# PHILO 171 - Logic Exam 2

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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
 Author: tiffanyscards ID: 11519 Card Set: PHILO 171 - Logic Exam 2 Updated: 2010-03-22 02:25:05 Tags: symbolic logic Folders: Description: Flashcards for exam 2 of Logic 171. Show Answers: