Tautology

From Logic

Revision as of 22:44, 18 June 2007

In Popositional 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 .

Tautologies versus validities

The use of 'tautology', however, can be extended to first-order logic since it includes propositional logic. It can be further extended to include sentences that are quantified in the following sense. Call any statement that is not a truth-functional compound (i.e. not a conjunction, disjunction, conditional, etc.) a 'Boolean atom'. Then every atomic sentence is a Boolean atom, as is every quantified sentence—i.e. those of the form Failed to parse (Can't write to or create math temp directory): \\forall x\\phi

or Failed to parse (Can't write to or create math temp directory): \\exists x\\phi

. For example, Failed to parse (Can't write to or create math temp directory): P(x)

and Failed to parse (Can't write to or create math temp directory): \\forall x(P(x)\\land Q(y))
are Boolean atoms, while Failed to parse (Can't write to or create math temp directory): \\forall xP(x)\\land Q(y)
is not. Then a statement of first-order logic is a tautology if the uniform relettering of each of its Boolean atoms yields a tautology in the propositional sense. Thus Failed to parse (Can't write to or create math temp directory): \\forall x(P(x) \\lor\\lnot P(x))
is not a tautology, since its Boolean relettering yields Failed to parse (Can't write to or create math temp directory): p

, while Failed to parse (Can't write to or create math temp directory): \\forall xP(x)\\lor\\lnot\\forall xP(x)

is a tautology. One could further extend this notion by taking statements to be equivalence classes of statements, each of which is closed under the property of its elements being variants of each other (e.g. ∀xP(x) is a variant of ∀yP(y), and likewise upon substituting any other variable for x in the former). Then the Boolean relettering of Failed to parse (Can't write to or create math temp directory): \\forall xP(x)\\lor\\lnot\\forall yP(y)
yields a tautology, since each disjunct falls under the same equivalence class.

Discovering tautologies

An effective procedure for checking whether a propositional formula is a tautology or not is by means of a Truth Table

References