PHILO 171 - Logic Exam 2

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tiffanyscards
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11519
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PHILO 171 - Logic Exam 2
Updated:
2010-03-21 22:25:05
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symbolic logic
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Flashcards for exam 2 of Logic 171.
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  1. TRUTH-FUNCTIONAL CONSISTENCY
    A finite set Gamma of sentences of SL is truth-functionally consistent if and only if Gamma has a truth-tree with at least one completed open branch.
  2. TRUTH-FUNCTIONAL INCONSISTENCY
    A finite set Gamma of sentences of SL is truth-functionally inconsistent if and only if Gamma has a closed truth-tree.
  3. TRUTH-FUNCTIONAL FALSITY
    A sentence P of SL is truth-functionally false if and only if the set {P} has a closed truth-tree.
  4. TRUTH-FUNCTIONAL TRUTH
    A sentence P of SL is truth-functionally true if and only if the set {not P} has a closed truth-tree.
  5. TRUTH-FUNCTIONAL INDETERMINACY
    A sentence P of SL is truth-functionally indeterminate if and only if neither the set {P} nor the set {not P} has a closed truth-tree.
  6. TRUTH-FUNCTIONAL EQUIVALENCE
    Sentences P and Q of SL are truth-functionally equivalent if and only if the set {not (P if and only if Q)} has a closed truth-tree.
  7. TRUTH-FUNCTIONAL ENTAILMENT
    A finite set Gamma of sentences of SL truth-functionally entails a sentence P of SL if and only if the set Gamma U {not P} has a closed truth-tree.
  8. TRUTH-FUNCTIONAL VALIDITY
    An argument of SL with a finite number of premises is truth-functionally valid if and only if the set consisting of the premises and the negation of the conclusion has a closed truth-tree.
  9. CLOSED BRANCH
    A branch containing both an atomic sentence and the negation of that sentence.
  10. CLOSED TRUTH-TREE
    A truth-tree each of whose branches is closed.
  11. OPEN BRANCH
    A truth-tree branch that is not closed.
  12. COMPLETED OPEN BRANCH
    An open truth-tree branch on which every sentence either is a literal or has been decomposed.
  13. COMPLETED TRUTH-TREE
    A truth-tree each of whose branches either is closed or is a completed open branch.
  14. OPEN TRUTH-TREE
    A truth-tree that is not closed.
  15. How do you use truth-trees to test for the consistency of a set of sentences?
    • Do a tree with all the set members on top.
    • If the tree is open, the set is consistent.
    • If the tree is closed, the set is inconsistent.
  16. How do you use truth-trees to test for the truth-functional truth of a sentence P?
    • Remember: P is truth-functionally true if and only if {not P} is inconsistent.
    • Do a tree for {not P}.
    • If the tree is closed, P is truth-functionally true.
    • If the tree is open, P is NOT truth-functionally true.
  17. How do you use truth-trees to test for the truth-functional falsehood of P?
    • Remember: P is truth-functionally false if and only if {P} is inconsistent.
    • Do a tree for {P}.
    • If the tree is closed, P is truth-functionally false.
    • If the tree is open, P is NOT truth-functionally false.
  18. How do you use truth-trees to test for the truth-functional indeterminacy of P?
    • Remember: P is truth-functionally indeterminate if and only if {P} is consistent and {not P} is consistent.
    • Do trees for both {P} and {not P}.
    • If both trees are open, P is indeterminate.
    • If not both trees are open, P is NOT indeterminate.
  19. What is an alternate way to use truth-trees to test for truth-functional truth, falsehood, and indeterminacy?
    • Do a truth-tree with sentence P at the top.
    • If the tree closes, then P is truth-functionally false.
    • If the tree does not close, list and count the number of TVAs on which the sentence P is true.
    • If the number of TVAs on which P is true is equal to the number of total possible TVAs (calculated from the number of atomic sentences involved), then P is truth-functionally true (since it is true on every TVA).
    • If the number of TVAs on which P is true is less than the number of total possible TVAs, then P is truth-functionally indeterminate (since then it is true on some but not all TVAs).
  20. How do you use truth-trees to test for the truth-functional equivalence of P and Q?
    • Remember: P and Q are truth-functionally equivalent if and only if {not (P if and only if Q)} is inconsistent.
    • Do a tree with {not (P if and only if Q)} at the top.
    • If the tree closes, they are equivalent.
    • If the tree is open, they are not.
  21. How do you use truth-trees to test for truth-functional entailment?
    • Remember: Set Gamma entails P if and only if Gamma U {not P} is inconsistent.
    • Do a tree for Gamma U {not P}.
    • If the tree closes, then Gamma entails P.
    • If the tree is open, then Gamma does NOT entail P.
  22. How do you use truth-trees to test for truth-functional validity?
    • Remember: An argument is valid if and only if the set consisting of the premises and the NEGATION of the conclusion is inconsistent.
    • Do a truth-tree for the set, consisting of the premises and the negation of conclusion.
    • If the tree closes, the argument is valid.
    • If the tree is open, the argument is invalid.
  23. True or False: From the open branches of a completed truth tree one can recover all the TVAs on which all members of the set being decomposed are true.
    True

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