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Natural Deduction: A Proof-Theoretical Study

Stockholm, Sweden: Dover Publications (1965)

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  1. Single-Assumption Systems in Proof-Theoretic Semantics.Leonardo Ceragioli - 2022 - Journal of Philosophical Logic 51 (5):1019-1054.
    Proof-theoretic semantics is an inferentialist theory of meaning, usually developed in a multiple-assumption and single-conclusion framework. In that framework, this theory seems unable to justify classical logic, so some authors have proposed a multiple-conclusion reformulation to accomplish this goal. In the first part of this paper, the debate originated by this proposal is briefly exposed and used to defend the diverging opinion that proof-theoretic semantics should always endorse a single-assumption and single-conclusion framework. In order to adopt this approach some of (...)
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  • The Correspondence Between Cut-Elimination and Normalization II.J. Zucker - 1974 - Annals of Mathematical Logic 7 (2):113.
  • Cut-Elimination and Normalization.J. Zucker - 1974 - Annals of Mathematical Logic 7 (1):1.
  • Natural Deduction Bottom Up.Ernst Zimmermann - 2021 - Journal of Logic, Language and Information 30 (3):601-631.
    The paper introduces a new type of rules into Natural Deduction, elimination rules by composition. Elimination rules by composition replace usual elimination rules in the style of disjunction elimination and give a more direct treatment of additive disjunction, multiplicative conjunction, existence quantifier and possibility modality. Elimination rules by composition have an enormous impact on proof-structures of deductions: they do not produce segments, deduction trees remain binary branching, there is no vacuous discharge, there is only few need of permutations. This new (...)
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  • Truth, Demonstration and Knowledge. A Classical Solution to the Paradox of Knowability.Elia Zardini - 2015 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 30 (3):365-392.
    After introducing semantic anti-realism and the paradox of knowability, the paper offers a reconstruction of the anti-realist argument from understanding. The proposed reconstruction validates an unrestricted principle to the effect that truth requires the existence of a certain kind of “demonstration”. The paper shows that that principle fails to imply the problematic instances of the original unrestricted feasible-knowability principle but that the overall view underlying the new principle still has unrestricted epistemic consequences. Appealing precisely to the paradox of knowability, the (...)
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  • Substructural approaches to paradox: an introduction to the special issue.Elia Zardini - 2021 - Synthese 199 (3):493-525.
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  • Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and Any Other Truth-Functional Connective).Richard Zach - 2015 - 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 versions of (...)
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  • Theory of Logical Calculi: Basic Theory of Consequence Operations.Ryszard Wójcicki - 1988 - Dordrecht, Boston and London: Kluwer Academic Publishers.
    The general aim of this book is to provide an elementary exposition of some basic concepts in terms of which both classical and non-dassicallogirs may be studied and appraised. Although quantificational logic is dealt with briefly in the last chapter, the discussion is chiefly concemed with propo gjtional cakuli. Still, the subject, as it stands today, cannot br covered in one book of reasonable length. Rather than to try to include in the volume as much as possible, I have put (...)
  • Subatomic Negation.Bartosz Więckowski - 2021 - Journal of Logic, Language and Information 30 (1):207-262.
    The operators of first-order logic, including negation, operate on whole formulae. This makes it unsuitable as a tool for the formal analysis of reasoning with non-sentential forms of negation such as predicate term negation. We extend its language with negation operators whose scope is more narrow than an atomic formula. Exploiting the usefulness of subatomic proof-theoretic considerations for the study of subatomic inferential structure, we define intuitionistic subatomic natural deduction systems which have several subatomic operators and an additional operator for (...)
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  • Subatomic Natural Deduction for a Naturalistic First-Order Language with Non-Primitive Identity.Bartosz Więckowski - 2016 - Journal of Logic, Language and Information 25 (2):215-268.
    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 property with respect (...)
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  • Rules for Subatomic Derivation.Bartosz Więckowski - 2011 - Review of Symbolic Logic 4 (2):219-236.
    In proof-theoretic semantics the meaning of an atomic sentence is usually determined by a set of derivations in an atomic system which contain that sentence as a conclusion (see, in particular, Prawitz, 1971, 1973). The paper critically discusses this standard approach and suggests an alternative account which proceeds in terms of subatomic introduction and elimination rules for atomic sentences. A simple subatomic normal form theorem by which this account of the semantics of atomic sentences and the terms from which they (...)
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  • A Consistent Theory of Attributes in a Logic Without Contraction.Richard B. White - 1993 - Studia Logica 52 (1):113 - 142.
    This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in (...)
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  • Typability and Type Checking in System F Are Equivalent and Undecidable.J. B. Wells - 1999 - Annals of Pure and Applied Logic 98 (1-3):111-156.
    Girard and Reynolds independently invented System F to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and (...)
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  • Revisiting Quine on Truth by Convention.Jared Warren - 2017 - Journal of Philosophical Logic 46 (2):119-139.
    In “Truth by Convention” W.V. Quine gave an influential argument against logical conventionalism. Even today his argument is often taken to decisively refute logical conventionalism. Here I break Quine’s arguments into two— the super-task argument and the regress argument—and argue that while these arguments together refute implausible explicit versions of conventionalism, they cannot be successfully mounted against a more plausible implicit version of conventionalism. Unlike some of his modern followers, Quine himself recognized this, but argued that implicit conventionalism was explanatorily (...)
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  • Rereading Gentzen.Jan Von Plato - 2003 - Synthese 137 (1-2):195 - 209.
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  • Normal Derivability in Classical Natural Deduction.Jan Von Plato & Annika Siders - 2012 - 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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  • A Sequent Calculus Isomorphic to Gentzen’s Natural Deduction.Jan von Plato - 2011 - 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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  • On the Semantics of the Universal Quantifier.Djordje Čubrić - 1997 - Annals of Pure and Applied Logic 87 (3):209-239.
    We investigate the universal fragment of intuitionistic logic focussing on equality of proofs. We give categorical models for that and prove several completeness results. One of them is a generalization of the well known Yoneda lemma and the other is an extension of Harvey Friedman's completeness result for typed lambda calculus.
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  • The Naturality of Natural Deduction.Luca Tranchini, Paolo Pistone & Mattia Petrolo - 2019 - Studia Logica 107 (1):195-231.
    Developing a suggestion by Russell, Prawitz showed how the usual natural deduction inference rules for disjunction, conjunction and absurdity can be derived using those for implication and the second order quantifier in propositional intuitionistic second order logic NI\. It is however well known that the translation does not preserve the relations of identity among derivations induced by the permutative conversions and immediate expansions for the definable connectives, at least when the equational theory of NI\ is assumed to consist only of (...)
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  • Stabilizing Quantum Disjunction.Luca Tranchini - 2018 - Journal of Philosophical Logic 47 (6):1029-1047.
    Since the appearance of Prior’s tonk, inferentialists tried to formulate conditions that a collection of inference rules for a logical constant has to satisfy in order to succeed in conferring an acceptable meaning to it. Dummett proposed a pair of conditions, dubbed ‘harmony’ and ‘stability’ that have been cashed out in terms of the existence of certain transformations on natural deduction derivations called reductions and expansions. A long standing open problem for this proposal is posed by quantum disjunction: although its (...)
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  • Proof-Theoretic Harmony: Towards an Intensional Account.Luca Tranchini - 2016 - Synthese 198 (Suppl 5):1145-1176.
    In this paper we argue that an account of proof-theoretic harmony based on reductions and expansions delivers an inferentialist picture of meaning which should be regarded as intensional, as opposed to other approaches to harmony that will be dubbed extensional. We show how the intensional account applies to any connective whose rules obey the inversion principle first proposed by Prawitz and Schroeder-Heister. In particular, by improving previous formulations of expansions, we solve a problem with quantum-disjunction first posed by Dummett. As (...)
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  • Proof, Meaning and Paradox: Some Remarks.Luca Tranchini - 2019 - Topoi 38 (3):591-603.
    In the present paper, the Fregean conception of proof-theoretic semantics that I developed elsewhere will be revised so as to better reflect the different roles played by open and closed derivations. I will argue that such a conception can deliver a semantic analysis of languages containing paradoxical expressions provided some of its basic tenets are liberalized. In particular, the notion of function underlying the Brouwer–Heyting–Kolmogorov explanation of implication should be understood as admitting functions to be partial. As argued in previous (...)
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  • Natural Deduction for Dual-Intuitionistic Logic.Luca Tranchini - 2012 - Studia Logica 100 (3):631-648.
    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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  • Mereotopology in 2nd-Order and Modal Extensions of Intuitionistic Propositional Logic.Paolo Torrini, John G. Stell & Brandon Bennett - 2002 - Journal of Applied Non-Classical Logics 12 (3-4):495-525.
    We show how mereotopological notions can be expressed by extending intuitionistic propositional logic with propositional quantification and a strong modal operator. We first prove completeness for the logics wrt Kripke models; then we trace the correspondence between Kripke models and topological spaces that have been enhanced with an explicit notion of expressible region. We show how some qualitative spatial notions can be expressed in topological terms. We use the semantical and topological results in order to show how in some extensions (...)
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  • In Defence of Deductive Inference.Paul Thagard - 1979 - Australasian Journal of Philosophy 57 (3):274 – 279.
  • Normalizability, Cut Eliminability and Paradox.Neil Tennant - 2016 - Synthese 199 (Suppl 3):597-616.
    This is a reply to the considerations advanced by Schroeder-Heister and Tranchini as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant and subsequently amended in Tennant. Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.
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  • Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural Logicism.Neil Tennant - 2014 - Philosophia Mathematica 22 (3):321-344.
    We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that could (...)
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  • Intuitionistic Mathematics Does Not Needex Falso Quodlibet.Neil Tennant - 1994 - Topoi 13 (2):127-133.
    We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consistent set of (...)
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  • Existence and Identity in Free Logic: A Problem for Inferentialism?Neil Tennant - 2007 - Mind 116 (464):1055-1078.
    Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantifier and the identity predicate. The final part of the investigation brings out an underlying order in a basic family of free logics.
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  • Cut for Core Logic.Neil Tennant - 2012 - Review of Symbolic Logic 5 (3):450-479.
    The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain.
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  • A Proof-Theoretic Approach to Entailment.N. Tennant - 1980 - Journal of Philosophical Logic 9 (2):185 - 209.
  • A New Unified Account of Truth and Paradox.N. Tennant - 2015 - Mind 124 (494):571-605.
    I propose an anti-realist account of truth and paradox according to which the logico-semantic paradoxes are not genuine inconsistencies. The ‘global’ proofs of absurdity associated with these paradoxes cannot be brought into normal form. The account combines epistemicism about truth with a proof-theoretic diagnosis of paradoxicality. The aim is to combine a substantive philosophical account of truth with a more rigorous and technical diagnosis of the source of paradox for further consideration by logicians. Core Logic plays a central role in (...)
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  • Aristotle’s Syllogistic and Core Logic.Neil Tennant - 2014 - History and Philosophy of Logic 35 (2):120-147.
    I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is (...)
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  • Globalization of Intui Tionistic Set Theory.Gaisi Takeuti & Satoko Titani - 1987 - Annals of Pure and Applied Logic 33 (C):195-211.
  • Some Weak Variants of the Existence and Disjunction Properties in Intermediate Predicate Logics.Nobu-Yuki Suzuki - 2017 - Bulletin of the Section of Logic 46 (1/2).
    We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.
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  • Semantic Values for Natural Deduction Derivations.Göran Sundholm - 2006 - 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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  • Implicit Epistemic Aspects of Constructive Logic.Göran Sundholm - 1997 - Journal of Logic, Language and Information 6 (2):191-212.
    In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite (...)
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  • A Completeness Proof for an Infinitary Tense-Logic.B. G. Sundholm - 1977 - Theoria 43 (1):47-51.
  • Reasoning Processes in Propositional Logic.Claes Strannegård, Simon Ulfsbäcker, David Hedqvist & Tommy Gärling - 2010 - Journal of Logic, Language and Information 19 (3):283-314.
    We conducted a computer-based psychological experiment in which a random mix of 40 tautologies and 40 non-tautologies were presented to the participants, who were asked to determine which ones of the formulas were tautologies. The participants were eight university students in computer science who had received tuition in propositional logic. The formulas appeared one by one, a time-limit of 45 s applied to each formula and no aids were allowed. For each formula we recorded the proportion of the participants who (...)
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  • Reasoning About Truth in First-Order Logic.Claes Strannegård, Fredrik Engström, Abdul Rahim Nizamani & Lance Rips - 2013 - Journal of Logic, Language and Information 22 (1):115-137.
    First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when (...)
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  • What Harmony Could and Could Not Be.Florian Steinberger - 2011 - Australasian Journal of Philosophy 89 (4):617 - 639.
    The notion of harmony has played a pivotal role in a number of debates in the philosophy of logic. Yet there is little agreement as to how the requirement of harmony should be spelled out in detail or even what purpose it is to serve. Most, if not all, conceptions of harmony can already be found in Michael Dummett's seminal discussion of the matter in The Logical Basis of Metaphysics. Hence, if we wish to gain a better understanding of the (...)
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  • Varieties of Relevant S5.Shawn Standefer - forthcoming - Logic and Logical Philosophy:1-28.
    In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal (...)
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  • Harmonising Natural Deduction.Barry Hartley Slater - 2008 - 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 for Contradiction, then we can (...)
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  • A Note on the Relation Between Formal and Informal Proof.Jörgen Sjögren - 2010 - Acta Analytica 25 (4):447-458.
    Using Carnap’s concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof and informal, mathematical proof.
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  • Normalization Proof for Peano Arithmetic.Annika Siders - 2015 - Archive for Mathematical Logic 54 (7-8):921-940.
    A proof of normalization for a classical system of Peano Arithmetic formulated in natural deduction is given. The classical rule of the system is the rule for indirect proof restricted to atomic formulas. This rule does not, due to the restriction, interfere with the standard detour conversions. The convertible detours, numerical inductions and instances of indirect proof concluding falsity are reduced in a way that decreases a vector assigned to the derivation. By interpreting the expressions of the vectors as ordinals (...)
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  • On the Explanatory Power of Truth in Logic.Gila Sher - 2018 - Philosophical Issues 28 (1):348-373.
    Philosophers are divided on whether the proof- or truth-theoretic approach to logic is more fruitful. The paper demonstrates the considerable explanatory power of a truth-based approach to logic by showing that and how it can provide (i) an explanatory characterization —both semantic and proof-theoretical—of logical inference, (ii) an explanatory criterion for logical constants and operators, (iii) an explanatory account of logic’s role (function) in knowledge, as well as explanations of (iv) the characteristic features of logic —formality, strong modal force, generality, (...)
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  • Truth-Values as Labels: A General Recipe for Labelled Deduction.Cristina Sernadas, Luca Viganò, João Rasga & Amílcar Sernadas - 2003 - Journal of Applied Non-Classical Logics 13 (3-4):277-315.
    We introduce a general recipe for presenting non-classical logics in a modular and uniform way as labelled deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truth-values. More specifically, the main idea underlying our approach is the use of algebras of truth-values, whose operators reflect the semantics we have in mind, as the labelling algebras of our labelled deduction systems. The “truth-values as (...)
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  • Normalization and Excluded Middle. I.Jonathan P. Seldin - 1989 - Studia Logica 48 (2):193 - 217.
    The usual rule used to obtain natural deduction formulations of classical logic from intuitionistic logic, namely.
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  • Arbitrary Truth-Value Functions and Natural Deduction.Krister Segerberg - 1983 - Mathematical Logic Quarterly 29 (11):557-564.
  • Hyperdoctrines, Natural Deduction and the Beck Condition.Robert A. G. Seely - 1983 - Mathematical Logic Quarterly 29 (10):505-542.