Tautologies and Contradictions A tautology is a statement that is always true, regardless of the truth values of the other statements in a logical system...
A tautology is a statement that is always true, regardless of the truth values of the other statements in a logical system. In other words, a tautology is a statement that is true by virtue of its own truth value, not because it is true based on some other set of facts.
For example, the statement "(A ∧ B) ∧ (A ∨ C)" is a tautology. This statement is true regardless of whether A, B, or C are true or false.
A contradiction is a statement that is always false. In other words, a contradiction is a statement that is false regardless of the truth values of the other statements in a logical system.
For example, the statement "A ∧ (¬A)" is a contradiction. This statement is false regardless of whether A is true or false.
Tautologies and contradictions are closely related to the concept of logical equivalence. Two statements are logically equivalent if they are both tautologies or both contradictions. This means that they have the same truth value, regardless of the truth values of the other statements in a logical system.
For example, the statements "(A ∨ B) ∧ (A ∨ C)" and "(A ∧ B) ∧ (¬C)" are logically equivalent. This is because both statements are tautologies, meaning they are true regardless of the truth values of the other statements.
Tautologies and contradictions are important concepts in logic because they help to define the truth values of statements and to determine when two statements are logically equivalent