Propositional Logic: Truth Functions and Truth Tables
Connectives and truth conditions
Propositional logic treats simple propositions as truth-functional units. Negation (¬P) reverses truth value. Conjunction (P ∧ Q) is true only when both conjuncts are true. Disjunction (P ∨ Q) is inclusive: false only when both disjuncts are false. The conditional (P → Q) is false only when P is true and Q is false. The biconditional (P ↔ Q) is true when truth values match.
Truth tables
A truth table lists every assignment of truth values to atomic propositions. A formula with all true final values is a tautology; all false, a contradiction; mixed, contingent. An argument form is valid when there is no row in which all premises are true and the conclusion false.
Worked example
Test (P → Q) ∧ P → Q. In the row P true, Q false, P → Q is false, so the conjunction is false; the conditional has a false antecedent and is true. In every row where (P → Q) ∧ P is true, Q is true. Hence the form is valid: modus ponens.
Traps that cost marks
- Treating “or” as exclusive when the standard symbol is inclusive.
- Thinking a conditional is false whenever Q is false; only T → F is false.
- Ignoring operator scope and precedence.
- Confusing a contingent formula with a tautology.
- Checking only one row instead of all relevant rows.