Laws of Logic

Mathematical Logic · Axiom Academy

Master the fundamental laws of propositional logic through animated truth table proofs. These laws form the foundation for logical reasoning and Boolean algebra. The identity laws state that ANDing with truth (T) or ORing with falsehood (F) leaves a proposition unchanged—just like multiplying by 1 or adding 0 in arithmetic. ORing with truth always gives truth (since at least one disjunct is true), and ANDing with falsehood always gives falsehood (since at least one conjunct is false). The word "idempotent" means "same power." Just as x · x ≠ x in algebra (except for special values), in logic p ∧ p = p and p ∨ p = p always hold. Repeating a statement doesn't change its truth value. Just as −(−x) = x in arithmetic, ¬¬p ≡ p in logic. Two negations cancel each other out. "It is not the case that it is not raining" means "It is raining." Just as 3 + 5 = 5 + 3 in arithmetic, logical operations are symmetric. "It's raining AND it's cold" means the same as "It's cold AND it's raining." Just as (2 + 3) + 4 = 2 + (3 + 4), we have (p ∧ q) ∧ r ≡ p ∧ (q ∧ r). This means we can write p ∧ q ∧ r without ambiguity. In arithmetic, multiplication distributes over addition: a(b + c) = ab + ac. In logic, both operations distribute over each other! This is more powerful than in arithmetic, where we don't have a + (b · c) = (a + b)(a + c).

This is the written version of the interactive lesson above. See the full Mathematical Logic course.