Chapter Three

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Chapter Three

• Truth Tables

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1. Computing Truth-Values

• We can use truth tables to determine the
  truth-value of any compound sentence containing
  one of the five truth-functional sentence
  connectives.
• This method can also be used to determine the
  truth-value of more complicated sentences.
• This procedure is called truth table analysis.

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2. Logical Form

• The assignment of a truth-value to a compound
  sentence from the truth-values of its atomic
  constituents is called a valuation.
• Expressions that contain only sentence variables
  and sentence connectives are called sentence
  forms.
• If we replace sentence variables with sentence
  constants we end up with a substitution instance
  of the original sentence form.
• Logical form is not the same as logical
  equivalence.

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3. Tautologies, Contradictions, and Contingent
Sentences

• A sentence that is true in virtue of its logical
  form is a tautology.
• Contradictions are sentences that cannot possibly
  be true.
• The form of a contradiction is a contradictory
  sentence form.

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Tautologies, Contradictions, and Contingent
Sentences, continued

• A contingent statement is one whose form has at
  least one T and one F in its truth table.

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4. Logical Equivalences

• When two statements are logically equivalent, the
  truth-value of one determines the truth-value of
  the other. That is, each has the same truth-value
  under the same truth conditions.

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5. Truth Table Test of Validity

• We can use truth tables to determine if any
  argument in sentential logic is valid.
• Recall An argument is valid if and only if it is
  not possible for its premises to all be true
  while its conclusion is false.
• So, if an argument is valid there will be no line
  in a truth table in which all the premises are
  true and the conclusion false.

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Truth Table Test of Validity, continued

• We can test an argument for validity by
  conjoining the
• premises into the antecedent of a conditional,
  putting the
• conclusion as the consequent, and testing its
  form to see if
• it is a tautological form. If it is, the
  argument is valid.

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Truth Table Test of Validity, continued

• If the corresponding conditional, or test
  statement form, of an argument is a tautology,
  then premises are said to logically imply or
  entail the conclusion of the argument.

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Truth Table Test of Validity, continued

• A logical implication is a tautology who main
  connective is a horseshoe.
• A counterexample is an assignment of truth-values
  that will yield true premises and a false
  conclusion.

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6. Truth Table Test of Consistency

• We can use a truth table to check for consistency
  by constructing the truth table for the
  conjunction of the forms and looking for a line
  on which all the substitutions are true. The set
  is consistent if and only if there is such a line.

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7. Validity and Consistency

• The counterexample set of an argument consists of
  the premises of the argument together with the
  denial of the conclusion.
• If the counterexample set is consistent then the
  argument is invalid.
• All arguments with inconsistent premises are
  valid.

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8. The Short Truth Table Test for Invalidity

• All it takes to show that an argument is invalid
  is a single counterexamplea single line of a
  truth table on which the premises are all true
  and the conclusion false.
• It is often possible to produce such a
  counterexample by assigning a truth-value to the
  entire sentence and then working to find the
  appropriate assignment of truth-values to the
  atomic constituents.

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The Short Truth Table Test for Invalidity,
continued

• We find the logical form of the argument, then
  assign an F to the conclusion and the try to
  assign a T to each premise.
• If we can do this the argument is invalid.

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9. The Short Truth Table Test for Consistency

• All it takes to show that a set of sentences is
• consistent is to produce a single line in a
  truth table
• that makes them all true.

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10. A Method of Justification for the Truth Tables

• We can build the truth table for any connective
  given the following information
• A set of intuitively valid and invalid arguments
• In a valid argument, if the premises are true the
  conclusion must be true
• In an invalid argument, there must be the
  possibility that the premises are true and the
  conclusion false

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A Method of Justification for the Truth Tables,
continued

• To justify a truth table, find valid and invalid
  argument forms, using the connective in question,
  that force the lines of the truth table.

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Key Terms

• Contingent statement
• Contingent sentence form
• Contradiction
• Contradictory sentence form
• Corresponding conditional of an argument
• Counterexample of an argument
• Counterexample set
• Logical equivalence
• Logically equivalent
• Logical implication

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Key Terms, continued

• Logically implies
• Sentence form
• Substitution instance
• Tautologous sentence form
• Tautology
• Test statement form
• Truth table analysis
• Valuation

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Key Terms, continued

• Truth-function
• Truth-functional operator
• Truth table
• Truth-value
• Wedge
• Well-formed