Tautology

A tautology in logic is a formula that is always true on any valuation or interpretation of its terms. They are also sometimes called valid formulas (not to be confused with a valid argument) or logical truths

The most obvious and commonly used example of a tautology is the formula A ∨ ¬A. Under any valuation, whether A is true or A is false, A or not-A will always be a true statement.

We can easily verify that A ∨ ¬A is a tautology by means of a truth table:

A tautology may otherwise be defined as a formula that is satisfied under every possible valuation.

We may formally indicate that a formula, φ, is a tautology by φ. The symbol ⊤ or the letter “T” is also used to indicate a tautology.

“Tautology”, the term

The term tautology is originally used in rhetoric to refer to statements that are in-themselves redundant. For example, the phrase “unsolved mystery” is a rhetorical tautology because any mystery is unsolved — the adjective is unneccessary and adds no meaning to the phrase. The same thing commonly happens with acronyms. For example, the acronym ATM means “automatic teller machine”. So, the common phrase “ATM machine” is a rhetorical tautology, as they are essentially saying “automated teller machine machine”. The same thing happens with “PDF format”, “PIN number” and “UPC code”.

This is, of course, different from a logical tautology, but the term migrated to logic by the claim that logical tautologies are essentially meaningless statements. They provide no information, or at least no new information. To make the tautologous claim “There is either a hat on my head, or there is not a hat on my head” is to say something true while saying nothing meaningful. We learn nothing from that statement about the state of hats and heads.

The use of the term tautology in propositional logic can be attributed to Wittgenstein.