Definition
Propositional logic is the branch of formal logic that treats whole statements — propositions — as indivisible atoms and studies how they combine into larger statements through truth-functional connectives: conjunction (and), disjunction (or), negation (not), the material conditional (if-then), and the biconditional (if and only if). It is the simplest rigorous system of logic, and the layer on which richer systems like predicate logic and modal logic are built.
The system's defining feature is that it ignores the internal structure of propositions. The statement "Socrates is mortal" is represented by a single letter, often P. Whatever makes Socrates the subject or mortal the predicate is invisible to propositional logic. What matters is only how P combines with other propositions and whether the resulting compound is true or false under every possible assignment of truth values to its atoms.
Why it matters
How it works
The system is built from atomic propositions (typically written as P, Q, R) and a small set of connectives that combine them into compound formulas. Each connective is defined by a truth table — a complete enumeration of how the truth value of the compound depends on the truth values of its parts. The conjunction P and Q is true only when both conjuncts are true. The disjunction P or Q is true when at least one is. The conditional P implies Q is false only when P is true and Q is false. Because every connective is truth-functional, the truth value of any formula, however complex, is fully determined by the values assigned to its atoms.
Validity in propositional logic is defined as truth-preservation: an argument is valid when there is no assignment of truth values that makes every premise true and the conclusion false. This is checkable by exhaustive truth-table enumeration, or — more efficiently for large formulas — by proof systems like natural deduction or resolution. The system is both sound (every provable formula is true in every interpretation) and complete (every formula true in every interpretation is provable), which makes it the gold standard of well-behaved logical systems.