Truth Functional Properties on Truth Trees
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1 Truth Functional Properties on Truth Trees
2 Truth-functional: Truth Falsity Indeterminacy Entailment Validity Equivalence Consistency
3 A set of sentences of SL is truth-functionally consistent if and only if there is at least one truth value assignment [of the constituents of the set of sentences] on which all the members of the set are true. We now know how to check for consistency using a tree, and can recover specific truthvalue assignments on which all members of a given set come out true (if the set is consistent).
4
5 A sentence of SL is truth-functionally false if and only if it is false on every possible truth-value assignment of its constituents. A sentences, is truthfunctionally false if and only if { } is truthfunctionally inconsistent. Since inconsistent sets are sets whose members can never all be true at the same time, and since { } has only one member, that member must never be true for the set to be inconsistent. Definition Explained via consistency
6 A sentence of SL is t.f. false if and only if { } has a closed truth tree (meaning { } is inconsistent). If the tree for { } closes, it means that it is impossible for to be true.
7 A sentence of SL is truth-functionally true if and only if it is true on every possible truth-value assignment of its constituents. Definition A sentence is truthfunctionally true if and only if {~ } is truthfunctionally inconsistent. Since inconsistent sets are sets whose members can never all be true, and since {~ } has only one member and is a negation, must be always true for ~ to be never true. Explained via Consistency
8 A sentence of SL is t.f. true if and only if {~ } has a closed truth tree (meaning {~ } is inconsistent). If the tree for {~ } closes, it means that it is impossible for ~ to be true, which in turn means it is impossible for to be false.
9 A sentence of SL is truth-functionally indeterminate if and only if it is neither truth-functionally true nor truthfunctionally false. A sentence is truthfunctionally indeterminate if and only if both {~ } and { } are truth-functionally consistent. Since is truth functionally true or false if one of the above sets is inconsistent. Definition Explained via consistency
10 A sentence of SL is t.f. indeterminate if and only if neither {~ } nor { } has a closed truth tree. If the tree for {~ } is open and the tree for { } is open, then it means that can be either true or false.
11 Sentences and of SL are truthfunctionally equivalent if and only if there is no truth value assignment [for the components of and ] on which and have different truth-values. Sentences and of SL are truth-functionally equivalent if and only if {~( )} is truthfunctionally inconsistent Since only truthfunctionally false sentences make sets of which they are sole members inconsistent, the negation of a sentence asserting that and have different truth-values being truth functionally false means that and must have the same truth-value. Definition Explained via consistency
12 Sentences and are equivalent if and only if {~( )} has a closed truth tree. If and are equivalent, then ( ) is truth functionally true because the two sentences always have the same truth-value. That would make ~( ) truth functionally false. So to check for equivalence of any two sentences of SL on a tree, join them with a biconditional, negate the biconditional, and check for consistency of the set with that negated biconditional as its only member.
13 A set of sentences of SL truthfunctionally entails a sentence if and only if there is no truthvalue assignment on which every member of is true and is false. if and only if {~ } is truthfunctionally inconsistent. Definition Explained via Consistency
14 A finite set entails a sentence if and only if the set {~ } has a closed tree (is inconsistent). So to check if some finite set entails some sentence, represent each member of the set along with the negation of what you re checking to see whether the set entails. If the table closes, then it is impossible for all members of the set to be true while false. is
15 Since validity is simply a special case of entailment, the same procedure can demonstrate that validity can be described in terms of consistency. If an argument is valid, then the union of the set of its premises and the negation of its conclusion will form a truth-functionally inconsistent set.
16 An argument with a finite number of premises is valid if and only if the set consisting of all and only its premises and the negation of the conclusion has a closed tree. A tree that closes when you include every premise and the negation of the conclusion means that the conclusion cannot be false while the premises are true.
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