On Herbrand’s Theorem for Intuitionistic Logic

Abstract

Abstract. In this paper we reduce the question of validity of a first-order intuitionistic formula without equality to generating ground instances of this formula and then checking 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. This result can be seen as a form of the Herbrand theorem, and so it provides grounds for further theoretical investigation of computer-oriented intuitionistic calculi.

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

“…But in the "pure" intuitionistic and classical cases (when Mod m = / 0), the validity of formulas has the well-known definitions and Theorems 3 and 4 can be reformulated as follows (cf. [2], [3], and [4]). …”

“…They use analogs of the notions of a Herbrand extension and a Herbrand quasi-universe considered in [2] (also see [3] and [4]). …”

“…Also draw your attention to the fact that the research is made on the base of deducibility being used in sequent calculi and we do not 1 The form A does not require skolemization and there exists a possibility to obtain its analog for some of non-classical logics, but there is doubt whether it will be a good basis for producing 'transparent" Herbrand-type reduction theorems. That is why the approach from [3] and [2] based on admissibility and compatibility seems more preferable: it permits to demonstrate a clear-cut difference between the classical and intuitionistic modal cases. use any model-theoretical approach and a relating technique in the paper.…”

“…Since skolemization is not a valid operation for many logics, in particular, for intuitionistic logic, there is no possibility to obtain the B and C forms even for intuitionistic logic. That is why the author's notions of admissibility and compatibility [2] (see also [3] and [4]) permitted to avoid skolemization and made it possible to construct the Herbrand-type skolemizationfree reduction of deducibility in first-order logic to deducibility in its propositional fragment for classical and intuitionistic modal cases 1 . The research presented below demonstrates that admissibility and compatibility allow achieving the purpose under consideration.…”

Evidence Algorithm and System for Automated Deduction: A Retrospective View

“…Now, the native prover of the current (English) SAD possesses the same features. 6 Later, a theoretical answer on this question was obtained in a number of papers of A.Lyaletski (see, for example, [77,86]); from this point of view, some researches on Herbrand theorems ( [78,79,82,90]) also may seem to be interesting.…”