Search results for 'Calculus' (try it on Scholar)

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  1. Intuitionistic Sentential Calculus (1990). 1. Intuitionistic Sentential Calculus with Iden-Tity. Bulletin of the Section of Logic 19 (3):92-99.score: 180.0
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  2. Moritz Cordes & Friedrich Reinmuth, A Speech Act Calculus. A Pragmatised Natural Deduction Calculus and its Meta-Theory.score: 24.0
    Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper (...)
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  3. Henry Somers-Hall (2010). Hegel and Deleuze on the Metaphysical Interpretation of the Calculus. Continental Philosophy Review 42 (4):555-572.score: 24.0
    The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G. W. F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this (...)
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  4. Gerhard Lakemeyer (2010). The Situation Calculus: A Case for Modal Logic. [REVIEW] Journal of Logic, Language and Information 19 (4):431-450.score: 24.0
    The situation calculus is one of the most established formalisms for reasoning about action and change. In this paper we will review the basics of Reiter’s version of the situation calculus, show how knowledge and time have been addressed in this framework, and point to some of the weaknesses of the situation calculus with respect to time. We then present a modal version of the situation calculus where these problems can be overcome with relative ease and (...)
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  5. Emmanuel Haven (2011). Itô's Lemma with Quantum Calculus (Q-Calculus): Some Implications. [REVIEW] Foundations of Physics 41 (3):529-537.score: 24.0
    q-derivatives are part of so called quantum calculus. In this paper we investigate how such derivatives can possibly be used in Itô’s lemma. This leads us to consider how such derivatives can be used in a social science setting. We conclude that in a Itô Lemma setting we cannot use a macroscopic version of the Heisenberg uncertainty principle with q-derivatives.
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  6. Carlos Castro (2010). On Nonlinear Quantum Mechanics, Noncommutative Phase Spaces, Fractal-Scale Calculus and Vacuum Energy. Foundations of Physics 40 (11):1712-1730.score: 24.0
    A (to our knowledge) novel Generalized Nonlinear Schrödinger equation based on the modifications of Nottale-Cresson’s fractal-scale calculus and resulting from the noncommutativity of the phase space coordinates is explicitly derived. The modifications to the ground state energy of a harmonic oscillator yields the observed value of the vacuum energy density. In the concluding remarks we discuss how nonlinear and nonlocal QM wave equations arise naturally from this fractal-scale calculus formalism which may have a key role in the final (...)
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  7. A. H. Louie & I. W. Richardson (2006). A Phenomenological Calculus for Anisotropic Systems. Axiomathes 16 (1-2):215-243.score: 24.0
    The phenomenological calculus is a relational paradigm for complex systems, closely related in substance and spirit to Robert Rosen’s own approach. Its mathematical language is multilinear algebra. The epistemological exploration continues in this paper, with the expansion of the phenomenological calculus into the realm of anisotropy.
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  8. J. Roger Hindley (1986). Introduction to Combinators and [Lambda]-Calculus. Cambridge University Press.score: 24.0
    Combinatory logic and lambda-conversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of 'operation' instead of 'set'. They have now developed into linguistic tools, useful in several branches of logic and computer science, especially in the study of programming languages. These notes form a simple introduction to the two topics, suitable for a reader who has no previous knowledge of combinatory logic, but has taken an undergraduate course in predicate calculus and (...)
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  9. Anna Zamansky, Nissim Francez & Yoad Winter (2006). A 'Natural Logic' Inference System Using the Lambek Calculus. Journal of Logic, Language and Information 15 (3):273-295.score: 24.0
    This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem (1997), Sánchez (1991) and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek-based system we propose extends the system by Fyodorov et~al. (2003), which is based on the Ajdukiewicz/Bar-Hillel (AB) calculus Bar Hillel, (1964). This (...)
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  10. Luca Alberucci & Alessandro Facchini (2009). On Modal Μ -Calculus and Gödel-Löb Logic. Studia Logica 91 (2):145 - 169.score: 24.0
    We show that the modal µ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modal µ~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new (...)
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  11. Michał Kozak (2009). Distributive Full Lambek Calculus has the Finite Model Property. Studia Logica 91 (2):201 - 216.score: 24.0
    We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1].
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  12. Lloyd Humberstone (2007). Investigations Into a Left-Structural Right-Substructural Sequent Calculus. Journal of Logic, Language and Information 16 (2):141-171.score: 24.0
    We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these (...)
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  13. Mark Schlatter & Ken Aizawa (2008). Walter Pitts and “a Logical Calculus”. Synthese 162 (2):235 - 250.score: 24.0
    Many years after the publication of “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Warren McCulloch gave Walter Pitts credit for contributing his knowledge of modular mathematics to their joint project. In 1941 I presented my notions on the flow of information through ranks of neurons to Rashevsky’s seminar in the Committee on Mathematical Biology of the University of Chicago and met Walter Pitts, who then was about seventeen years old. He was working on a (...)
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  14. Maria Bulińska (2009). On the Complexity of Nonassociative Lambek Calculus with Unit. Studia Logica 93 (1):1 - 14.score: 24.0
    Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). (...)
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  15. Kentaro Kikuchi & Katsumi Sasaki (2003). A Cut-Free Gentzen Formulation of Basic Propositional Calculus. Journal of Logic, Language and Information 12 (2):213-225.score: 24.0
    We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination theorem (...)
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  16. Grigori Mints (2012). Effective Cut-Elimination for a Fragment of Modal Mu-Calculus. Studia Logica 100 (1-2):279-287.score: 24.0
    A non-effective cut-elimination proof for modal mu-calculus has been given by G. Jäger, M. Kretz and T. Studer. Later an effective proof has been given for a subsystem M 1 with non-iterated fixpoints and positive endsequents. Using a new device we give an effective cut-elimination proof for M 1 without restriction to positive sequents.
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  17. Aarne Ranta (1998). Syntactic Calculus with Dependent Types. Journal of Logic, Language and Information 7 (4):413-431.score: 24.0
    The aim of this study is to look at the the syntactic calculus of Bar-Hillel and Lambek, including semantic interpretation, from the point of view of constructive type theory. The syntactic calculus is given a formalization that makes it possible to implement it in a type-theoretical proof editor. Such an implementation combines formal syntax and formal semantics, and makes the type-theoretical tools of automatic and interactive reasoning available in grammar.In the formalization, the use of the dependent types of (...)
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  18. Piotr Kulicki (2012). An Axiomatisation of a Pure Calculus of Names. Studia Logica 100 (5):921-946.score: 24.0
    A calculus of names is a logical theory describing relations between names. By a pure calculus of names we mean a quantifier-free formulation of such a theory, based on classical propositional calculus. An axiomatisation of a pure calculus of names is presented and its completeness is discussed. It is shown that the axiomatisation is complete in three different ways: with respect to a set theoretical model, with respect to Leśniewski's Ontology and in a sense defined with (...)
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  19. Jose G. Vargas (2008). The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus. Foundations of Physics 38 (7):610-647.score: 24.0
    In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons.The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his (...) by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent.A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms. (shrink)
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  20. Wendy MacCaull (1998). Relational Semantics and a Relational Proof System for Full Lambek Calculus. Journal of Symbolic Logic 63 (2):623-637.score: 24.0
    In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style (...)
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  21. Wojciech Zielonka (2001). Cut-Rule Axiomatization of the Syntactic Calculus L. Journal of Logic, Language and Information 10 (2):339-352.score: 24.0
    In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L 0 of L. This modification, introduced in the early 1980s (see, (...)
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  22. Wojciech Zielonka (2000). Cut-Rule Axiomatization of the Syntactic Calculus NL. Journal of Logic, Language and Information 9 (3):339-352.score: 24.0
    An axiomatics of the product-free syntactic calculus L ofLambek has been presented whose only rule is the cut rule. It was alsoproved that there is no finite axiomatics of that kind. The proofs weresubsequently simplified. Analogous results for the nonassociativevariant NL of L were obtained by Kandulski. InLambek's original version of the calculus, sequent antecedents arerequired to be nonempty. By removing this restriction, we obtain theextensions L 0 and NL 0 ofL and NL, respectively. Later, the finiteaxiomatization problem (...)
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  23. Hajnal Andréka & Szabolcs Mikulás (1994). Lambek Calculus and its Relational Semantics: Completeness and Incompleteness. [REVIEW] Journal of Logic, Language and Information 3 (1):1-37.score: 24.0
    The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version (...)
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  24. Michael Kaminski & Nissim Francez (2014). Relational Semantics of the Lambek Calculus Extended with Classical Propositional Logic. Studia Logica 102 (3):479-497.score: 24.0
    We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.
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  25. Makoto Kanazawa (1992). The Lambek Calculus Enriched with Additional Connectives. Journal of Logic, Language and Information 1 (2):141-171.score: 24.0
    Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.
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  26. Karim Nour (2002). Non Deterministic Classical Logic: The $Lambdamu^{++}$-Calculus. Mathematical Logic Quarterly 48 (3):357-366.score: 24.0
    In this paper, we present an extension of $lambdamu$-calculus called $lambdamu^{++}$-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or.
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  27. Michael Gabbay (2011). A Proof-Theoretic Treatment of Λ-Reduction with Cut-Elimination: Λ-Calculus as a Logic Programming Language. Journal of Symbolic Logic 76 (2):673 - 699.score: 24.0
    We build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic. We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).
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  28. Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part I. Archive for Mathematical Logic 52 (1-2):173-212.score: 24.0
    This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${\neg}$ , parallel conjunction ${\wedge}$ , parallel disjunction ${\vee}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.
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  29. Andréa Lopari? (2011). Valuation Semantics for Intuitionic Propositional Calculus and Some of its Subcalculi. Principia 14 (1):125-33.score: 24.0
    In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus). Algorithms based in our definitions yields decision methods for these calculi. DOI:10.5007/1808-1711.2010v14n1p125.
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  30. Glyn Morrill, Oriol Valentín & Mario Fadda (2011). The Displacement Calculus. Journal of Logic, Language and Information 20 (1):1-48.score: 24.0
    If all dependent expressions were adjacent some variety of immediate constituent analysis would suffice for grammar, but syntactic and semantic mismatches are characteristic of natural language; indeed this is a, or the, central problem in grammar. Logical categorial grammar reduces grammar to logic: an expression is well-formed if and only if an associated sequent is a theorem of a categorial logic. The paradigmatic categorial logic is the Lambek calculus, but being a logic of concatenation the Lambek calculus can (...)
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  31. Koji Nakazawa & Hiroto Naya (forthcoming). Strong Reduction of Combinatory Calculus with Streams. Studia Logica:1-13.score: 24.0
    This paper gives the strong reduction of the combinatory calculus SCL, which was introduced as a combinatory calculus corresponding to the untyped Lambda-mu calculus. It proves the confluence of the strong reduction. By the confluence, it also proves the conservativity of the extensional equality of SCL over the combinatory calculus CL, and the consistency of SCL.
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  32. Peter Schroeder-Heister (2011). Implications-as-Rules Vs. Implications-as-Links: An Alternative Implication-Left Schema for the Sequent Calculus. [REVIEW] Journal of Philosophical Logic 40 (1):95 - 101.score: 24.0
    The interpretation of implications as rules motivates a different left-introduction schema for implication in the sequent calculus, which is conceptually more basic than the implication-left schema proposed by Gentzen. Corresponding to results obtained for systems with higher-level rules, it enjoys the subformula property and cut elimination in a weak form.
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  33. Wojciech Zielonka (2002). On Reduction Systems Equivalent to the Lambek Calculus with the Empty String. Studia Logica 71 (1):31-46.score: 24.0
    The paper continues a series of results on cut-rule axiomatizability of the Lambek calculus. It provides a complete solution of a problem which was solved partially in one of the author''s earlier papers. It is proved that the product-free Lambek Calculus with the empty string (L 0) is not finitely axiomatizable if the only rule of inference admitted is Lambek''s cut rule. The proof makes use of the (infinitely) cut-rule axiomatized calculus C designed by the author exactly (...)
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  34. H. P. Barendregt (1984). The Lambda Calculus: Its Syntax and Semantics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..score: 24.0
    The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.
     
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  35. Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part II. Archive for Mathematical Logic 52 (1-2):213-259.score: 24.0
    This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${{\neg}}$ , parallel conjunction ${{\wedge}}$ , parallel disjunction ${{\vee}}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the previous) Part I containing preliminaries and a soundness proof, and (the present) Part II containing a completeness proof.
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  36. Kazushige Terui (2007). Light Affine Lambda Calculus and Polynomial Time Strong Normalization. Archive for Mathematical Logic 46 (3-4):253-280.score: 24.0
    Light Linear Logic (LLL) and Intuitionistic Light Affine Logic (ILAL) are logics that capture polynomial time computation. It is known that every polynomial time function can be represented by a proof of these logics via the proofs-as-programs correspondence. Furthermore, there is a reduction strategy which normalizes a given proof in polynomial time. Given the latter polynomial time “weak” normalization theorem, it is natural to ask whether a “strong” form of polynomial time normalization theorem holds or not. In this paper, we (...)
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  37. Michael J. Carroll (1976). On Interpreting the S5 Propositional Calculus: An Essay in Philosophical Logic. Dissertation, University of Iowascore: 21.0
    Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.
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  38. David Hestenes (2005). Gauge Theory Gravity with Geometric Calculus. Foundations of Physics 35 (6):903-970.score: 21.0
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  39. Mati Pentus (1994). The Conjoinability Relation in Lambek Calculus and Linear Logic. Journal of Logic, Language and Information 3 (2):121-140.score: 21.0
    In 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: x y means that there exists a sequence x=x1,...,xn=y (n 1), such thatx i x i+1 or xi+ x i (1 i n). He pointed out thatx y if and only if there is joinz such thatx z andy z. This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girard's and Yetter's linear (...)
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  40. Karim Nour (2002). Non Deterministic Classical Logic: The Λμ++ ‐Calculus. Mathematical Logic Quarterly 48 (3):357-366.score: 21.0
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  41. H. P. Barendregt (2013). Lambda Calculus with Types. Cambridge University Press.score: 21.0
    This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.
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  42. Douglas Bridges & Hajime Ishihara (1994). Absolute Continuity and the Uniqueness of the Constructive Functional Calculus. Mathematical Logic Quarterly 40 (4):519-527.score: 21.0
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  43. Emil Jerábek & Phuong Nguyen (2011). Simulating Non-Prenex Cuts in Quantified Propositional Calculus. Mathematical Logic Quarterly 57 (5):524-532.score: 21.0
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  44. Nikolai Pankrat'ev (1994). On the Completeness of the Lambek Calculus with Respect to Relativized Relational Semantics. Journal of Logic, Language and Information 3 (3):233-246.score: 21.0
    Recently M. Szabolcs [12] has shown that many substructural logics including Lambek CalculusL are complete with respect to relativized Relational Semantics. The current paper proves that it is sufficient forL to consider a relativization to the relation x dividesy in some fixed semigroupG.
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  45. Ernst Zimmermann (2010). Full Lambek Calculus in Natural Deduction. Mathematical Logic Quarterly 56 (1):85-88.score: 21.0
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  46. Marcelo Da Silva Corrêa & Edward Hermann Haeusler (1997). A Concrete Categorical Model for the Lambek Syntactic Calculus. Mathematical Logic Quarterly 43 (1):49-59.score: 21.0
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  47. Maciej Kandulski (1993). Normal Form of Derivations in the Nonassociative and Commutative Lambek Calculus with Product. Mathematical Logic Quarterly 39 (1):103-114.score: 21.0
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  48. K. Kikuchi (2002). Dual-Context Sequent Calculus and Strict Implication. Mathematical Logic Quarterly 48 (1):87-92.score: 21.0
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  49. J. Von Plato (2003). Translations From Natural Deduction to Sequent Calculus. Mathematical Logic Quarterly 49 (5):435.score: 21.0
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  50. Benedetto Intrigila (1993). The Basis Decision Problem in Λ‐Calculus. Mathematical Logic Quarterly 39 (1):178-180.score: 21.0
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