Propositional logic builds complex statements from simple ones using connectives. Each connective has a precise meaning — a rule for when the compound statement is true or false — regardless of what the component propositions are about.
~
Negation
Negation
also: ~P · not P · it is not the case that P
Negation flips the truth value of a single proposition. True becomes false, false becomes true. It is the only unary connective — it operates on just one statement.
True when P is false · False when P is true
"It is not raining." "It is not the case that it is raining."
~R (where R = "It is raining")
"She is not happy."
~H
∧
Conjunction
Conjunction
also: P · Q · P and Q · both P and Q
Conjunction joins two propositions with "and." The whole statement is true only when both conjuncts are true. If either is false, the conjunction is false.
True only when both P and Q are true
"It is raining and it is cold."
R ∧ C
"She studied hard and passed the exam."
S ∧ P
∨
Disjunction
Disjunction
also: P ∨ Q · P or Q · at least one of P, Q
Disjunction joins two propositions with "or." In logic, "or" is inclusive — the statement is true when at least one disjunct is true, including when both are true. It is only false when both are false.
False only when both P and Q are false
"It is raining or it is snowing."
R ∨ S
"You need a ticket or a pass to enter."
T ∨ P
→
Conditional
Conditional
also: P ⊃ Q · if P then Q · P only if Q
The conditional expresses an "if…then" relationship. P is the antecedent, Q is the consequent. The whole statement is false only when the antecedent is true and the consequent is false — a true condition that fails to deliver. In all other cases it is true.
False only when P is true and Q is false
"If it is raining, then the ground is wet."
R → W
"If you study, you will pass."
S → P
↔
Biconditional
Biconditional
also: P ≡ Q · P if and only if Q · P iff Q
The biconditional says P and Q have the same truth value. It is true when both are true or both are false — and false when they differ. Think of it as a conditional running in both directions: "if P then Q, and if Q then P."
True when P and Q have the same truth value
"You pass if and only if you score above 70."
P ↔ S
"The light is on if and only if the switch is up."
L ↔ U
On the inclusive "or": In everyday English, "or" is sometimes exclusive ("soup or salad" — not both). In propositional logic, disjunction is always inclusive — both being true is fine. If you need to express exclusive or, it takes a more complex formula: (P ∨ Q) ∧ ~(P ∧ Q).