Term
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Definition
Sentence P of SL is truth-functionally true iff P is true on EVERY truth-value assignment
-Sentence that is logically true on truth function grounds
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Term
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Definition
Sentence P of sentence logic is truth functionally false if and only if P is false on every truth value assignment
-aka a contradiction or contradictory sentence
-senetence that is logically false on truth functional grounds
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D
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~ D
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D & ~ D
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T
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F
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T
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T
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F
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F
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T
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F
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T
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F
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F
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F
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T
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F
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Term
| Truth Functional Indeterminacy |
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Definition
Sentence of P of Sentence Logic is truth-functionally indeterminate if and only if P is neither truth functionally true or truth functionally false
- true on at least one value truth value assignment; false on at least one truth-value assignment
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Term
| Shortened Truth Tables are Appropriate to: |
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Definition
-Show that a sentence is not truth functionally false, you need only show it true on one truth value assignmenttrue
-Show that a sentence is not truth functionally true, need only to show it false on one truth-value assignment
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Term
| Truth Functional Equivalence |
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Definition
P and Q are truth functionally equivalent if and only if there is no truth value assignemnt on which P and Q have different truth values
-Truth-functionally true sentences are truth-functionally equivalent (all true); truth-functionally false sentences are truth-functionally equivalent (all false), while truth-functionally indeterminate may be and may fail to be. |
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Term
| Truth functional consistency for a set of sentences |
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Definition
A set of sentences in sentence logic is truth functionally consistent if and only if there is AT LEAST ONE truth value assignment on which all the memebers of the set are true
While a set of sentences is truth functionally inconsistent if and only if it is not truth functionally consistent
Can use short method in order to determine if a set is truth functionally consistent by assigning T to all the sentences and seeing if it can be done, if it can it is truth functionally consistent |
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Term
| Truth Functional Entailment |
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Definition
A set of (GAMMA half t thing) sentences os sentence logic truth functionally entails a sentence P if and only if there is no truth value assignment on which every memebr of (GAMMA) is true and P is false
Γ ⊨ P is read Γ truth-functionally entails P
{A, B & C } ⊨ ‘B’ reads { A, B & C } truth-functionally entails ‘B’ |
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Term
| Example of Truth Functional Entailment |
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Definition
Γ and P. {A, B&C} ⊨ B is shown here:
↓ ↓ ↓
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A B C
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A B & C B
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T
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T
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T
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T
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T
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T
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T
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T
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T
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T
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F
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T
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T
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F
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F
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T
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T
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F
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T
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T
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F
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F
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T
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F
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T
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F
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F
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T
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F
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F
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F
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F
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F
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T
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T
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F
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T
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T
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T
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T
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F
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T
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F
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F
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T
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F
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F
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T
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F
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F
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T
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F
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F
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F
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T
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F
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F
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F
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F
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F
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F
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F
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F
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F
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Term
| Truth Functional validity |
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Definition
-An argument of sentence logic is truth functionally valid if and only if there is no truth value assignment on which all the premises are true and the conclusion is false
-An argument of sentence logic is truth functionally invalid if and only if it is not truth functionally valid
Example:
F ≡ G
F v G
F & G
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F G
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F ≡ G F v G F & G
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T
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T
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T
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T
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T
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T
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T
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T
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T
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T
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T
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T
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F
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T
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F
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F
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T
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T
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F
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T
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F
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F
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F
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T
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F
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F
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T
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F
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T
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T
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F
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F
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T
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F
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F
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F
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T
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F
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F
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F
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F
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F
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F
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F
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Term
Trth Functional Invalidity- Example for:
D ≡ (~ W v G)
G ≡ ~ D_______
~ D |
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Definition
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D G W
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D ≡ (~ W v G) G ≡ ~ D ~ D
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T
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T
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T
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T
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T
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F
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T
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T
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T
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T
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F
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F
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T
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F
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T
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T
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T
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F
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T
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T
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T
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F
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T
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T
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T
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F
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F
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T
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F
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T
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T
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F
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T
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T
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F
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F
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T
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F
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F
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F
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T
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F
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T
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F
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T
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T
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F
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F
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T
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T
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T
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F
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T
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F
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F
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T
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F
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T
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F
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T
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F
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T
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T
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F
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F
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F
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T
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T
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T
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T
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T
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T
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F
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T
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F
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F
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T
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F
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F
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F
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T
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F
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T
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T
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T
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T
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T
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F
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T
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F
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F
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F
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T
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F
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T
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F
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T
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F
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F
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F
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F
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