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  1. [author unknown], .
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  2. Peter Aczel, Harold Simmons & S. S. Wainer (eds.) (1992). Proof Theory: A Selection of Papers From the Leeds Proof Theory Programme, 1990. Cambridge University Press.
    This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.
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  3. Henry Africk (1972). A Proof Theoretic Proof of Scott's General Interpolation Theorem. Journal of Symbolic Logic 37 (4):683-695.
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  4. Mojtaba Aghaei & Mohammad Ardeshir (2001). Gentzen-Style Axiomatizations for Some Conservative Extensions of Basic Propositional Logic. Studia Logica 68 (2):263-285.
    We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.
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  5. Marc Aiguier & Delphine Longuet (2010). Some General Results About Proof Normalization. Logica Universalis 4 (1):1-29.
    In this paper, we provide a general setting under which results of normalization of proof trees such as, for instance, the logicality result in equational reasoning and the cut-elimination property in sequent or natural deduction calculi, can be unified and generalized. This is achieved by giving simple conditions which are sufficient to ensure that such normalization results hold, and which can be automatically checked since they are syntactical. These conditions are based on basic properties of elementary combinations of inference rules (...)
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  6. Natasha Alechina & Michiel van Lambalgen (1996). Generalized Quantification as Substructural Logic. Journal of Symbolic Logic 61 (3):1006-1044.
    We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...)
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  7. Toshiyasu Arai (2002). Review: Wilfried Buchholz, Notation Systems for Infinitary Derivations ; Wilfried Buchholz, Explaining Gentzen's Consistency Proof Within Infinitary Proof Theory ; Sergei Tupailo, Finitary Reductions for Local Predicativity, I: Recursively Regular Ordinals. [REVIEW] Bulletin of Symbolic Logic 8 (3):437-439.
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  8. Andrew Arana (2010). Proof Theory in Philosophy of Mathematics. Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
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  9. Andrew Arana (2009). On Formally Measuring and Eliminating Extraneous Notions in Proofs. Philosophia Mathematica 17 (2):208–219.
    Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen’s cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
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  10. S. Artemov, B. Kushner, G. Mints, E. Nogina & A. Troelstra (1999). In Memoriam: Albert G. Dragalin, 1941-1998. Bulletin of Symbolic Logic 5 (3):389-391.
  11. 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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  12. Jeremy Avigad, “Clarifying the Nature of the Infinite”: The Development of Metamathematics and Proof Theory.
    We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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  13. Jeremy Avigad, Proof Theory.
    At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
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  14. Jeremy Avigad (2010). Proof Theory. Gödel and the Metamathematical Tradition. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  15. Jeremy Avigad (2004). Forcing in Proof Theory. Bulletin of Symbolic Logic 10 (3):305-333.
    Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...)
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  16. A. Avron (1998). Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening. Journal of Symbolic Logic 63 (3):831-859.
    We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...)
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  17. Arnon Avron, A Simple Proof of Completeness and Cut-Elimination for Propositional G¨ Odel Logic.
    We provide a constructive, direct, and simple proof of the completeness of the cut-free part of the hypersequential calculus for G¨odel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). We then extend the results and proofs to derivations from assumptions, showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions.
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  18. Arnon Avron, Canonical Constructive Systems ⋆.
    We define the notions of a canonical inference rule and a canonical system in the framework of single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and prove that such a canonical system is non-trivial iff it is coherent (where coherence is a constructive condition). Next we develop a general non-deterministic Kripke-style semantics for such systems, and show that every constructive canonical system (i.e. coherent canonical single-conclusion system) induces a class of non-deterministic Kripke-style frames for which it is strongly sound and (...)
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  19. Arnon Avron, The Method of Hypersequents in the Proof Theory of Propositional Non-Classical Logics.
    Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers (...)
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  20. Arnon Avron, The Semantics and Proof Theory of Linear Logic.
    Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we shall investigate (...)
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  21. Arnon Avron, Formulas for Which Contraction is Admissible.
    A formula A is said to have the contraction property in a logic L i whenever A;A;? `L B (when ? is a multiset) also A;? `L B. In MLL and in MALL without the additive constants a formula has the contractionproperty i it is a theorem. Adding the mix rule does not change this fact. In MALL (with or without mix) and in a ne logic A has the contraction property i either A is provable or A is equivalent (...)
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  22. Arnon Avron, Multi-Valued Calculi for Logics Based on Non-Determinism.
    Non-deterministic matrices (Nmatrices) are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one (the latter is new here). We use the Rasiowa-Sikorski (R-S) decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such (...)
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  23. Arnon Avron (1991). A Note of Provability, Truth and Existence. Journal of Philosophical Logic 20 (4):403 - 409.
  24. Arnon Avron (1991). Natural 3-Valued Logics--Characterization and Proof Theory. Journal of Symbolic Logic 56 (1):276-294.
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  25. Arnon Avron (1990). Relevance and Paraconsistency---A New Approach. II. The Formal Systems. Notre Dame Journal of Formal Logic 31 (2):169-202.
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  26. Arnon Avron (1990). Relevance and Paraconsistency---A New Approach. II. The Formal Systems. Notre Dame Journal of Formal Logic 31 (2):169-202.
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  27. Arnon Avron (1989). Gentzenizing Schroeder-Heister's Natural Extension of Natural Deduction. Notre Dame Journal of Formal Logic 31 (1):127-135.
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  28. Arnon Avron, Jonathan Ben-Naim & Beata Konikowska (2007). Cut-Free Ordinary Sequent Calculi for Logics Having Generalized Finite-Valued Semantics. Logica Universalis 1 (1):41-70.
    . The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general (...)
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  29. Arnon Avron & Beata Konikowska (2009). Proof Systems for Reasoning About Computation Errors. Studia Logica 91 (2):273 - 293.
    In the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options (...)
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  30. Arnon Avron & Beata Konikowska (2001). Decomposition Proof Systems for Gödel-Dummett Logics. Studia Logica 69 (2):197-219.
    The main goal of the paper is to suggest some analytic proof systems for LC and its finite-valued counterparts which are suitable for proof-search. This goal is achieved through following the general Rasiowa-Sikorski methodology for constructing analytic proof systems for semantically-defined logics. All the systems presented here are terminating, contraction-free, and based on invertible rules, which have a local character and at most two premises.
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  31. 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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  32. David Bresolin, Joanna Golinska-Pilarek & Ewa Orlowska (2006). Relational Dual Tableaux for Interval Temporal Logics. Journal of Applied Non-Classical Logics 16 (3-4):251–277.
    Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...)
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  33. Samuel R. Buss (ed.) (1998). Handbook of Proof Theory. Elsevier.
    This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; (...)
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  34. Michael J. Carroll (1978). An Axiomatization of S13. Philosophia 8 (2-3):381-382.
    Specifies an axiomatization of the system S13 of modal logic. Referenced in Cocchiarella & Freund "Modal Logic: an Introduction to its Syntax and Semantics", Oxford University Press, 2008.
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  35. Cesare Cozzo (forthcoming). Inference and Compulsion. In E. Moriconi (ed.), Second Pisa Colloquium in Logic,Language and Epistemology. ETS.
    What is an inference? Logicians and philosophers have proposed various conceptions of inference. I shall first highlight seven features that contribute to distinguish these conceptions. I shall then compare three conceptions to see which of them best explains the special force that compels us to accept the conclusion of an inference, if we accept its premises.
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  36. William Craig (1957). Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory. Journal of Symbolic Logic 22 (3):269-285.
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  37. Roy Dyckhoff (2010). Positive Logic with Adjoint Modalities: Proof Theory, Semantics, and Reasoning About Information. Review of Symbolic Logic 3 (3):351-373.
    We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4, and S5, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of (...)
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  38. Solomon Feferman, The Proof Theory of Classical and Constructive Inductive Definitions. A 40 Year Saga, 1968-2008.
    1. Pohlers and The Problem. I first met Wolfram Pohlers at a workshop on proof theory organized by Walter Felscher that was held in Tübingen in early April, 1973. Among others at that workshop relevant to the work surveyed here were Kurt Schütte, Wolfram’s teacher in Munich, and Wolfram’s fellow student Wilfried Buchholz. This is not meant to slight in the least the many other fine logicians who participated there.2 In Tübingen I gave a couple of survey lectures on (...)
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  39. Solomon Feferman (2000). Does Reductive Proof Theory Have a Viable Rationale? Erkenntnis 53 (1-2):63-96.
    The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, (...)
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  40. Christian G. Fermüller & George Metcalfe (2009). Giles's Game and the Proof Theory of Łukasiewicz Logic. Studia Logica 92 (1):27 - 61.
    In the 1970s, Robin Giles introduced a game combining Lorenzen-style dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in Łukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In (...)
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  41. James Franklin (1996). Proof in Mathematics. Quakers Hill Press.
    A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
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  42. Dov M. Gabbay (2000). Goal-Directed Proof Theory. Kluwer Academic.
    Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural logics. The book (...)
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  43. Dov Gabbay & Ruth Kempson (1996). Language and Proof Theory. Journal of Logic, Language and Information 5 (3-4):247-251.
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  44. Joanna Golinska-Pilarek (2007). Rasiowa-Sikorski Proof System for the Non-Fregean Sentential Logic SCI. Journal of Applied Non-Classical Logics 17 (4):509–517.
    The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.
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  45. Joanna Golinska-Pilarek, Angel Mora & Emilio Munoz Velasco (2008). An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-Closeness and Distance. In Tu-Bao Ho & Zhi-Hua Zhou (eds.), PRICAI 2008: Trends in Artificial Intelligence. Springer. 128--139.
    We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some (...)
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  46. Joanna Golinska-Pilarek & Emilio Munoz Velasco (2012). Reasoning with Qualitative Velocity: Towards a Hybrid Approach. In Emilio Corchado, Vaclav Snasel, Ajith Abraham, Michał Woźniak, Manuel Grana & Sung-Bae Cho (eds.), Hybrid Artificial Intelligent Systems. Springer. 635--646.
    Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning (...)
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  47. Joanna Golinska-Pilarek & Emilio Munoz Velasco (2009). Relational Approach for a Logic for Order of Magnitude Qualitative Reasoning with Negligibility Non-Closeness and Distance. Logic Journal of IGPL 17 (4):375–394.
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  48. Joanna Golinska-Pilarek, Emilio Munoz Velasco & Angel Mora (2011). A New Deduction System for Deciding Validity in Modal Logic K. Logic Journal of IGPL 19 (2): 425-434.
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  49. Joanna Golinska-Pilarek & Emilio Munoz-Velasco (2009). Dual Tableau for a Multimodal Logic for Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility. International Journal of Computer Mathematics 86 (10-11):1707–1718.
  50. Joanna Golinska-Pilarek, Emilio Munoz-Velasco & Angel Mora (2012). Relational Dual Tableau Decision Procedure for Modal Logic K. Logic Journal of IGPL 20 (4):747-756.
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