Overview
Logical connectives — also called logical operators or propositional connectives — are symbols that link propositions together to form compound propositions. The truth value of a compound proposition is determined entirely by the truth values of its components and the connective used.
In classical propositional logic, every proposition is either true or false, and connectives are defined by their truth-functional behavior: the output depends only on the inputs.
{T, F} and connectives are truth-functional.
Conjunction (∧)
The conjunction of two propositions P and Q, written P ∧ Q (read "P and Q"), is true if and only if both P and Q are true.
P ∧ Q ≡ true iff P = true and Q = true
P ∧ Q ≡ false otherwiseTruth Table
| P | Q | P ∧ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
P ∧ Q = "It is raining and I have an umbrella," which is true only when both conditions hold simultaneously.
Disjunction (∨)
The disjunction of P and Q, written P ∨ Q (read "P or Q"), is true when at least one of the propositions is true. This is the inclusive or.
P ∨ Q ≡ false iff P = false and Q = false
P ∨ Q ≡ true otherwiseTruth Table
| P | Q | P ∨ Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Negation (¬)
Negation is the unary connective that inverts the truth value of a proposition. The negation of P, written ¬P (also written ~P or !P), is true exactly when P is false.
¬P ≡ true iff P = false
¬P ≡ false iff P = true¬¬P ≡ P) holds in classical logic but is rejected in intuitionistic logic, where it is replaced by the weaker rule that ¬¬P does not imply P.
Implication (→)
Material implication, written P → Q (read "if P then Q" or "P implies Q"), is false only when P is true and Q is false. This is the material conditional and differs from everyday causal implication.
P → Q ≡ ¬P ∨ Q
P → Q is false iff P = true and Q = false
P → Q is true otherwiseTruth Table
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
P → Q is true regardless of Q. This is called vacuous truth and is a well-known counterintuitive feature of material implication.
Biconditional (↔)
The biconditional P ↔ Q (read "P if and only if Q", abbreviated "P iff Q") is true when P and Q have the same truth value.
P ↔ Q ≡ (P → Q) ∧ (Q → P)
P ↔ Q ≡ true iff P = QTruth Table
| P | Q | P ↔ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Operator Precedence
When parentheses are absent, connectives bind according to a standard precedence order, highest to lowest:
1. ¬ (negation — highest, unary)
2. ∧ (conjunction)
3. ∨ (disjunction)
4. → (implication — right-associative)
5. ↔ (biconditional — lowest)Thus ¬P ∧ Q → R is parsed as (¬P ∧ Q) → R, and P → Q → R as P → (Q → R).
Key Laws
Connectives satisfy a rich system of identities that enable algebraic manipulation of logical formulas:
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬(P ∨ Q) ≡ ¬P ∧ ¬QP ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)P ∧ (P ∨ Q) ≡ P
P ∨ (P ∧ Q) ≡ P