1.3. Connectives
Connectives build compound propositions from simpler ones, and Table 1.3.1 names the five.
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 |
Table 1.3.1. The five connectives, with their symbols and readings.
The truth value of a compound proposition depends only on the truth values of its parts. Table 1.3.2 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 |
Table 1.3.2. The truth values of the five connectives.
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.
The disjunction column deserves the same attention. Disjunction is inclusive, so P ∨ Q is true when both disjuncts are.