Search results for 'Natural Deduction' (try it on Scholar)

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  1. Moritz Cordes & Friedrich Reinmuth, A Speech Act Calculus. A Pragmatised Natural Deduction Calculus and its Meta-Theory.score: 240.0
    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 (...)
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  2. Susan Rogerson (2007). Natural Deduction and Curry's Paradox. Journal of Philosophical Logic 36 (2):155 - 179.score: 240.0
    Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.
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  3. Torben BraÜner (2005). Natural Deduction for First-Order Hybrid Logic. Journal of Logic, Language and Information 14 (2):173-198.score: 240.0
    This is a companion paper to Braüner (2004b, Journal of Logic and Computation 14, 329–353) where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove (...)
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  4. Nissim Francez (2014). Harmony in Multiple-Conclusion Natural-Deduction. Logica Universalis 8 (2):215-259.score: 240.0
    The paper studies the extension of harmony and stability, major themes in proof-theoretic semantics, from single-conclusion natural-deduction systems to multiple-conclusions natural-deduction, independently of classical logic. An extension of the method of obtaining harmoniously-induced general elimination rules from given introduction rules is suggested, taking into account sub-structurality. Finally, the reductions and expansions of the multiple-conclusions natural-deduction representation of classical logic are formulated.
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  5. Torben Braüner (2004). Two Natural Deduction Systems for Hybrid Logic: A Comparison. [REVIEW] Journal of Logic, Language and Information 13 (1):1-23.score: 240.0
    In this paper two different natural deduction systems forhybrid logic are compared and contrasted.One of the systems was originally given by the author of the presentpaper whereasthe other system under consideration is a modifiedversion of a natural deductionsystem given by Jerry Seligman.We give translations in both directions between the systems,and moreover, we devise a set of reduction rules forthe latter system bytranslation of already known reduction rules for the former system.
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  6. Allen P. Hazen & Francis Jeffry Pelletier (forthcoming). Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different. Studia Logica:1-40.score: 240.0
    Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives (which leads to a view of semantics called ‘inferentialism’). The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. (...)
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  7. Jan von Plato (forthcoming). From Axiomatic Logic to Natural Deduction. Studia Logica:1-18.score: 240.0
    Recently discovered documents have shown how Gentzen had arrived at the final form of natural deduction, namely by trying out a great number of alternative formulations. What led him to natural deduction in the first place, other than the general idea of studying “mathematical inference as it appears in practice,” is not indicated anywhere in his publications or preserved manuscripts. It is suggested that formal work in axiomatic logic lies behind the birth of Gentzen’s natural (...)
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  8. Jonathan Payne (forthcoming). Natural Deduction for Modal Logic with a Backtracking Operator. Journal of Philosophical Logic:1-22.score: 240.0
    Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal (...)
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  9. Norihiro Kamide & Motohiko Mouri (2007). Natural Deduction Systems for Some Non-Commutative Logics. Logic and Logical Philosophy 16 (2-3):105-146.score: 240.0
    Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL.
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  10. Hans Lycke (2009). Fitch-Style Natural Deduction for Modal Paralogics. Logique Et Analyse 207:193-218.score: 240.0
    In this paper, I will present a Fitch–style natural deduction proof theory for modal paralogics (modal logics with gaps and/or gluts for negation). Besides the standard classical subproofs, the presented proof theory also contains modal subproofs, which express what would follow from a hypothesis, in case it would be true in some arbitrary world.
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  11. Barry Hartley Slater (2008). Harmonising Natural Deduction. Synthese 163 (2):187 - 198.score: 240.0
    Prawitz proved a theorem, formalising 'harmony' in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, (...)
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  12. Ernst Zimmermann (2010). Full Lambek Calculus in Natural Deduction. Mathematical Logic Quarterly 56 (1):85-88.score: 210.0
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  13. J. Von Plato (2003). Translations From Natural Deduction to Sequent Calculus. Mathematical Logic Quarterly 49 (5):435.score: 210.0
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  14. Annika Kanckos (2010). Consistency of Heyting Arithmetic in Natural Deduction. Mathematical Logic Quarterly 56 (6):611-624.score: 210.0
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  15. J. Von Plato (2000). A Problem of Normal Form in Natural Deduction. Mathematical Logic Quarterly 46 (1):121-124.score: 210.0
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  16. William P. Bechtel (1994). Natural Deduction in Connectionist Systems. Synthese 101 (3):433-463.score: 192.0
    The relation between logic and thought has long been controversial, but has recently influenced theorizing about the nature of mental processes in cognitive science. One prominent tradition argues that to explain the systematicity of thought we must posit syntactically structured representations inside the cognitive system which can be operated upon by structure sensitive rules similar to those employed in systems of natural deduction. I have argued elsewhere that the systematicity of human thought might better be explained as resulting (...)
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  17. Edgar Jose Andrade & Edward Samuel Becerra (2008). Establishing Connections Between Aristotle's Natural Deduction and First-Order Logic. History and Philosophy of Logic 29 (4):309-325.score: 180.0
    This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the (...)
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  18. Hartley Slater (2008). Harmonising Natural Deduction. Synthese 163 (2):187 - 198.score: 180.0
    Prawitz proved a theorem, formalising 'harmony' in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, (...)
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  19. Francis Jeffry Pelletier, A History of Natural Deduction and Elementary Logic Textbooks.score: 180.0
    In 1934 a most singular event occurred. Two papers were published on a topic that had (apparently) never before been written about, the authors had never been in contact with one another, and they had (apparently) no common intellectual background that would otherwise account for their mutual interest in this topic.1 These two papers formed the basis for a movement in logic which is by now the most common way of teaching elementary logic by far, and indeed is perhaps all (...)
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  20. Peter Milne (2010). Subformula and Separation Properties in Natural Deduction Via Small Kripke Models. Review of Symbolic Logic 3 (2):175-227.score: 180.0
    Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaumlmarck.
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  21. Francis Jeffry Pelletier (1999). A Brief History of Natural Deduction. History and Philosophy of Logic 20 (1):1-31.score: 180.0
    Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Ja?kowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers (...)
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  22. Sara Negri & Jan von Plato (2001). Sequent Calculus in Natural Deduction Style. Journal of Symbolic Logic 66 (4):1803-1816.score: 180.0
    A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula (...)
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  23. Göran Sundholm (2006). Semantic Values for Natural Deduction Derivations. Synthese 148 (3):623 - 638.score: 180.0
    Drawing upon Martin-Löf’s semantic framework for his constructive type theory, semantic values are assigned also to natural-deduction derivations, while observing the crucial distinction between (logical) consequence among propositions and inference among judgements. Derivations in Gentzen’s (1934–5) format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects (of propositions), whereas derivations in Gentzen’s (1936) sequential format are (blue-prints for) proof-acts.
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  24. Allard Tamminga & Koji Tanaka (1999). A Natural Deduction System for First Degree Entailment. Notre Dame Journal of Formal Logic 40 (2):258-272.score: 180.0
    This paper is concerned with a natural deduction system for First Degree Entailment (FDE). First, we exhibit a brief history of FDE and of combined systems whose underlying idea is used in developing the natural deduction system. Then, after presenting the language and a semantics of FDE, we develop a natural deduction system for FDE. We then prove soundness and completeness of the system with respect to the semantics. The system neatly represents the four-valued (...)
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  25. Yannis Delmas-Rigoutsos (1997). A Double Deduction System for Quantum Logic Based on Natural Deduction. Journal of Philosophical Logic 26 (1):57-67.score: 180.0
    The author presents a deduction system for Quantum Logic. This system is a combination of a natural deduction system and rules based on the relation of compatibility. This relation is the logical correspondant of the commutativity of observables in Quantum Mechanics or perpendicularity in Hilbert spaces. Contrary to the system proposed by Gibbins and Cutland, the natural deduction part of the system is pure: no algebraic artefact is added. The rules of the system are the (...)
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  26. Wilfried Sieg & John Byrnes (1998). Normal Natural Deduction Proofs (in Classical Logic). Studia Logica 60 (1):67-106.score: 180.0
    Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? (...)
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  27. David Basin, Seán Matthews & Luca Viganò (1998). Natural Deduction for Non-Classical Logics. Studia Logica 60 (1):119-160.score: 180.0
    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, (...)
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  28. Frederic B. Fitch (1973). Natural Deduction Rules for English. Philosophical Studies 24 (2):89 - 104.score: 180.0
    A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as the c, a (an) c, some c, any c, and every c, where c represents a common noun. These latter expressions are called quantifiers, and other expressions of the form that c or that c itself, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), (...)
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  29. J. A. Burgess & I. L. Humberstone (1987). Natural Deduction Rules for a Logic of Vagueness. Erkenntnis 27 (2):197-229.score: 180.0
    Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded middle.
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  30. James W. Garson (2001). Natural Semantics: Why Natural Deduction is Intuitionistic. Theoria 67 (2):114-139.score: 180.0
    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 (...)
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  31. John M. Martin (1997). Aristotle'S Natural Deduction Reconsidered. History and Philosophy of Logic 18 (1):1-15.score: 180.0
    John Corcoran?s natural deduction system for Aristotle?s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz?s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role (...)
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  32. Michel Parigot (1997). Proofs of Strong Normalisation for Second Order Classical Natural Deduction. Journal of Symbolic Logic 62 (4):1461-1479.score: 180.0
    We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.
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  33. René David & Karim Nour (2003). A Short Proof of the Strong Normalization of Classical Natural Deduction with Disjunction. Journal of Symbolic Logic 68 (4):1277-1288.score: 180.0
    We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.
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  34. Peter Schroeder-Heister (1984). A Natural Extension of Natural Deduction. Journal of Symbolic Logic 49 (4):1284-1300.score: 180.0
    The framework of natural deduction is extended by permitting rules as assumptions which may be discharged in the course of a derivation. this leads to the concept of rules of higher levels and to a general schema for introduction and elimination rules for arbitrary n-ary sentential operators. with respect to this schema, (functional) completeness "or", "if..then" and absurdity is proved.
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  35. Luca Tranchini (2012). Natural Deduction for Dual-Intuitionistic Logic. Studia Logica 100 (3):631-648.score: 180.0
    We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.
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  36. Paul C. Gilmore (1986). Natural Deduction Based Set Theories: A New Resolution of the Old Paradoxes. Journal of Symbolic Logic 51 (2):393-411.score: 180.0
    The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as (...)
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  37. Andrzej Indrzejczak (2003). A Labelled Natural Deduction System for Linear Temporal Logic. Studia Logica 75 (3):345 - 376.score: 180.0
    The paper is devoted to the concise description of some Natural Deduction System (ND for short) for Linear Temporal Logic. The system's distinctive feature is that it is labelled and analytical. Labels convey necessary semantic information connected with the rules for temporal functors while the analytical character of the rules lets the system work as a decision procedure. It makes it more similar to Labelled Tableau Systems than to standard Natural Deduction. In fact, our solution of (...)
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  38. Koji Nakazawa & Makoto Tatsuta (2003). Corrigendum to "Strong Normalization Proof with CPS-Translation for Second Order Classical Natural Deduction". Journal of Symbolic Logic 68 (4):1415-1416.score: 180.0
    This paper points out an error of Parigot's proof of strong normalization of second order classical natural deduction by the CPS-translation, discusses erasing-continuation of the CPS-translation, and corrects that proof by using the notion of augmentations.
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  39. Koji Nakazawa & Makoto Tatsuta (2003). Strong Normalization Proof with CPS-Translation for Second Order Classical Natural Deduction. Journal of Symbolic Logic 68 (3):851-859.score: 180.0
    This paper points out an error of Parigot's proof of strong normalization of second order classical natural deduction by the CPS-translation, discusses erasing-continuation of the CPS-translation, and corrects that proof by using the notion of augmentations.
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  40. Francis Jeffry Pelletier (1998). Automated Natural Deduction in Thinker. Studia Logica 60 (1):3-43.score: 180.0
    Although resolution-based inference is perhaps the industry standard in automated theorem proving, there have always been systems that employed a different format. For example, the Logic Theorist of 1957 produced proofs by using an axiomatic system, and the proofs it generated would be considered legitimate axiomatic proofs; Wang’s systems of the late 1950’s employed a Gentzen-sequent proof strategy; Beth’s systems written about the same time employed his semantic tableaux method; and Prawitz’s systems of again about the same time are often (...)
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  41. Michael Gabbay, Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning.score: 180.0
    In this paper we make some observations about Natural Deduction derivations [Prawitz, 1965, van Dalen, 1986, Bell and Machover, 1977]. We assume the reader is familiar with it and with proof-theory in general. Our development will be simple, even simple-minded, and concrete. However, it will also be evident that general ideas motivate our examples, and we think both our specific examples and the ideas behind them are interesting and may be useful to some readers. In a sentence, the (...)
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  42. John Pollock, Natural Deduction.score: 180.0
    Most automated theorem provers are clausal-form provers based on variants of resolutionrefutation. In my [1990], I described the theorem prover OSCAR that was based instead on natural deduction. Some limited evidence was given suggesting that OSCAR was suprisingly efficient. The evidence consisted of a handful of problems for which published data was available describing the performance of other theorem provers. This evidence was suggestive, but based upon too meager a comparison to be conclusive. The question remained, “How does (...)
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  43. Claudio Cerrato (1994). Natural Deduction Based Upon Strict Implication for Normal Modal Logics. Notre Dame Journal of Formal Logic 35 (4):471-495.score: 180.0
    We present systems of Natural Deduction based on Strict Implication for the main normal modal logics between K and S5. In this work we consider Strict Implication as the main modal operator, and establish a natural correspondence between Strict Implication and strict subproofs.
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  44. K. Broda, M. Finger & A. Russo (1999). Labelled Natural Deduction for Substructural Logics. Logic Journal of the Igpl 7 (3):283-318.score: 180.0
    In this paper a uniform methodology to perform natural\ndeduction over the family of linear, relevance and intuitionistic\nlogics is proposed. The methodology follows the Labelled\nDeductive Systems (LDS) discipline, where the deductive process\nmanipulates {\em declarative units} -- formulas {\em labelled}\naccording to a {\em labelling algebra}. In the system described\nhere, labels are either ground terms or variables of a given {\em\nlabelling language} and inference rules manipulate formulas and\nlabels simultaneously, generating (whenever necessary)\nconstraints on the labels used in the rules. A set of (...)\ndeduction style inference rules is given, and the notion of a\n{\em derivation} is defined which associates a labelled natural\ndeduction style ``structural derivation'' with a set of generated\nconstraints. Algorithmic procedures, based on a technique called\n{\em resource abduction\/}, are defined to solve the constraints\ngenerated within a structural derivation, and their termination\nconditions discussed. A natural deduction derivation is then\ndefined to be {\em correct} with respect to a given substructural\nlogic, if, under the condition that the algorithmic procedures\nterminate, the associated set of constraints is satisfied with\nrespect to the underlying labelling algebra. Finally, soundness\nand completeness of the natural deduction system are proved with\nrespect to the LKE tableaux system \cite{daga:LAD}.\footnote{Full\nversion of a paper presented at the {\em 3rd Workshop on Logic,\nLanguage, Information and Computation\/} ({\em WoLLIC'96\/}), May\n8--10, Salvador (Bahia), Brazil, organised by Univ.\ Federal de\nPernambuco (UFPE) and Univ.\ Federal da Bahia (UFBA), and\nsponsored by IGPL, FoLLI, and ASL.}. (shrink)
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  45. J. L. Mackie (1958). The Rules of Natural Deduction. Analysis 19 (2):27 - 35.score: 180.0
    This article is a clarification of different procedures in natural deduction: universal instantiation, Universal generalisation, Existential generalisation, And existential instantiation. The author discusses rules concerning universal generalisation from copi's "symbolic logic". (staff).
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  46. Tor Sandqvist (2012). The Subformula Property In Classical Natural Deduction Established Constructively. Review of Symbolic Logic 5 (4):710-719.score: 180.0
    A constructive proof is provided for the claim that classical first-order logic admits of a natural deduction formulation featuring the subformula property.
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  47. Maria Da Paz N. Medeiros (2006). A New S4 Classical Modal Logic in Natural Deduction. Journal of Symbolic Logic 71 (3):799 - 809.score: 180.0
    We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.
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  48. Jan von Plato & Annika Siders (2012). Normal Derivability in Classical Natural Deduction. Review of Symbolic Logic 5 (2):205-211.score: 180.0
    A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.
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  49. Wilfried Sieg & Saverio Cittadini, Normal Natural Deduction Proof (In Non-Classical Logics).score: 180.0
    Wilfred Sieg and Saverio Cittadini. Normal Natural Deduction Proof (In Non-Classical Logics.
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  50. Sara Negri (2002). A Normalizing System of Natural Deduction for Intuitionistic Linear Logic. Archive for Mathematical Logic 41 (8):789-810.score: 180.0
    The main result of this paper is a normalizing system of natural deduction for the full language of intuitionistic linear logic. No explicit weakening or contraction rules for -formulas are needed. By the systematic use of general elimination rules a correspondence between normal derivations and cut-free derivations in sequent calculus is obtained. Normalization and the subformula property for normal derivations follow through translation to sequent calculus and cut-elimination.
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