Introduction to Logic

 Satisfaction

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.

φ ¬φ
1 0
0 1

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