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  1. Ben Ellison, Jonathan Fleischmann, Dan McGinn & Wim Ruitenburg (2008). Quantifier Elimination for a Class of Intuitionistic Theories. Notre Dame Journal of Formal Logic 49 (3):281-293.
    From classical, Fraïissé-homogeneous, ($\leq \omega$)-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.
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  2. Ben Ellison, Jonathan Fleischmann, Dan McGinn & Wim Ruitenburg (2007). Kripke Submodels and Universal Sentences. Mathematical Logic Quarterly 53 (3):311-320.
    We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in (...)
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  3. Mohammad Ardeshir, Wim Ruitenburg & Saeed Salehi (2003). Intuitionistic Axiomatizations for Bounded Extension Kripke Models. Annals of Pure and Applied Logic 124 (1-3):267-285.
    We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of cofinal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show that Heyting arithmetic is strongly complete for its class of end-extension models. Cofinal extension models of HA are models of Peano arithmetic.
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  4. Mohammad Ardeshir & Wim Ruitenburg (2001). Basic Propositional Calculus II. Interpolation. Archive for Mathematical Logic 40 (5):349-364.
    Let ℒ and ? be propositional languages over Basic Propositional Calculus, and ℳ = ℒ∩?. Weprove two different but interrelated interpolation theorems. First, suppose that Π is a sequent theory over ℒ, and Σ∪ {C⇒C′} is a set of sequents over ?, such that Π,Σ⊢C⇒C′. Then there is a sequent theory Φ over ℳ such that Π⊢Φ and Φ, Σ⊢C⇒C′. Second, let A be a formula over ℒ, and C 1, C 2 be formulas over ?, such that A∧C 1⊢C (...)
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  5. Wim Ruitenburg (1999). Basic Logic, K4, and Persistence. Studia Logica 63 (3):343-352.
    We characterize the first-order formulas with one free variable that are preserved under bisimulation and persistence or strong persistence over the class of Kripke models with transitive frames and unary persistent predicates.
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  6. Mohammad Ardeshir & Wim Ruitenburg (1998). Basic Propositional Calculus I. Mathematical Logic Quarterly 44 (3):317-343.
    We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B is a formula (...)
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  7. Wim Ruitenburg (1998). Basic Predicate Calculus. Notre Dame Journal of Formal Logic 39 (1):18-46.
    We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.
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  8. Wim Ruitenburg (1991). Inequality in Constructive Mathematics. Notre Dame Journal of Formal Logic 32 (4):533-553.
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  9. Paul Bankston & Wim Ruitenburg (1990). Notions of Relative Ubiquity for Invariant Sets of Relational Structures. Journal of Symbolic Logic 55 (3):948-986.
    Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, (...)
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  10. Daniel Lascar, Ray Mines, Fred Richman & Wim Ruitenburg (1990). Stabilité En Théorie des Modèles. Journal of Symbolic Logic 55 (2):883-886.
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  11. Wim Ruitenburg (1984). On the Period of Sequences (an(P)) in Intuitionistic Propositional Calculus. Journal of Symbolic Logic 49 (3):892 - 899.
    In classical propositional calculus for each proposition A(p) the following holds: $\vdash A(p) \leftrightarrow A^3(p)$ . In this paper we consider what remains of this in the intuitionistic case. It turns out that for each proposition A(p) the following holds: there is an n ∈ N such that $\vdash A^n(p) \leftrightarrow A^{n + 2}(p)$ . As a byproduct of the proof we give some theorems which may be useful elsewhere in propositional calculus.
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