Tautology

In logic, a tautology is a formula that is true under every possible assignment of truth values to its propositional variables. In classical logic, its truth depends on its logical structure rather than on the particular facts represented by its components. A standard example is “It is raining or it is not raining,” represented as $p \lor \neg p$. The term also has a distinct ordinary-language meaning: unnecessary repetition of the same meaning in different words. These logical and linguistic uses should not be confused. (openlogicproject.org)

Definition and truth-table evaluation

In propositional logic, variables such as $p$ and $q$ stand for propositions. A valuation assigns each variable either true or false; the values of compound formulas then follow from the rules for their connectives. Under this semantics, a formula $A$ is a tautology precisely when every valuation makes it true. This is conventionally written $\models A$, indicating that $A$ holds without premises. (upload.wikimedia.org)

A truth table provides a direct test. For the law of excluded middle, the table is:

Because the final column contains only true values, the formula is a tautology. Showing that a formula is not a tautology requires only one countervaluation: an assignment under which it is false. Thus $p \lor q$ is not a tautology, because it is false when both variables are false. The test concerns all permitted assignments, not merely the truth values that happen to obtain in a particular situation. (openlogicproject.org)

Examples and related classifications

Other classical examples include $p \to p$, expressing self-implication, and $\neg(p \land \neg p)$, expressing noncontradiction. A less immediately obvious example is

Here the arrow represents material implication, which is false only when its antecedent is true and its consequent false. The whole formula is always true: whenever its antecedent holds, both $p$ and $p \to q$ hold, requiring $q$ to hold as well. It expresses the truth-preserving structure of modus ponens. (logicmatters.net)

A contradiction is false under every valuation; $p \land \neg p$ is an example. A contingent formula is true under some valuations and false under others. Satisfiability requires truth under at least one valuation, so every tautology is satisfiable, but not every satisfiable formula is tautological. In classical propositional logic, $A$ is a tautology exactly when $\neg A$ is unsatisfiable. These distinctions classify formulas by their complete range of possible evaluations rather than their actual truth alone. (forallx.openlogicproject.org)

Consequence, equivalence, and proof

Tautology is closely connected with logical validity, but formulas and arguments are different objects. An argument is valid when no valuation makes all its premises true and its conclusion false. For finitely many premises $P_1,\ldots,P_n$ and conclusion $C$, this amounts to the conditional

being a tautology. The conclusion need not itself be tautological: the argument from $p$ and $p\to q$ to $q$ is valid, although $q$ alone can be false. This distinction is fundamental to deductive reasoning. (logicmatters.net)

Two formulas are logically equivalent when they receive the same truth value under every valuation. Equivalently, their biconditional is a tautology. This gives a precise way to establish that differently written formulas express the same truth-functional conditions. (forallx.openlogicproject.org)

Tautology is a semantic notion, whereas a theorem is a syntactic notion: something derivable within a specified formal system. In a sound and complete classical propositional calculus, the formulas derivable without premises are exactly the tautologies. Soundness ensures that derivations do not produce non-tautologies from no premises; completeness ensures that every tautology has a formal proof. Such proofs may use natural deduction rather than truth tables. (forallx.openlogicproject.org)

Scope and alternative logics

In first-order logic, validity requires truth in every interpretation of predicates, names, and quantifiers. Under a narrower terminology, “tautology” is reserved for validity arising solely from truth-functional structure. For example, $\forall x\,P(x)\to P(a)$ is first-order valid, but treating its two components as independent propositional variables produces $p\to q$, which is not tautological. Quantifier structure therefore establishes validities beyond propositional tautologies. (forallx.openlogicproject.org)

The relevant logical framework matters. Intuitionistic logic does not generally validate $p\lor\neg p$, although it validates particular instances and the double-negated formula $\neg\neg(p\lor\neg p)$. In many-valued systems, validity is defined using their permitted valuations and designated truth values. Consequently, a classical tautology need not remain valid when the semantics changes. (builds.openlogicproject.org)

Philosophical and linguistic uses

Ludwig Wittgenstein gave tautologies a central role in the Tractatus Logico-Philosophicus. In propositions 4.46–4.463, he distinguishes tautologies from statements that represent particular possible situations. A tautology permits every possible situation and therefore supplies no factual information about which situation obtains. Nevertheless, it belongs to logical symbolism rather than being a meaningless string. Proposition 6.1 identifies the propositions of logic as tautologies. This account connects logical necessity with the structure of representation. (gutenberg.org)

In ordinary language, “tautology” instead denotes redundant wording that repeats a meaning. That usage concerns expression, not truth under every valuation. Verbal repetition need not constitute a logical tautology, and a logical tautology need not repeat any words. The two senses distinguish redundancy in phrasing from unconditional truth within a logical framework. (dictionary.cambridge.org)