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