A completeness theorem for higher-order intuitionistic logic
an intuitionistic proof
pp. 107-124
in: Dimiter G. Skordev (ed), Mathematical logic and its applications, Berlin, Springer, 1987Abstract
The Goedel's famous completeness theorem for predicate logic is the foundation of modern model theory in classic logic as well as in intuitionistic one. Moreover, this theorem is intensively used in the theory of mechanical theorem proving [1] and in the theory of logic programming [2]. So there is an insistent aspiration for constructive treating of this theorem. Usual proofs of the completeness theorem are founded on considerations of maximal consistent sets of formulas and are nonconstructive (see, for example, [1] for classical case and [3] for intuitionistic logic). The situation is specially urgent in the intuitionistic case because this logic is intended for effective treating of logical connectives and is used usually in computer science as an instrument for getting a program from a constructive (= intuitionistic) proof of a given formula. A nonconstructive completeness proof in this situation can serve only as a general indication for possible success of a given proof-searching procedure, just as a constructive proof provides precise bounds for complexity of searching.