Herbrand's theorem [SEP] named after
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 formula_2 quantifier-free. This version of Herbrand's theorem states that
relevant substitution instances of formula_2 in the right side of the sequent, thus resulting in a proof of formula_13, from which the Herbrand disjunction can be obtained.
However, sequent calculus and cut-elimination were not known at the time of Herbrand's theorem, and Herbrand had to prove his theorem in a more complicated way.
Generalizations of Herbrand's theorem.
- Herbrand's theorem has been extended to higher-order logic by using expansion-tree proofs. The deep representation of expansion-tree proofs corresponds to