Tautologies and contradictionsA compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology.Example: ? ∨ ¬?is a tautology. A compound proposition that is always false, regardless of the truth values of the individual propositions involved, is called a contradiction.Example: ? ∧ ¬?is a contradiction. A compound statement that is neither a tautology nor a contradiction is called a contingency.
Logical equivalenceWe call two algebraic expressions equal if they have the same value for each possible value of the input variables. For example, we say𝑥2− 1 = (𝑥 + 1)(𝑥 − 1)because for all real numbers 𝑥, the left side and the right side have the same value. Correspondingly, we should call two compound statements ?and ?“equal” if they always share the same truth value. However, the convention is to call them logically equivalentinstead, and to use the symbol ≡to represent logical equivalence.Using the biconditional and the concept of a tautologythat we just introduced, we can formally define logical equivalence as follows:? ≡ ?means that ? ↔ ?is a tautology.An alternative definition is that ? ≡ ?means that ?and ?share the same truth table.
“Equivalent” vs “Equal”Do not use the words “equivalent” and “equal” interchangeably . Equivalenceapplies to statements. Two quantities that are the same are equal. If you fail to keep this distinction in mind when dealing with statements that are about quantities, you are in danger of making wholly nonsensical statements. For example, the inequality 2 < x < 3 is equivalentto 4 < 2x < 6. They always share the same truth value, no matter what value is inserted for x. However, if you connect them with an equal sign, then you are making a totally different statement because the equal sign connects quantities, not statements.The notation2 < x < 3 = 4 < 2x < 6 does NOT mean that 2 < x < 3 is equivalent to 4 < 2x < 6. Rather, it is the false statement2 < x and x < 3 and 3 = 4 and 4 < 2x and 2x < 6.