Truth tables, connectives, tautologies, and equivalence
Propositional logic is the study of how truth values combine through logical connectives: AND, OR, NOT, IMPLIES, and IF-AND-ONLY-IF. Every compound proposition has a definite truth value determined entirely by the truth values of its atomic parts. This is the foundation of all formal reasoning.
In this lesson, you will build truth tables, explore each connective interactively, and check whether two formulas are logically equivalent.
A truth table lists every possible combination of truth values for the variables and evaluates the formula for each. If the result column is all true, the formula is a tautology. If all false, a contradiction. Otherwise, it is a contingency.
Select a formula to generate its truth table. Tautologies are always true; contradictions always false.
Key insight: Truth tables provide a mechanical, foolproof method for determining validity. The cost is exponential: n variables require 2ⁿ rows.
Each logical connective has a precise truth-functional definition. The most counterintuitive is material implication (→): "if P then Q" is false only when P is true and Q is false. A false premise implies anything. Toggle inputs to build intuition for each connective.
Click the colored circles to toggle inputs between True and False. The output updates automatically based on the connective.
Key insight: Material implication differs from everyday "if-then" because it has no causal requirement. "If 2+2=5 then the moon is made of cheese" is logically true.
Two formulas are logically equivalent if they have the same truth value under every assignment. For example, p → q is equivalent to ¬p ∨ q. Compare truth tables column by column to verify equivalences or find counterexamples.
Compare two formulas row by row. Try p \u2192 q vs \u00ACp \u2228 q (equivalent), or \u00AC(p \u2227 q) vs \u00ACp \u2228 \u00ACq (De Morgan).
Key insight: De Morgan's laws, double negation, and implication elimination are among the most important equivalences. They form the basis for converting formulas into normal forms.