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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. |
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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. (...) |
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We present non-clausal resolution systems for propositional modal logics whose Kripke models do not involve symmetry, and for first order versions whose Kripke models do not involve constant domains. We give systems for K, T , K4 and S4; other logics are also possible. Our systems do not require preliminary reduction to a normal form and, in the first order case, intermingle resolution steps with Skolemization steps. |
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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 (...) |
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