F (for falsity). So by using the propositional variables A and B, the binary connectives and representing disjunction and conjunction respectively, and the unary connective representing negation, the following formula can be obtained:. A valuation here must assign to each of A and B either T or F. But no matter how this assignment is made, the overall formula will come out true. For if the first conjunction is not satisfied by a particular valuation, then one of A and B is assigned F, which will make one of the following disjunct to be assigned T. Definition and examples A formula of propositional logic is a tautology if the formula itself is always true, regardless of which valuation is used for the propositional variables. There are infinitely many tautologies. Examples include: ("A or not A"), the law of excluded middle. This formula has only one propositional variable, A. Any valuation for this formula must, by definition, assign A one of the truth values true or false, and assign A the other truth value. For instance, "The cat is black or the cat is not black". ("if A implies B, then not-B implies not-A", and vice versa), which expresses the law of contraposition. For instance, "If it's a book, it is blue; if it's not blue, it's not a book." ("if not-A implies both B and its negation not-B, then not-A must be false, then A must be true"), which is the principle known as reductio ad absurdum. For instance, "If it's not blue, it's a book, if it's not blue, it's also not a book, so it is blue." ("if not both A and B, then not-A or not-B", and vice versa), which is known as De Morgan's law. "If it is not both blue and a book, then it's either not a book or it's not blue" ("if A implies B and B implies C, then A implies C"), which is the principle known as syllogism. "If it's a book, then it's blue and if it's blue, then it's on that shelf, then if it's a book, it's on that shelf." ("if at least one of A or B is true, and each implies C, then C must be true as well"), which is the principle known as proof by cases. "Books and blue things are on that shelf. If it's either a book or it's blue, it's on