In mathematical logic, a tautology is an assertion or formula that is true in every possible interpretation. An example would be “x=y or x≠y”. It is an unfalsifiable claim that is made in the realm of mathematics. A similar example would be ‘either the ball is green, or the ball is not green’, which upholds its truth irrespective of the outcome because it is inherently true.
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The philosopher Ludwig Wittgenstein was the first to apply the term tautology to redundancies of propositional logic, borrowing from the concept of rhetoric, where it is a repetitive statement. In logic, a formula can be satisfied if it is true under at least one interpretation, and hence it is a formula whose negation is not able to be satisfied. In other words, it cannot be false at all, it just simply cannot be untrue.
Unsatisfiable statements, both via affirmation as well as negation, are known as contradictions. A formula that is neither a contradiction nor a tautology is considered logically contingent. Such a formula can be made either true or false on the basis of the values assigned to its propositional variables. Tautology is symbolized by ‘Vpq’ and contradiction by ‘Opq‘. The tee symbol is used to denote an arbitrary tautology, while the dual symbol is used to represent an arbitrary contradiction. In symbolism, the best way to describe it would be to denote it using the binary notations, 1 for true and 0 for false.
Tautologies are the main concept in propositional logic, where it is defined as a propositional formula true under any possible Boolean valuation of its variables. A key property of tautologies when it comes to propositional logic is that an effective method does exist for testing whether a given formula is always satisfied.
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