Tautologies and Contradictions

Mathematical Logic · Axiom Academy

LESSON Tautologies and Contradictions Understanding special types of propositions in propositional logic The most fundamental tautology is the Law of Excluded Middle : This says: either a proposition is true, or its negation is true. There is no middle ground. Let's verify this with a truth table animation: 2. Contradiction: Always False The simplest contradiction is a proposition and its negation joined by "and": This says: a proposition and its negation are both true simultaneously. This is impossible! Watch the truth table verification: Key examples of contradictions: False implication: p p q is always true (vacuously) 3. Contingency: Sometimes True, Sometimes False Most ordinary propositions are contingencies. For example: This implication is true in most cases, but false when p is true and q is false. Let's see this pattern in action: Classification method using truth tables: If the final column is all T → Tautology If the final column is all F → Contradiction If the final column has both T and F → Contingency 4. Famous Tautologies in Logic Certain tautologies appear repeatedly in mathematical proofs and logical reasoning. These are the workhorses of logic: Implication equivalence: p q p q Distributive laws: p (q r) (p q) (p r) Law of Syllogism: ((p q) (q r)) (p r) 5. Tautologies and Valid Arguments There is a deep connection between tautologies and valid logical arguments: For example, consider the classic Modus Ponens argument: Other tautological argument forms:

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