Propositional Logic: Truth Functions and Truth Tables
Truth functions
A compound proposition’s truth value depends only on the truth values of its components. The five connectives: negation (~P), conjunction (P • Q), disjunction (P ∨ Q, inclusive), conditional (P ⊃ Q) and biconditional (P ≡ Q).
Truth conditions
- ~P is true when P is false.
- P • Q is true only when both conjuncts are true.
- P ∨ Q is false only when both disjuncts are false.
- P ⊃ Q is false only when P is true and Q is false.
- P ≡ Q is true when both components have the same value.
Truth tables
With n distinct statement letters a table needs 2ⁿ rows. A form is a tautology if true on every row, a contradiction if false on every row, and contingent otherwise. An argument is valid when no row makes all premises true and the conclusion false.
Worked example
Test: P ⊃ Q, ~Q ∴ ~P. | P | Q | P ⊃ Q | ~Q | ~P | | T | T | T | F | F | | T | F | F | T | F | | F | T | T | F | T | | F | F | T | T | T | No row has both premises true and the conclusion false; the argument is valid (Modus Tollens).
Traps that cost marks
The material conditional is not causal: a false antecedent makes “If P then Q” true. “Unless” is inclusive disjunction, not exclusive. Identify the main operator before building columns; P ⊃ Q ∨ R without brackets is misread.