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  1. M. Abad, J. P. Díaz Varela, L. A. Rueda & A. M. Suardíaz (2000). Varieties of Three-Valued Heyting Algebras with a Quantifier. Studia Logica 65 (2):181-198.
    This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q 3 and we construct the lattice of subvarieties (...)
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  2. Andrew Aberdein & Stephen Read (2009). The Philosophy of Alternative Logics. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. 613-723.
    This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...)
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  3. Romà J. Adillon & Ventura Verdú (2000). On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion. Studia Logica 65 (1):11-30.
    In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is a (...)
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  4. Henry Africk (1992). Classical Logic, Intuitionistic Logic, and the Peirce Rule. Notre Dame Journal of Formal Logic 33 (2):229-235.
    A simple method is provided for translating proofs in Grentzen's LK into proofs in Gentzen's LJ with the Peirce rule adjoined. A consequence is a simpler cut elimination operator for LJ + Peirce that is primitive recursive.
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  5. M. Aghaei & M. Ardeshir (2000). A Bounded Translation of Intuitionistic Propositional Logic Into Basic Propositional Logic. Mathematical Logic Quarterly 46 (2):195-206.
    In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.
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  6. R. Alonderis (2013). A Proof-Search Procedure for Intuitionistic Propositional Logic. Archive for Mathematical Logic 52 (7-8):759-778.
    A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.
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  7. Takahito Aoto (1999). Uniqueness of Normal Proofs in Implicational Intuitionistic Logic. Journal of Logic, Language and Information 8 (2):217-242.
    A minimal theorem in a logic L is an L-theorem which is not a non-trivial substitution instance of another L-theorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique -normal proof in NJ whenever A is provable without (...)
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  8. Hiroshi Aoyama (2004). LK, LJ, Dual Intuitionistic Logic, and Quantum Logic. Notre Dame Journal of Formal Logic 45 (4):193-213.
    In this paper, we study the relationship among classical logic, intuitionistic logic, and quantum logic . These logics are related in an interesting way and are not far apart from each other, as is widely believed. The results in this paper show how they are related with each other through a dual intuitionistic logic . Our study is completely syntactical.
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  9. Mohammad Ardeshir (1999). A Translation of Intuitionistic Predicate Logic Into Basic Predicate Logic. Studia Logica 62 (3):341-352.
    Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.
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  10. Mohammad Ardeshir & Mojtaba Moniri (1998). Intuitionistic Open Induction and Least Number Principle and the Buss Operator. Notre Dame Journal of Formal Logic 39 (2):212-220.
    In "Intuitionistic validity in -normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory , that of all -normal Kripke structures for which he gave an r.e. axiomatization. In the language of arithmetic and denote PA plus Open Induction or Open LNP, and are their intuitionistic deductive closures. We show is recursively axiomatizable and , while . If proves PEM but not totality of a classically provably total Diophantine function of , then and so . A (...)
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  11. Mohammad Ardeshir & Rasoul Ramezanian (2010). The Double Negation of the Intermediate Value Theorem. Annals of Pure and Applied Logic 161 (6):737-744.
    In the context of intuitionistic analysis, we consider the set consisting of all continuous functions from [0,1] to such that =0 and =1, and the set consisting of ’s in where there exists x[0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have . However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema about intuitionistic (...)
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  12. Sergei Artemov & Rosalie Iemhoff (2007). The Basic Intuitionistic Logic of Proofs. Journal of Symbolic Logic 72 (2):439 - 451.
    The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.
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  13. Sergei Artemov & Tudor Protopopescu (2013). Discovering Knowability: A Semantic Analysis. Synthese 190 (16):3349-3376.
    In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle /all truths are knowable/, when expressed as a bi-modal principle F --> K♢F, yields an unacceptable omniscience property /all truths are known/. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it (...)
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  14. M. V. Atten (forthcoming). The Development of Intuitionistic Logic. Stanford Encyclopedia of Philosophy. The Meta-27here I Am Assuming That’Evidence’Provides the Basis for One’s Doxastic Justification. Additionally, I.
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  15. Jeremy Avigad, Algebraic Proofs of Cut Elimination.
    Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...)
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  16. Jeremy Avigad (2000). Interpreting Classical Theories in Constructive Ones. Journal of Symbolic Logic 65 (4):1785-1812.
    A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.
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  17. Arnon Avron (2005). A Non-Deterministic View on Non-Classical Negations. Studia Logica 80 (2-3):159 - 194.
    We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...)
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  18. S. Awodey & C. Butz (2000). Topological Completeness for Higher-Order Logic. Journal of Symbolic Logic 65 (3):1168-1182.
    Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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  19. Matthias Baaz & Rosalie Iemhoff (2008). On Skolemization in Constructive Theories. Journal of Symbolic Logic 73 (3):969-998.
    In this paper a method for the replacement, in formulas, of strong quantifiers by functions is introduced that can be considered as an alternative to Skolemization in the setting of constructive theories. A constructive extension of intuitionistic predicate logic that captures the notions of preorder and existence is introduced and the method, orderization, is shown to be sound and complete with respect to this logic. This implies an analogue of Herbrand's theorem for intuitionistic logic. The orderization method is applied to (...)
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  20. Matthias Baaz & Rosalie Iemhoff (2006). The Skolemization of Existential Quantifiers in Intuitionistic Logic. Annals of Pure and Applied Logic 142 (1):269-295.
    In this paper an alternative Skolemization method is introduced that, for a large class of formulas, is sound and complete with respect to intuitionistic logic. This class extends the class of formulas for which standard Skolemization is sound and complete and includes all formulas in which all strong quantifiers are existential. The method makes use of an existence predicate first introduced by Dana Scott.
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  21. Matthias Baaz & Rosalie Iemhoff (2006). Gentzen Calculi for the Existence Predicate. Studia Logica 82 (1):7 - 23.
    We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.
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  22. Seyed Bagheri & Massoud Pourmahdian (2006). Diagram Construction in Intuitionistic Logic. Logic Journal of the Igpl 14 (6):889-901.
    Every classical first order structure is coded in its diagram consisting of atomic sentences it satisfies. We study diagrams for the class of constant domain Kripke models and use it to define notions of submodel, reduction, expansion and ultraproduct for a ceratin subclass of it. In particular, we study conditions under which forcing is preserved by reductions and expansions.
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  23. David Basin, Seán Matthews & Luca Viganò (1998). Natural Deduction for Non-Classical Logics. Studia Logica 60 (1):119-160.
    We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness and (...)
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  24. Michael Beeson (1976). The Unprovability in Intuitionistic Formal Systems of the Continuity of Effective Operations on the Reals. Journal of Symbolic Logic 41 (1):18-24.
  25. W. Russell Belding (1971). Intuitionistic Negation. Notre Dame Journal of Formal Logic 12 (2):183-187.
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  26. N. D. Belnap, H. Leblanc & R. H. Thomason (1963). On Not Strengthening Intuitionistic Logic. Notre Dame Journal of Formal Logic 4 (4):313-320.
    tic sequenzen-kalkul of Gentzen, into rules for PCc, the classical sequenzenkalkul. We shall limit ourselves here to sequenzen or turnstile statements of the form A„A„..., A„ I- B, where A„A„..., A„(n ~ 0), and B are wffs consisting of propositional variables, zero or more of the connectives '5', "v', ' ', ')', and '=', and zero or more parentheses. One can pass from PCi to PCc by amending the intelim rules for ' a result of long standing, or by amending (...)
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  27. Johan Van Benthem (2009). The Information in Intuitionistic Logic. Synthese 167 (2):251 - 270.
    Issues about information spring up wherever one scratches the surface of logic. Here is a case that raises delicate issues of 'factual' versus 'procedural' information, or 'statics' versus 'dynamics'. What does intuitionistic logic, perhaps the earliest source of informational and procedural thinking in contemporary logic, really tell us about information? How does its view relate to its 'cousin' epistemic logic? We discuss connections between intuitionistic models and recent protocol models for dynamic-epistemic logic, as well as more general issues that emerge.
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  28. E. W. Beth & H. Leblanc (1960). A Note on the Intuitionistic and the Classical Proposition Calculus. Logique Et Analyse 3 (4):174-176.
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  29. Evert Willem Beth (1956). Semantic Construction of Intuitionistic Logic. Noord-Hollandsche Uitg. Mij.
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  30. G. M. Bierman & V. C. V. de Paiva (2000). On an Intuitionistic Modal Logic. Studia Logica 65 (3):383-416.
    In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also (...)
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  31. Rodolfo C. Ertola Biraben (2012). On Some Extensions of Intuitionistic Logic. Bulletin of the Section of Logic 41 (1/2):17-22.
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  32. Jaime Bohórquez V. (2008). Intuitionistic Logic According to Dijkstra's Calculus of Equational Deduction. Notre Dame Journal of Formal Logic 49 (4):361-384.
    Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert's style of proof and Gentzen's deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra's words, "letting the symbols do (...)
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  33. Branislav R. Boricic (1991). Interpolation Theorem for Intuitionistic S4. Bulletin of the Section of Logic 20 (1):2-6.
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  34. Donald Michael Brown (1993). Evolutionary Intermediacy, Ideal Judges, and Intuitionistic Logic. Dissertation, University of California, Los Angeles
    This dissertation consists of two related parts, one philosophical and the other mathematical. The philosophical part concerns the grounds available to biologists for believing the law of excluded middle to be violated in the course of the evolutionary origin of species. At bottom an incompleteness of the laws of nature, rather than the vagueness of any concept, is best seen to account for the tertium quids biologists postulate. Accordingly, there is an underlying symmetry between the source of the failure of (...)
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  35. Xavier Caicedo & Roberto Cignoli (2001). An Algebraic Approach to Intuitionistic Connectives. Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  36. Olivia Caramello (2014). Topologies for Intermediate Logics. Mathematical Logic Quarterly 60 (4-5):335-347.
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  37. Colin Caret & Ole Thomassen Hjortland (forthcoming). Logical Consequence: Its Nature, Structure, and Application. In Colin Caret & Ole Thomassen Hjortland (eds.), Foundations of Logical Consequence. Oxford University Press.
  38. G. Cattaneo, R. Giuntini & S. Pulmannovà (2000). Pre-BZ and Degenerate BZ Posets: Applications to Fuzzy Sets and Unsharp Quantum Theories. [REVIEW] Foundations of Physics 30 (10):1765-1799.
    Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the so-called (...)
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  39. L. Farinas Del Cerro (1983). Some results in intuitionistic modal logic. Logique Et Analyse 26 (2):219.
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  40. Alexander Chagrov & Michael Zakharyashchev (1991). The Disjunction Property of Intermediate Propositional Logics. Studia Logica 50 (2):189 - 216.
    This paper is a survey of results concerning the disjunction property, Halldén-completeness, and other related properties of intermediate prepositional logics and normal modal logics containing S4.
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  41. Saverio Cittadini, Intercalation Calculus for Intuitionistic Propositional Logic.
    Saverio Cittadini. Intercalation Calculus for Intuitionistic Propositional Logic.
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  42. Janusz Ciuciura (2000). Intuitionistic Discursive System (Ids). Bulletin of the Section of Logic 29 (1/2):57-62.
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  43. Juan M. Cornejo & Ignacio D. Viglizzo (2015). On Some Semi-Intuitionistic Logics. Studia Logica 103 (2):303-344.
    Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus (...)
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  44. Giovanna Corsi & Gabriele Tassi (2007). Intuitionistic Logic Freed of All Metarules. Journal of Symbolic Logic 72 (4):1204 - 1218.
    In this paper we present two calculi for intuitionistic logic. The first one, IG, is characterized by the fact that every proof-search terminates and termination is reached without jeopardizing the subformula property. As to the second one, SIC, proof-search terminates, the subformula property is preserved and moreover proof-search is performed without any recourse to metarules, in particular there is no need to back-track. As a consequence, proof-search in the calculus SIC is accomplished by a single tree as in classical logic.
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  45. M. J. Cresswell (2004). Possibility Semantics for Intuitionistic Logic. Australasian Journal of Logic 2:11-29.
    The paper investigates interpretations of propositional and firstorder logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for (...)
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  46. Bernd I. Dahn (1981). Partial Isomorphisms and Intuitionistic Logic. Studia Logica 40 (4):405 - 413.
    A game for testing the equivalence of Kripke models with respect to finitary and infinitary intuitionistic predicate logic is introduced and applied to discuss a concept of categoricity for intuitionistic theories.
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  47. A. J. Dale (1985). A completeness property of negationless intuitionist propositional logics. Logique Et Analyse 28 (9):79.
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  48. D. Dalen (1986). Glueing of Analysis Models in an Intuitionistic Setting. Studia Logica 45 (2):181 - 186.
    Beth models of analysis are used in model theoretic proofs of the disjunction and (numerical) existence property. By glueing strings of models one obtains a model that combines the properties of the given models. The method asks for a common generalization of Kripke and Beth models. The proof is carried out in intuitionistic analysis plus Markov's Principle. The main new feature is the external use of intuitionistic principles to prove their own preservation under glueing.
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  49. Carlo Dalla Pozza & Claudio Garola (1995). A Pragmatic Interpretation of Intuitionistic Propositional Logic. Erkenntnis 43 (1):81-109.
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  50. G. Dardzania (1977). Intuitionistic System Without Contraction. Bulletin of the Section of Logic 6 (1):2-6.
    The paper deals with a rst order intuitionistic predicate calculus with- out contraction. We examine the Hilbert variant of this calculus and an equivalent Genzen variant of it, prove that this calculus is decidable, evalu- ate its complexity of deduction and study its connection with the classical logics.
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