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If the truth value of a sentence is true, the truth value of its negation is false. If the truth value of a sentence is false, the truth value of its negation is true.
The truth value of a conjunction is true if and only if the truth value of its conjuncts are both true; otherwise, the truth value is false.
| φ |
ψ |
φ ∧ ψ |
| 1 |
1 |
1 |
| 1 |
0 |
0 |
| 0 |
1 |
0 |
| 0 |
0 |
0 |
The truth value of a disjunction is true if and only if the truth value of at least one its disjuncts is true; otherwise, the truth value is false. Note that this is the inclusive or interpretation of the ∨ operator and is differentiated from the exclusive or interpretation in which a disjunction is true if and only if an odd number of its disjuncts are true.
| φ |
ψ |
φ ∨ ψ |
| 1 |
1 |
1 |
| 1 |
0 |
1 |
| 0 |
1 |
1 |
| 0 |
0 |
0 |
The truth value of an implication is false if and only if its antecedent is true and its consequent is false; otherwise, the truth value is true. This is called material implication.
| φ |
ψ |
φ ⇒ ψ |
| 1 |
1 |
1 |
| 1 |
0 |
0 |
| 0 |
1 |
1 |
| 0 |
0 |
1 |
A biconditional is true if and only if the truth values of its constituents agree, i.e. they are either both true or both false.
| φ |
ψ |
φ ⇔ ψ |
| 1 |
1 |
1 |
| 1 |
0 |
0 |
| 0 |
1 |
0 |
| 0 |
0 |
1 |
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