tautology a proposition whose negation is inconsistent, or (self-) contradictory, e.g. ‘Socrates is Socrates’, ‘Every human is either male or nonmale’, ‘No human is both male and non-male’, ‘Every human is identical to itself’, ‘If Socrates is human then Socrates is human’. A proposition that is (or is logically equivalent to) the negation of a tautology is called a (self-)contradiction. According to classical logic, the property of being implied by its own negation is a necessary and sufficient condition for being a tautology and the property of implying its own negation is a necessary and sufficient condition for being a contradiction. Tautologies are logically necessary and contradictions are logically impossible.
Epistemically, every proposition that can be known to be true by purely logical reasoning is a tautology and every proposition that can be known to be false by purely logical reasoning is a contradiction. The converses of these two statements are both controversial among classical logicians. Every proposition in the same logical form as a tautology is a tautology and every proposition in the same logical form as a contradiction is a contradiction. For this reason sometimes a tautology is said to be true in virtue of form and a contradiction is said to be false in virtue of form; being a tautology and being a contradiction (tautologousness and contradictoriness) are formal properties. Since the logical form of a proposition is determined by its logical terms (‘every’, ‘some’, ‘is’, etc.), a tautology is sometimes said to be true in virtue of its logical terms and likewise mutatis mutandis for a contradiction.
Since tautologies do not exclude any logical possibilities they are sometimes said to be ’empty’ or ‘uninformative’; and there is a tendency even to deny that they are genuine propositions and that knowledge of them is genuine knowledge. Since each contradiction ‘includes’ (implies) all logical possibilities (which of course are jointly inconsistent), contradictions are sometimes said to be ‘overinformative.’ Tautologies and contradictions are sometimes said to be ‘useless,’ but for opposite reasons. More precisely, according to classical logic, being implied by each and every proposition is necessary and sufficient for being a tautology and, coordinately, implying each and every proposition is necessary and sufficient for being a contradiction.
Certain developments in mathematical logic, especially model theory and modal logic, seem to support use of Leibniz’s expression ‘true in all possible worlds’ in connection with tautologies. There is a special subclass of tautologies called truth-functional tautologies that are true in virtue of a special subclass of logical terms called truthfunctional connectives (‘and’, ‘or’, ‘not’, ‘if’, etc.). Some logical writings use ‘tautology’ exclusively for truth-functional tautologies and thus replace ‘tautology’ in its broad sense by another expression, e.g. ‘logical truth’. Tarski, Gödel, Russell, and many other logicians have used the word in its broad sense, but use of it in its narrow sense is widespread and entirely acceptable. Propositions known to be tautologies are often given as examples of a priori knowledge. In philosophy of mathematics, the logistic hypothesis of logicism is the proposition that every true proposition of pure mathematics is a tautology. Some writers make a sharp distinction between the formal property of being a tautology and the non-formal metalogical property of being a law of logic. For example, ‘One is one’ is not metalogical but it is a tautology, whereas ‘No tautology is a contradiction’ is metalogical but is not a tautology. See also LAWS OF THOUGHT, LOGICAL FORM , LOGICIS. J.Cor.