1.3. Connectives
Connectives build compound propositions from simpler ones.
Symbol | Name | Reading |
|---|---|---|
¬P | negation | not P |
P ∧ Q | conjunction | P and Q |
P ∨ Q | disjunction | P or Q |
P → Q | implication | if P then Q |
P ↔ Q | biconditional | P if and only if Q |
The truth value of a compound proposition depends only on the truth values of its parts. The table below defines the five connectives, with T for true and F for false.
P | Q | ¬P | P ∧ Q | P ∨ Q | P → Q | P ↔ Q |
|---|---|---|---|---|---|---|
T | T | F | T | T | T | T |
T | F | F | F | T | F | F |
F | T | T | F | T | T | F |
F | F | T | F | F | T | T |
Two rows of the implication column deserve attention. When P is false, P → Q is true regardless of Q. An implication claims nothing about cases where its antecedent fails, so those cases cannot refute it. Disjunction is inclusive, so P ∨ Q is true when both disjuncts are.