Herbrand Theorems: the Classical and Intuitionistic Cases

A unified approach is applied for the construction of sequent forms of the famous Herbrand theorem for first-order classical and intuitionistic logics without equality. The forms do not explore skolemization, have wording on deducibility, and as usual, provide a reduction of deducibility in the first-order logics to deducibility in their propositional fragments. They use the original notions of admissibility, compatibility, a Herbrand extension, and a Herbrand universe being constructed from constants, special variables, and functional symbols occurring in the signature of a formula under investigation. The ideas utilized in the research may be applied for the construction and theoretical investigations of various computer-oriented calculi for efficient logical inference search without skolemization in both classical and intuitionistic logics and provide some new technique for further development of methods for automated reasoning in non-classical logics.