Most automated theorem provers have been built around some version of resolution [4]. But resolution is an inherently Classical logic technique. Attempts to extend the method to other logics have tended to obscure its simplicity. In this paper we present a resolution style theorem prover for Intuitionistic logic that, we believe, retains many of the attractive features of Classical resolution. It is, of course, more complicated, but the complications can be given intuitive motivation. We note that a small change in (...) the system as presented here causes it to collapse back to a Classical resolution system. We present the system in some detail for the propositional case, including soundness and completeness proofs. For the first order version we are sketchier. (shrink)

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. (...) Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17]. (shrink)

A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.

The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are elementarily (...) inequivalent. Special properties of algebraically closed, existentially closed, finitely generic, and infinitely generic closure algebras are established along the way. (shrink)