Sequent forms of Herbrand theorem and their applications

Abstract

New sequent forms * of the famous Herbrand theorem are proved for first-order classical logic without equality. These forms use the original notion of an admissible substitution and a certain modification of the Herbrand universe, which is constructed from constants, special variables, and functional symbols occurring only in the signature of an initial theory. Other well-known forms of the Herbrand theorem are obtained as special cases of the sequent ones. Besides, the sequent forms give an approach to the construction and theoretical investigation of computer-oriented calculi for efficient logical inference search in the signature of an initial theory. In a comparably simple way, they provide us with some technique for proving the completeness and soundness of the calculi.

On Herbrand’s Theorem for Intuitionistic Logic

“…As a consequence of Theorem 1, Fφ is not deducible in in T J. Note Fφ is also not classically deducible as shown in [12].…”

“…This section develops the ideas suggested in [12]: Given a first-order intuitionistic formula, we generate ground instances of this formula and then check whether the instances are deducible in a propositional intuitionistic tableaux calculus, provided that the propositional proof is compatible with the way how the instances were generated. First, we introduce a specialised convolution calculus, which allows one to "gather" the required multiple occurrences of subformulae.…”

“…Now we introduce our modification of the notions of the multiplicity [2,25] and the Herbrand quasi-universe [12]. Let φ be a formula and φ 1 , .…”

“…The notion of a Herbrand quasi-universe, introduced in [12] and modified here for the intuitionistic case, plays the same role in our research as the usual Herbrand universe in the case of classical logic. Unlike the "usual" Herbrand universe, the quasi-universe also contains parameters in the case where strong variables occur in an initial sequent.…”

On Herbrand-like Theorems for Cut-free Modal Sequent Logics

“…To every sequent S in Tr, we assign the sequent ι Tr (S) or simply ι(S) (an analogue of what is called the spur of S in [18]) as follows:…”