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  1. 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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  2. 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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  3. 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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  4. Hiroshi Aoyama (2004). LK, LJ, Dual Intuitionistic Logic, and Quantum Logic. Notre Dame Journal of Formal Logic 45 (4):193-213.
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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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.
  10. W. Russell Belding (1971). Intuitionistic Negation. Notre Dame Journal of Formal Logic 12 (2):183-187.
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  11. 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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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. H. C. M. de Swart (1977). An Intuitionistically Plausible Interpretation of Intuitionistic Logic. Journal of Symbolic Logic 42 (4):564-578.
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  20. Wil Dekkers, Martin Bunder & Henk Barendregt (1998). Completeness of the Propositions-as-Types Interpretation of Intuitionistic Logic Into Illative Combinatory Logic. Journal of Symbolic Logic 63 (3):869-890.
    Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which (...)
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  21. David DeVidi & Graham Solomon (2001). Knowability and Intuitionistic Logic. Philosophia 28 (1-4):319-334.
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  22. Gustavo Fernández Díez (2000). Five Observations Concerning the Intended Meaning of the Intuitionistic Logical Constants. Journal of Philosophical Logic 29 (4):409-424.
    This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of 'proof'; (2) Kreisel's explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of 'proofs from premises' results in a loss of the inductive character of (...)
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  23. J. Diller & A. S. Troelstra (1984). Realizability and Intuitionistic Logic. Synthese 60 (2):253 - 282.
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  24. Adriano Dodó & João Marcos (2014). Negative Modalities, Consistency and Determinedness. Electronic Notes in Theoretical Computer Science 300:21-45.
    We study a modal language for negative operators—an intuitionistic-like negation and its paraconsistent dual—added to (bounded) distributive lattices. For each non-classical negation an extra operator is hereby adjoined in order to allow for standard logical inferences to be opportunely restored. We present abstract characterizations and exhibit the main properties of each kind of negative modality, as well as of the associated connectives that express consistency and determinedness at the object-language level. Appropriate sequent-style proof systems and adequate kripke semantics are also (...)
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  25. Kosta Došen (1981). A Reduction of Classical Propositional Logic to the Conjunction-Negation Fragment of an Intuitionistic Relevant Logic. Journal of Philosophical Logic 10 (4):399 - 408.
  26. Michael Dummett (1998). Truth From the Constructive Standpoint. Theoria 64 (2-3):122-138.
  27. Michael Dummett (1973). The Philosophical Basis of Intuitionistic Logic. In , Truth and Other Enigmas. Duckworth. 215--247.
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  28. Michael A. E. Dummett (2000). Elements of Intuitionism. Oxford University Press.
    This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer's proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. Readers (...)
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  29. Michael Dummett & J. N. Crossley (eds.) (1963). Formal Systems and Recursive Functions. North Holland.
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  30. Roy Dyckhoff (1992). Contraction-Free Sequent Calculi for Intuitionistic Logic. Journal of Symbolic Logic 57 (3):795-807.
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  31. Roy Dyckhoff & Sara Negri (2000). Admissibility of Structural Rules for Contraction-Free Systems of Intuitionistic Logic. Journal of Symbolic Logic 65 (4):1499-1518.
    We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.
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  32. Roy Dyckhoff & Luis Pinto (1998). Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic. Studia Logica 60 (1):107-118.
    We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.
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  33. David Fernandez (2006). A Polynomial Translation of S4 Into Intuitionistic Logic. Journal of Symbolic Logic 71 (3):989 - 1001.
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  34. Melvin Fitting, Resolution for Intuitionistic Logic.
    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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  35. Melvin Fitting (1969). Intuitionistic Logic, Model Theory and Forcing. Amsterdam, North-Holland Pub. Co..
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  36. Jonathan Fleischmann (2010). Syntactic Preservation Theorems for Intuitionistic Predicate Logic. Notre Dame Journal of Formal Logic 51 (2):225-245.
    We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich Theorem, which characterizes the (...)
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  37. Josep M. Font (1986). Modality and Possibility in Some Intuitionistic Modal Logics. Notre Dame Journal of Formal Logic 27 (4):533-546.
  38. Harvey Friedman (1973). The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic. Journal of Symbolic Logic 38 (2):315-319.
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  39. Dov M. Gabbay (1977). A New Version of Beth Semantics for Intuitionistic Logic. Journal of Symbolic Logic 42 (2):306-308.
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  40. Dov M. Gabbay (1977). Craig Interpolation Theorem for Intuitionistic Logic and Extensions Part III. Journal of Symbolic Logic 42 (2):269-271.
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  41. James W. Garson (2001). Natural Semantics: Why Natural Deduction is Intuitionistic. Theoria 67 (2):114-139.
    In this paper investigates how natural deduction rules define connective meaning by presenting a new method for reading semantical conditions from rules called natural semantics. Natural semantics explains why the natural deduction rules are profoundly intuitionistic. Rules for conjunction, implication, disjunction and equivalence all express intuitionistic rather than classical truth conditions. Furthermore, standard rules for negation violate essential conservation requirements for having a natural semantics. The standard rules simply do not assign a meaning to the negation sign. Intuitionistic negation fares (...)
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  42. Silvio Ghilardi (1999). Unification in Intuitionistic Logic. Journal of Symbolic Logic 64 (2):859-880.
    We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.
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  43. Paul C. Gilmore (1953). The Effect of Griss's Criticism of the Intuitionistic Logic on Deducative Theories Formalized Within the Intuitionistic Logic. Amsterdam, Drukkerij Holland.
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  44. Rob Goldblatt (1978). Arithmetical Necessity, Provability and Intuitionistic Logic. Theoria 44 (1):38-46.
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  45. Santos Gonçalo (forthcoming). Numbers and Everything. Philosophia Mathematica.
    I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.
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  46. Rajeev Gore, A Cut-Free Sequent Calculus for Bi-Intuitionistic Logic.
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  47. Rajeev Gore, Linda Postniece & Alwen Tiu, Cut-Elimination and Proof-Search for Bi-Intuitionistic Logic Using Nested Sequents.
    We propose a new sequent calculus for bi intuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cut elimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cut elimination proof. We then present the derived calculus, and then present a proof (...)
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  48. Yuri Gurevich (1977). Intuitionistic Logic with Strong Negation. Studia Logica 36 (1-2):49 - 59.
    This paper is a reaction to the following remark by grzegorczyk: "the compound sentences are not a product of experiment. they arise from reasoning. this concerns also negations; we see that the lemon is yellow, we do not see that it is not blue." generally, in science the truth is ascertained as indirectly as falsehood. an example: a litmus-paper is used to verify the sentence "the solution is acid." this approach gives rise to a (very intuitionistic indeed) conservative extension of (...)
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  49. J. H. Harris (1982). What's So Logical About the “Logical” Axioms? Studia Logica 41 (2-3):159 - 171.
    Intuitionists and classical logicians use in common a large number of the logical axioms, even though they supposedly mean different things by the logical connectives and quantifiers — conquans for short. But Wittgenstein says The meaning of a word is its use in the language. We prove that in a definite sense the intuitionistic axioms do indeed characterize the logical conquans, both for the intuitionist and the classical logician.
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  50. Jaakko Hintikka (2001). Intuitionistic Logic as Epistemic Logic. Synthese 127 (1-2):7 - 19.
    In the present day and age, it seems that every constructivist philosopher of mathematics and her brother wants to be known as an intuitionist. In this paper, It will be shown that such a self-identification is in most cases mistaken. For one thing, not any old (or new) constructivism is intuitionism because not any old relevant construction is carried out mentally in intuition, as Brouwer envisaged. (edited).
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