Herbrand's Theorem

Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic, the simpler version shown here, restricted to formulas in prenex form containing only existential quantifiers became more popular.

Let

be a formula of first-order logic with

quantifier-free.

Then

is valid if and only if there exists a finite sequence of terms:, with

and ,

such that

is valid. If it is valid,

is called a Herbrand disjunction for

.

Informally: a formula in prenex form containing existential quantifiers only is provable (valid) in first-order logic if and only if a disjunction composed of substitution instances of the quantifier-free subformula of is a tautology (propositionally derivable).

The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization. Conversion to prenex form can be avoided, if structural Herbrandization is performed. Herbrandization can be avoided by imposing additional restrictions on the variable dependencies allowed in the Herbrand disjunction.