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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.
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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.
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TRUTH-FUNCTIONAL FALSITY
A sentence P of SL is truth-functionally false if and only if the set {P} has a closed truth-tree.
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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.
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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.
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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.
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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.
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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.
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CLOSED BRANCH
A branch containing both an atomic sentence and the negation of that sentence.
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CLOSED TRUTH-TREE
A truth-tree each of whose branches is closed.
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OPEN BRANCH
A truth-tree branch that is not closed.
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COMPLETED OPEN BRANCH
An open truth-tree branch on which every sentence either is a literal or has been decomposed.
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COMPLETED TRUTH-TREE
A truth-tree each of whose branches either is closed or is a completed open branch.
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OPEN TRUTH-TREE
A truth-tree that is not closed.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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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