is it probable to the degree 0.Z that x is G

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Statistical Syllogism

• Whats the difference between an inductive
  generalization and a statistical syllogism?
• Inductive generalizations reason from particular
  observations to a general claim about a class.
• Statistical syllogisms reason from a general
  claim about a class to a claim about a particular
  individual.

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Statistical Syllogism

• E.g.,
• 60 of students at STFX believe in God.
• Bob is a student at STFX.
• Therefore, there is a .6 degree of probability
  that Bob believes in God.

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Statistical Syllogism

• When assessing these arguments, ask
• Is there any additional information about x that
  has not been included in the premises?
• E.g. Bob is President of Catholic League of
  Students (prob that he believes in God
  increases).
• E.g., Bob is President of the Atheists for the
  Environment Society (Prob that he believes in God
  decreases).

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Induction by Confirmation

• Induction can be used to support a hypothesis or
  theory by providing confirming instances of that
  hypothesis or theory.
• When we propose a theory or hypothesis, there are
  certain things that ought to be observed if it is
  actually true (or probable).
• These are called observation statements. If we
  do observe what the theory predicts, then we have
  confirmed the theory.

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Induction by Confirmation

• Induction by Confirmation then has the following
  form
• If h then o
• o
• It is probable that h
• NB similar to the formal fallacy of affirming
  the consequent

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Induction by Confirmation

• E.g. (203)
• If the theory of general relativity is true, then
  it follows that light rays passing near the sun
  will bend.
• During the solar eclipse of 1919 it was observed
  that light rays passing near the sun did bend.
• It is probable therefore that the theory of
  general relativity is true.

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Induction by Confirmation

• When assessing these arguments, ask
• Is the number of confirming instances relatively
  high?
• In general, the more confirming instances the
  better the theory.
• Are there any disconfirming instances?
• Any disconfirming instance refutes the theory.

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Induction by Confirmation

• Disconfirming instances are regarded as
  refutations of a theory because such a refutation
  takes this form
• If h then o
• Not-o
• Therefore not-h
• That is, a disconfirming instances refutes a
  theory because we are dealing with a deductively
  valid argument form Modus Tollens (denying the
  consequent).

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Analogical Reasoning

• Analogical reasoning works by comparing things
  which are similar (analogous) and concluding that
  properties or relations that one thing has must
  also be present in the other.

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Analogical Reasoning

• E.g.,
• Last year I put some fertilizer on my
  strawberries and in the fall got about 20 per
  cent more strawberries. You should do the same
  with your strawberries, since you got the same
  kind of soil. Youll probably get more
  strawberries too.

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Analogical Reasoning

• Analogies compare two cases the subject case,
  and the analogue case.
• The subject case is the case about which we are
  trying to derive a conclusion (fertilizer on your
  soil)
• The analogue case is the case about which we are
  more familiar (fertilizer on my soil).

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Analogical Reasoning

• The conclusion in an analogy makes a claim about
  the subject case, and in particular states that
  the subject case will (probably) have the target
  feature.
• The target feature (increase in strawberry
  production) is the feature that is present in the
  analogue case, and it is being concluded that it
  (probably) is in the subject case.

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Analogical Reasoning

• There are two kinds of analogical arguments
• Analogical Argument by Properties
• Analogical Argument by Relations

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Analogical Argument by Properties

• Analogical Argument by Properties has the
  following form
• x has A, B, C. analogue case
• y has A, B. subject case
• It is probable therefore that y has C target
  feature

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Analogical Argument by Properties

• E.g.,
• Canada geese are water birds that nest in Canada
  in the early spring and migrate south to warmer
  climates for the winter months. Ducks are also
  water birds that nest in Canada in early spring.
  Therefore, ducks probably migrate south for the
  winter, too.

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Analogical Argument by Properties

• P1 analogue case Canada geese are water birds
  that nest in Canada in the early spring and
  migrate south to warmer climates for the winter
  months.
• P2 subject case Ducks are also water birds
  that nest in Canada in early spring.
• Conclusion Therefore, ducks probably migrate
  south for the winter target feature, too.

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Analogical Argument by Relations

• Analogical Argument by Relations has the
  following form
• x is to y analogue case as a is to b subject
  case.
• x is R to y.
• It is probable therefore that a is R to b target
  feature

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Analogical Argument by Relations

• E.g.,
• The proposal to give clean needles to prison
  inmates to stop the spread of AIDS from the use
  of dirty needles is ridiculous. It is like
  giving bank robbers normal bullets to stop them
  from using dum-dum bullets, which are much more
  damaging to the victim.

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Analogical Argument by Relations

• P1 Dum-dum bullets are to normal bullets (as
  used by bank robbers) analogue case as dirty
  needles are to clean (as used by prison inmates)
  subject case.
• P2 Although dum-dum bullets are much more
  damaging to the victim, normal bullets still kill
  their victims. Further, the role of police
  officers is to stop bank robbers, not prevent the
  harms they cause.
• Conclusion Although dirty needles are more
  damaging to the victim (addicts are likely to get
  HIV, etc.), clean needles can be just as damaging
  (e.g., overdoses). Further, the role of prison
  officials is to stop drug use, not prevent the
  harms caused by it.

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Analogical Reasoning

• When assessing these arguments, ask
• Are the analogue case and the target case
  relevantly similar?
• The more similar the two cases are, the stronger
  the analogy.
• Of course, everything is similar to everything
  else in some respect. You are looking for strong
  similarities.