Tautology
From Logic
In Propositional Logic, a tautology is a statement that is truth-functionally valid—i.e. it is universally true, or true in every interpretation). For example, the statement "If it rains, then it rains" is a tautology. Every theorem of propositional logic is a tautology, and so we can equivalently define 'tautology' as any theorem of propositional logic—i.e. any statement that is deducible from the empty set in some system of deduction of propositional logic, such as a natural deduction system. The term is often mistakenly applied to any validity (or theorem) of first-order logic, though it applies only to a proper subset of such validities. The term was originally introduced by Ludwig Wittgenstein.
The negation of a tautology is clearly a contradiction, and the negation of a contradiction is clearly a tautology. A sentence that is neither a tautology (always true) nor a contradiction (always false) is logically contingent, i.e., possible of being true or false .
Discovering tautologies
An effective procedure for checking whether a propositional formula is a tautology or not is by means of a Truth Table
