Propositional Logic

Truth tables, connectives, tautologies, and equivalence

Propositional Logic

Every statement is either true or false, and a handful of connectives decide how those truth values combine. Pick formulas, flip inputs, and watch the truth tables fill in.

Truth Table Generator

Pick a formula and read its result column. All true means atautology, all false a contradiction, and a mix a contingency.

pqp ∧ qFFFFTFTFFTTTContingency

Select a formula to generate its truth table. Tautologies are always true; contradictions always false.

Go deeper:Truth tables

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.

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.

Key insight: Truth tables provide a mechanical, foolproof method for determining validity. The cost is exponential: n variables require 2ⁿ rows.

Connective Explorer

Click the inputs on each card to toggle them. Watch IMPLIES closely: it is false only when the first input is true and the second is false.

AND (∧)TF∧FBoth must be trueclick inputs to toggleOR (∨)TF∨TAt least one must be trueclick inputs to toggleNOT (¬)TFFlips the truth valueclick inputs to toggleIMPLIES (→)TF→FIf A then BThe only false case: T → Fclick inputs to toggleIFF (↔)TF↔FBoth must matchclick inputs to toggleXOR (⊕)TF⊕TExactly one must be trueclick inputs to toggle

Click the colored circles to toggle inputs between True and False. The output updates automatically based on the connective.

Go deeper:Material implication

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.

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.

Logical Equivalence Checker

Choose Formula A and Formula B and compare their columns row by row. If they match on every row, the two formulas are equivalent.

pqAB=?FFTT✓FTTT✓TFFF✓TTTT✓Logically Equivalent ✓

Compare two formulas row by row. Try p → q vs ¬p ∨ q (equivalent), or ¬(p ∧ q) vs ¬p ∨ ¬q (De Morgan).

Go deeper:Logical equivalence

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.

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.

Key Takeaways

  • Propositional logic is truth-functional: the truth value of a compound depends only on the truth values of its parts.
  • Truth tables mechanically determine tautologies, contradictions, and contingencies.
  • Material implication is true whenever the premise is false -- this is the hardest connective to internalize.
  • Logical equivalence means identical truth tables; it is the foundation of formula simplification.

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