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

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  1. Norihiro Kamide & Motohiko Mouri (2007). Natural Deduction Systems for Some Non-Commutative Logics. Logic and Logical Philosophy 16 (2-3):105-146.
    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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  2. John Corcoran (1974). Aristotle's Natural Deduction System. In Ancient Logic and its Modern Interpretations. Boston,Reidel 85--131.
    This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".
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  3.  14
    Allen P. Hazen & Francis Jeffry Pelletier (2014). Gentzen and Jaśkowski Natural Deduction: Fundamentally Similar but Importantly Different. Studia Logica 102 (6):1103-1142.
    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 . 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. We close with a demonstration of this latter (...)
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  4.  7
    Tin Perkov (forthcoming). Natural Deduction for Modal Logic of Judgment Aggregation. Journal of Logic, Language and Information:1-20.
    We can formalize judgments as logical formulas. Judgment aggregation deals with judgments of several agents, which need to be aggregated to a collective judgment. There are several logical formalizations of judgment aggregation. This paper focuses on a modal formalization which nicely expresses classical properties of judgment aggregation rules and famous results of social choice theory, like Arrow’s impossibility theorem. A natural deduction system for modal logic of judgment aggregation is presented in this paper. (...)
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  5. Moritz Cordes & Friedrich Reinmuth, A Speech Act Calculus. A Pragmatised Natural Deduction Calculus and its Meta-Theory.
    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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  6.  39
    Torben BraÜner (2005). Natural Deduction for First-Order Hybrid Logic. Journal of Logic, Language and Information 14 (2):173-198.
    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 (...)
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  7.  18
    Nissim Francez (2014). Harmony in Multiple-Conclusion Natural-Deduction. Logica Universalis 8 (2):215-259.
    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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  8.  17
    Torben Braüner (2004). Two Natural Deduction Systems for Hybrid Logic: A Comparison. [REVIEW] Journal of Logic, Language and Information 13 (1):1-23.
    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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  9.  4
    Francesca Poggiolesi (forthcoming). Natural Deduction Calculi and Sequent Calculi for Counterfactual Logics. Studia Logica:1-34.
    In this paper we present labelled sequent calculi and labelled natural deduction calculi for the counterfactual logics CK + {ID, MP}. As for the sequent calculi we prove, in a semantic manner, that the cut-rule is admissible. As for the natural deduction calculi we prove, in a purely syntactic way, the normalization theorem. Finally, we demonstrate that both calculi are sound and complete with respect to Nute semantics [12] and that the natural deduction (...)
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  10.  18
    Richard Zach (2016). Natural Deduction for the Sheffer Stroke and Peirce’s Arrow. Journal of Philosophical Logic 45 (2):183-197.
    Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic (...)
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  11.  2
    Bartosz Więckowski (forthcoming). Subatomic Natural Deduction for a Naturalistic First-Order Language with Non-Primitive Identity. Journal of Logic, Language and Information:1-54.
    A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language. Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula (...)
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  12.  35
    Jonathan Payne (2015). Natural Deduction for Modal Logic with a Backtracking Operator. Journal of Philosophical Logic 44 (3):237-258.
    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 (...)
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  13.  7
    J. Von Plato (2003). Translations From Natural Deduction to Sequent Calculus. Mathematical Logic Quarterly 49 (5):435.
    Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which (...)
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  14.  10
    Hans Lycke (2009). Fitch-Style Natural Deduction for Modal Paralogics. Logique Et Analyse 207:193-218.
    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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  15.  7
    Ernst Zimmermann (2010). Full Lambek Calculus in Natural Deduction. Mathematical Logic Quarterly 56 (1):85-88.
    A formulation of Full Lambek Calculus in the framework of natural deduction is given.
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  16.  8
    Carlo Cellucci (1995). On Quine's Approach to Natural Deduction'. In Paolo Leonardi & Marco Santambrogio (eds.), On Quine: New Essays. Cambridge University Press 314--335.
    This article examines Quine's original proposal for a natural deduction calculus including an existential specification rule, it argues that it introduces a new paradigm of natural deduction alternative to Gentzen's but has some substantial defects. As an alternative the article puts forward a system of sequent natural deduction.
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  17.  5
    J. Von Plato (2000). A Problem of Normal Form in Natural Deduction. Mathematical Logic Quarterly 46 (1):121-124.
    Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal. It is shown that this problem is caused by a non-normality inherent in the usual modus ponens rule.
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  18.  16
    Jan von Plato (2014). From Axiomatic Logic to Natural Deduction. Studia Logica 102 (6):1167-1184.
    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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  19.  53
    Susan Rogerson (2007). Natural Deduction and Curry's Paradox. Journal of Philosophical Logic 36 (2):155 - 179.
    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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  20.  4
    Barry Hartley Slater (2008). Harmonising Natural Deduction. Synthese 163 (2):187 - 198.
    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 (...)
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  21.  4
    Annika Kanckos (2010). Consistency of Heyting Arithmetic in Natural Deduction. Mathematical Logic Quarterly 56 (6):611-624.
    A proof of the consistency of Heyting arithmetic formulated in natural deduction is given. The proof is a reduction procedure for derivations of falsity and a vector assignment, such that each reduction reduces the vector. By an interpretation of the expressions of the vectors as ordinals each derivation of falsity is assigned an ordinal less than ε 0, thus proving termination of the procedure.
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  22.  30
    Dag Prawitz (1965/2006). Natural Deduction: A Proof-Theoretical Study. Dover Publications.
    This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.
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  23. William P. Bechtel (1994). Natural Deduction in Connectionist Systems. Synthese 101 (3):433-463.
    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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  24. Peter Schroeder-Heister (1984). A Natural Extension of Natural Deduction. Journal of Symbolic Logic 49 (4):1284-1300.
    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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  25.  4
    Jan von Plato (2001). Natural Deduction with General Elimination Rules. Archive for Mathematical Logic 40 (7):541-567.
    The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free (...)
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  26.  69
    Allard Tamminga (1994). Logics of Rejection: Two Systems of Natural Deduction. Logique Et Analyse 146:169-208.
    This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the two (...)
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  27. Sara Negri & Jan von Plato (2001). Sequent Calculus in Natural Deduction Style. Journal of Symbolic Logic 66 (4):1803-1816.
    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 (...)
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  28.  4
    Ralph Matthes (2005). Non-Strictly Positive Fixed Points for Classical Natural Deduction. Annals of Pure and Applied Logic 133 (1):205-230.
    Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions.It covers second-order universal quantification and also the extension of the logic with fixed points of non-strictly positive operators, which appears to be a new result.Finally, the relation to Parigot’s strictly positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained.
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  29.  30
    Michel Parigot (1997). Proofs of Strong Normalisation for Second Order Classical Natural Deduction. Journal of Symbolic Logic 62 (4):1461-1479.
    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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  30.  25
    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.
    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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  31.  30
    David Basin, Seán Matthews & Luca Viganò (1998). Natural Deduction for Non-Classical Logics. Studia Logica 60 (1):119-160.
    We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, (...)
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  32. 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.
    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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  33.  23
    John M. Martin (1997). Aristotle'S Natural Deduction Reconsidered. History and Philosophy of Logic 18 (1):1-15.
    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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  34. 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.
    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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  35.  50
    Francis Jeffry Pelletier (1999). A Brief History of Natural Deduction. History and Philosophy of Logic 20 (1):1-31.
    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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  36.  13
    Jan von Plato & Annika Siders (2012). Normal Derivability in Classical Natural Deduction. Review of Symbolic Logic 5 (2):205-211.
    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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  37.  9
    Andrzej Indrzejczak, Natural Deduction.
    Natural Deduction Natural Deduction is a common name for the class of proof systems composed of simple and self-evident inference rules based upon methods of proof and traditional ways of reasoning that have been applied since antiquity in deductive practice. The first formal ND systems were independently constructed in the 1930s by G. Gentzen and S. Jaśkowski and … Continue reading Natural Deduction →.
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  38.  31
    J. A. Burgess & I. L. Humberstone (1987). Natural Deduction Rules for a Logic of Vagueness. Erkenntnis 27 (2):197-229.
    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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  39.  33
    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 (...)
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  40.  6
    Sara Negri (2002). A Normalizing System of Natural Deduction for Intuitionistic Linear Logic. Archive for Mathematical Logic 41 (8):789-810.
    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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  41.  3
    Y. Andou (2003). Church–Rosser Property of a Simple Reduction for Full First-Order Classical Natural Deduction. Annals of Pure and Applied Logic 119 (1-3):225-237.
    A system of typed terms which corresponds with the classical natural deduction with one conclusion and full logical symbols is defined. Church–Rosser property of the system is proved using an extended method of parallel reduction.
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  42.  6
    Jan von Plato (2011). A Sequent Calculus Isomorphic to Gentzen's Natural Deduction. Review of Symbolic Logic 4 (1):43-53.
    Gentzens natural deduction. Thereby the appearance of the cuts in translation is explained.
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  43.  5
    Karim Nour & Khelifa Saber (2006). A Semantical Proof of the Strong Normalization Theorem for Full Propositional Classical Natural Deduction. Archive for Mathematical Logic 45 (3):357-364.
    We give in this paper a short semantical proof of the strong normalization for full propositional classical natural deduction. This proof is an adaptation of reducibility candidates introduced by J.-Y. Girard and simplified to the classical case by M. Parigot.
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  44.  63
    Allard Tamminga & Koji Tanaka (1999). A Natural Deduction System for First Degree Entailment. Notre Dame Journal of Formal Logic 40 (2):258-272.
    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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  45.  6
    Maria Da Paz N. Medeiros (2006). A New S4 Classical Modal Logic in Natural Deduction. Journal of Symbolic Logic 71 (3):799 - 809.
    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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  46.  42
    Francis Jeffry Pelletier, A History of Natural Deduction and Elementary Logic Textbooks.
    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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  47.  45
    J. L. Mackie (1958). The Rules of Natural Deduction. Analysis 19 (2):27 - 35.
    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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  48.  5
    Ana Teresa Martins & Lilia Ramalho Martins (2008). Full Classical S5 in Natural Deduction with Weak Normalization. Annals of Pure and Applied Logic 152 (1):132-147.
    Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on Mathematics, 2006] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic only for a language without logical or, there exists and with a restricted application of reduction ad absurdum. Reduction steps related to logical or, (...)
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  49.  33
    Göran Sundholm (2006). Semantic Values for Natural Deduction Derivations. Synthese 148 (3):623-638.
    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 consequence among propositions and inference among judgements. Derivations in Gentzen’s format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects, whereas derivations in Gentzen’s sequential format are proof-acts.
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  50.  3
    Carlo Cellucci (1992). Existential Instantiation and Normalization in Sequent Natural Deduction. Annals of Pure and Applied Logic 58 (2):111-148.
    ellucci, C., Existential instantiation and normalization in sequent natural deduction, Annals of Pure and Applied Logic 58 111–148. A sequent conclusion natural deduction system is introduced in which classical logic is treated per se, not as a special case of intuitionistic logic. The system includes an existential instantiation rule and involves restrictions on the discharge rules. Contrary to the standard formula conclusion natural deduction systems for classical logic, its normal derivations satisfy both the subformula (...)
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