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TRUTHFUNCTIONAL CONSISTENCY
A finite set Gamma of sentences of SL is truthfunctionally consistent if and only if Gamma has a truthtree with at least one completed open branch.

TRUTHFUNCTIONAL INCONSISTENCY
A finite set Gamma of sentences of SL is truthfunctionally inconsistent if and only if Gamma has a closed truthtree.

TRUTHFUNCTIONAL FALSITY
A sentence P of SL is truthfunctionally false if and only if the set {P} has a closed truthtree.

TRUTHFUNCTIONAL TRUTH
A sentence P of SL is truthfunctionally true if and only if the set {not P} has a closed truthtree.

TRUTHFUNCTIONAL INDETERMINACY
A sentence P of SL is truthfunctionally indeterminate if and only if neither the set {P} nor the set {not P} has a closed truthtree.

TRUTHFUNCTIONAL EQUIVALENCE
Sentences P and Q of SL are truthfunctionally equivalent if and only if the set {not (P if and only if Q)} has a closed truthtree.

TRUTHFUNCTIONAL ENTAILMENT
A finite set Gamma of sentences of SL truthfunctionally entails a sentence P of SL if and only if the set Gamma U {not P} has a closed truthtree.

TRUTHFUNCTIONAL VALIDITY
An argument of SL with a finite number of premises is truthfunctionally valid if and only if the set consisting of the premises and the negation of the conclusion has a closed truthtree.

CLOSED BRANCH
A branch containing both an atomic sentence and the negation of that sentence.

CLOSED TRUTHTREE
A truthtree each of whose branches is closed.

OPEN BRANCH
A truthtree branch that is not closed.

COMPLETED OPEN BRANCH
An open truthtree branch on which every sentence either is a literal or has been decomposed.

COMPLETED TRUTHTREE
A truthtree each of whose branches either is closed or is a completed open branch.

OPEN TRUTHTREE
A truthtree that is not closed.

How do you use truthtrees 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.

How do you use truthtrees to test for the truthfunctional truth of a sentence P?
 Remember: P is truthfunctionally true if and only if {not P} is inconsistent.
 Do a tree for {not P}.
 If the tree is closed, P is truthfunctionally true.
 If the tree is open, P is NOT truthfunctionally true.

How do you use truthtrees to test for the truthfunctional falsehood of P?
 Remember: P is truthfunctionally false if and only if {P} is inconsistent.
 Do a tree for {P}.
 If the tree is closed, P is truthfunctionally false.
 If the tree is open, P is NOT truthfunctionally false.

How do you use truthtrees to test for the truthfunctional indeterminacy of P?
 Remember: P is truthfunctionally 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.

What is an alternate way to use truthtrees to test for truthfunctional truth, falsehood, and indeterminacy?
 Do a truthtree with sentence P at the top.
 If the tree closes, then P is truthfunctionally 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 truthfunctionally 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 truthfunctionally indeterminate (since then it is true on some but not all TVAs).

How do you use truthtrees to test for the truthfunctional equivalence of P and Q?
 Remember: P and Q are truthfunctionally 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.

How do you use truthtrees to test for truthfunctional 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.

How do you use truthtrees to test for truthfunctional 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 truthtree 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.

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

