Chapter Twelve

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Chapter Twelve

• Predicate Logic Truth Trees

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1. Introductory Remarks

• The trees for sentential logic give us
  decidabilitythere is a mechanical decision
  procedure that a machine could follow to
  determine the validity or invalidity of each
  argument in sentential logic.

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Introductory Remarks, continued

• The truth trees for predicate logic do not give
  us decidability as there can be no such decision
  procedure for predicate logic.
• This is called Churchs undecidability result.

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Introductory Remarks, continued

• If an argument in predicate logic is valid, a
  machine will be able to decide it is valid in a
  finite number of steps.
• But if an argument is invalid a machine might not
  be able to show it is invalid in a finite number
  of steps.

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Introductory Remarks, continued

• Given a tree in predicate logic, three things
  might occur
• All paths will close, so the argument is valid.
• There will be at least one open path, and no way
  to apply the tree rules to any line in that path,
  so the argument is invalid.
• The tree may seem to grow infinitely, in which
  case we cannot determine if the argument is
  invalid.

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Introductory Remarks, continued

• Since we cannot predict if we have an infinitely
  growing tree we cannot know whether a particular
  argument that meets condition (3) is invalid or
  not.

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2. General Features of the Method

• We use a form of indirect proof. We begin by
  testing an argument by listing its premises and
  the negation of its conclusion.
• The only new rules we need are two of the four QN
  rules and UI and EI.
• UI and EI will always be applied to constants.

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General Features of the Method, continued

• If we incorporate the identity sign into our
  symbolism trees become more difficult to
  construct.

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3. Specific Examples of the Method

• There are four new rules for predicate trees that
  supplement those for sentential trees these
  concern the use of denial, connectives, UI and EI.

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Specific Examples of the Method, continued

• There are two methods for doing predicate trees
• The adherence to a prescribed order.
• The unrestricted order.

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4. Some Advantages of the Trees

• For longer natural deduction proofs trees will
  usually involve fewer steps.
• We can also break down certain sentences more
  easily than we can in proofs.

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5. Example of an Invalid Argument with at Least
One Open Path

• If we apply the tree rules until we can no longer
  apply them and end with at least one open path we
  have an invalid argument.
• We can then read off the truth-values of the
  atomic sentences and construct a counterexample
  to the argument.

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6. Metatheoretic Results

• Invalidity in a domain An argument is invalid in
  a domain if we can find a counterexample in it.
• A domain with n members is said to be of
  cardinality n.

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Metatheoretic Results, continued

1. The tree method will mechanically yield a correct
   decision on every argument on which it yields any
   decision at all, and it will yield a correct
   decision on all valid arguments.

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Metatheoretic Results, continued

• 2. We should be able to figure out a method such
  that with the method of expansion we could know
  that if we choose a domain of a certain size, a
  valid argument will show up valid for the domain
  and hence for all domains of cardinality greater
  than zero.

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Metatheoretic Results, continued

• 3. If the argument we are testing by trees is
  invalid, the method may fail, and the method of
  expression equally fails. Even for some argument
  we know to be invalid, the truth tree method will
  not yield a decision.

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Metatheoretic Results, continued

• Points 1-3 are in effect Churchs undecidability
  results.

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7. Strategy and Accounting

• When predicate truth trees branch, the rules
  apply serially to each open path.

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Key Terms

• Cardinality of a domain
• Churchs undecidability result
• Flowchart for predicate trees
• Infinitely growing tree
• Invalid argument
• Invalidity in a domain