Logic: Chapter One: Sections V VI

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Logic Chapter One Sections V VI

• Jay Odenbaugh
• Philosophy
• Lewis and Clark College

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V. Reconstructing Arguments

• As we have seen, arguments often do not contain
  indicator words.
• Thus, we must determine what the premises and
  conclusion are from context. One can insert
  indicator words into the argument to see if the
  meaning is the same or different.

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Incomplete Arguments (Enthymemes)

• Another problem with reconstructing arguments is
  that some arguments are incomplete.
• An argument is incomplete if there is at least
  one missing premise.
• If an argument is incomplete, then at first
  glance, the argument might appear weak. However,
  to properly evaluate the argument, we must
  identify the missing premise(s).

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Generalizations

• Many incomplete arguments omit either universal
  or statistical generalizations.
• Lets let A and B be general terms or what
  are called predicates (they stand for classes or
  sets of objects)
• Universal generalizations All A are B No A
  are B
• Statistical generalizations Most A are B
  (some, usually, seldom, frequently, rarely,
  etc.)

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Square of Opposition

• All As are Bs No As are Bs
• Some As are Bs Some As are not Bs.
• What are the logical relations between these
  generalizations?

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Contradictions

• Let P and Q be statements. Then
• P and Q are contradictory just in case there are
  mutually exclusive.
• P and Q are mutually exclusive just in case if P
  is true, the Q must be false and if Q is true,
  then P must be false
• In our square of opposition, which statements are
  contradictory?

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Contradictions, Cont.

• In our square of opposition, there are two pairs
  of contradictions
• All As are Bs and Some As are not Bs.
• No As are Bs and Some As are Bs.
• If everything which is A is also B, then there
  can be no A which is not also a B.
• Similarly, if nothing which is an A is a B, then
  there can be no A which is a B.

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Contradictions, Cont.

• Here is an example of a contradiction
• All of Sylvester Stallone movies are ridiculous.
• Some of Sylvester Stallone movies are not
  ridiculous
• Notice that if the former is true, then the
  latter must be false.
• Likewise, if the latter is true, then the former
  must be true.

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Contradictions, Cont.

• Here is the other example of a contradiction
• No Sylvester Stallone movie is ridiculous.
• Some Sylvester Stallone movies are ridiculous.
• If the former is true, then the latter must be
  false.
• If the latter is true, the former must be false.

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Contraries

• Besides contradictions, our square of opposition
  also contains contraries.
• P and Q are contraries just in case both P and Q
  cannot be true but they both can be false.
• Which statements in our square of opposition are
  contraries?

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Contraries, Cont.

• In our square of opposition, the contraries are
• All As are Bs and No As are Bs.
• If every things which is an A is also a B, then
  it cannot be true that nothing is both an A and a
  B.
• However, both statements may be false - it may be
  true that some As are Bs and some are not.

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Contraries, Cont.

• Here is an example of contraries
• All Hondas are easy to steal.
• No Hondas are easy to steal.
• Notice that they cannot both be true but they can
  both be false.

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Subcontraries

• Besides contradictions and contraries, our square
  of opposition finally contains subcontraries
• P and Q are subcontraries just in case they both
  can be true but that cannot both be false.
• Which statements are subcontraries in our square
  of opposition?

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Subcontraries, Cont.

• In our square of opposition, the subcontraries
  are
• Some As are Bs and Some As are not Bs.
• Some As can be B and some might not be Bs.
• However, for anything which is an A, it must
  either be a B or not a B.

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Subcontraries, Cont.

• Here is an example of subcontraries
• Some Lewis and Clark students are democrats.
• Some Lewis and Clark students are not democrats.
• It is possible that some Lewis and Clark students
  are democrats and some are not.
• However, for any Lewis and Clark student, they
  are either a democrat or they are not.

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Summary

• So, we have the following relationships
• Contradictory
• All A are B, Some A are not B
• No As are Bs, Some As are Bs.
• Contrary All As are Bs, No As are Bs
• Subcontrary Some As are Bs, Some As are not Bs.

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Statistical Generalizations

• Statistical generalizations have the following
  form
• Most As are Bs (or many, some, frequently, etc.)
• For simplicity, one can think of generalizations
  as forming a continuum. A generalization can be
  construed as
• n of A are B

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Statistical Generalizations, Cont.

• In the limiting cases, either n 100 or n 0.
• 100 of As are Bs ? universal generalization
• 0 of As are Bs ? universal generalization
• If 100 lt n lt 0, then we have a statistical
  generalization.
• n of As are Bs ? statistical generalization

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Incomplete Arguments

• Arguments are often incomplete because the
  missing premise(s) is taken to be obvious.
  However,
• What we take to be obvious may not be so to
  everyone,
• What everyone takes to be obvious may be false.
• When you reconstruct arguments, use a principle
  of charity do not attribute false beliefs or
  unjustified beliefs to someone unless there is no
  alternative.

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Important Skills for Reconstructing Arguments

• Be sensitive to different uses of language.
• Recognize when evidence is required to support an
  assertion.
• Be aware of the difference between the truth of
  statements and the support they would provide
  some other statement if they were true.
• Recognize arguments, identify their parts, supply
  missing premises, and distinguish between the
  premises and conclusion.

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Homework, 1.4

• 2. College graduates earn more than those who
  have never been to college.
• 4. Most hockey players are missing at least one
  tooth.
• 6. Zero percent of the ingredients of Brand X
  breakfast cereal are toxic.

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Homework, 1.5

• 3. Discriminationrefusing to admit someone to a
  public place of business on grounds of race, sex,
  or religionis against federal law. However, the
  law also respects the rights of people to
  associate privately with whomever they wish.
• Last year the national organization of Jaycees
  voted to kick out the women members they had
  previously recruited. They even voted to kick out
  the chapters who wouldnt kick out the women they
  had previously invited.
• But in Minnesota, the courts ruled that the
  Jaycees were actually operating as a public
  business. They were soliciting members, and open
  to anyone so long as that anyone was a male. So,
  this duty designated public business was
  forbidden to discriminate.

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Homework, 1.5 Cont.

• Discriminationrefusing to admit someone to a
  public place of business on grounds of race, sex,
  or religionis against federal law.
• However, the law also respects the rights of
  people to associate privately with whomever they
  wish.
• Last year the national organization of Jaycees
  voted to kick out the women members they had
  previously recruited.
• They even voted to kick out the chapters who
  wouldnt kick out the women they had previously
  invited.
• But in Minnesota, the courts ruled that the
  Jaycees were actually operating as a public
  business.
• They were soliciting members, and open to anyone
  so long as that anyone was a male.
• So, this duty designated public business was
  forbidden to discriminate.

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Homework 1.5

• Discriminationrefusing to admit someone to a
  public place of business on grounds of race, sex,
  or religionis against federal law.
• However, the law also respects the rights of
  people to associate privately with whomever they
  wish.
• Last year the national organization of Jaycees
  voted to kick out the women members they had
  previously recruited.
• But in Minnesota, the courts ruled that the
  Jaycees were actually operating as a public
  business.
• They were soliciting members, and open to anyone
  so long as that anyone was a male.
• __________________________________________________
  _________
• So, this duty designated public business was
  forbidden to discriminate.