Section 3'6 Arguments and Truth Tables

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Section 3.6Arguments and Truth Tables

• Objectives
• Use truth tables to determine validity.
• Recognize and use forms of valid and invalid
  arguments.

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Arguments

• An Argument consists of two parts
• Premises the given statements.
• Conclusion the result determined by the truth
  of the premises.
• Valid Argument The conclusion is true whenever
  the premises are assumed to be true.
• Invalid Argument Also called a fallacy
• Truth tables can be used to test validity.

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Example 1The Menendez case

• Did Eric and Lyle Menendez kill their parents to
  get the inheritance or as a result of years of
  abuse?
• Premise 1 If children murder their parents in
  cold blood, they deserve to be punished to the
  full extent of the law.
• .
• Premise 2 These children murdered their parents
  in cold blood.
• Conclusion Therefore, these children deserve to
  be punished to the full extent of the law.

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Example 1 continuedThe Menendez case

• Solution
• p Children murder their parents in cold blood.
• q They deserve to be punished to the full
  extent of the law.
• Write the argument in symbolic form
• Premise 1 p ? q If children under their
  parents in cold
• blood, they
  deserve to be punished
• to the full
  extent of the law.
• Premise 2 p These children murdered
  their parents in cold blood.
• Conclusion ? q Therefore, these children
  deserve to be punished to the full extent
  of the law.

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Example 1 continued

• Rewriting as a conditional statement and forming
  the truth table (p ? q) ? p ? q
• This conditional statement is a tautology. This
  argument is called direct reasoning and is valid.

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Testing the validity of an Argument with a Truth
Table

• Use a letter to represent each simple statement
  in the argument.
• Express the premises and the conclusion
  symbolically.
• Write a symbolic conditional statement of the
  form
• (premise 1) ? (premise 2)?? (premise n)
  ?conclusion,
• where n is the number of premeses.
• Construct a truth table for the conditional
  statement in step 3.
• If the final column of the truth table has all
  trues, the conditional statement is a tautology
  and the argument is valid. If the final column
  does not have all trues, the conditional
  statement is not a tautology and the argument is
  invalid.

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Example 2Determining Validity with a Truth Table

• Determine if the following argument is valid
• I cant have anything more to do with the
  operation. If I did, Id have to lie to the
  Ambassador. And I cant do that
• Henry Bromell
• Solution
• We can express the argument as follows
• If I had anything more to do with the operation.
  Id have to lie to the Ambassador.
• I cant lie to the Ambassador
  .
• Therefore, I cant have anything more to do with
  the operation.

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Example 2 continued

• Step 1 Use a letter to represent each statement
  in the argument
• p I have more to do with the operation
• q I have to lie to the Ambassador.
• Step 2 Express the premises and the conclusion
  symbolically.
• p ? q If I had anything more to do with the
  operation, Id have to lie to the Ambassador.
• q I cant lie to the Ambassador.
  .
• ? p Therefore, I cant have anything more
  to do with the operation.

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Example 2 continued

• Step 3. Write a symbolic statement
• (p ? q) ? q ? p
• Step 4. Construct the truth table.
• The form of this argument is called
  contrapositive reasoning.
• It is a valid argument.

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Standard Forms of Arguments
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Example 3Determining Validity Without Truth
Tables

• Determine whether this argument is valid or
  invalid
• There is no need for surgery because if there is
  a tumor then
• there is a need for surgery but there is no
  tumor.
• Solution
• p There is a tumor
• q There is a need for surgery.
• Expressed symbolically
• If there is a tumor then there is need for
  surgery. p ? q
• There is no tumor.
  . p
• Therefore, there is no need for surgery.
  ? q
• The argument is in the form of the fallacy of the
  inverse and
• is therefore, invalid.