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  1. Truth Values and Proof Theory.Greg Restall - 2009 - Studia Logica 92 (2):241-264.
    I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence.
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  • Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research.Yaroslav Shramko - 2005 - Studia Logica 80 (2-3):347-367.
    We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to (...)
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  • Meaning Approached Via Proofs.Dag Prawitz - 2006 - Synthese 148 (3):507-524.
    According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...)
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  • Co-constructive logic for proofs and refutations.James Trafford - 2014 - Studia Humana 3 (4):22-40.
    This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...)
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  • (1 other version)From Brouwer to Hilbert: the debate on the foundations of mathematics in the 1920s.Paolo Mancosu (ed.) - 1998 - New York: Oxford University Press.
    From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...)
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  • Conceptions of truth in intuitionism.Panu Raatikainen - 2004 - History and Philosophy of Logic 25 (2):131--45.
    Intuitionism’s disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.
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  • Ryszard Wójcicki. Theory of logical calculi. Basis theory of consequence operations. Synthese library, vol. 199. Kluwer Academic Publishers, Dordrecht, Boston, and London, 1988, xviii + 473 pp. [REVIEW]Stephen L. Bloom - 1990 - Journal of Symbolic Logic 55 (3):1324-1326.
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  • Why Conclusions Should Remain Single.Florian Steinberger - 2011 - Journal of Philosophical Logic 40 (3):333-355.
    This paper argues that logical inferentialists should reject multiple-conclusion logics. Logical inferentialism is the position that the meanings of the logical constants are determined by the rules of inference they obey. As such, logical inferentialism requires a proof-theoretic framework within which to operate. However, in order to fulfil its semantic duties, a deductive system has to be suitably connected to our inferential practices. I argue that, contrary to an established tradition, multiple-conclusion systems are ill-suited for this purpose because they fail (...)
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  • Reply to Dag Prawitz.Michael Dummett - 1987 - In Barry Taylor (ed.), Michael Dummett: contributions to philosophy. Hingham, MA, USA: Kluwer Academic Publishers. pp. 281--316.
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  • A short note on intuitionistic propositional logic with multiple conclusions.Valéria de Paiva & Luiz Pereira - 2005 - Manuscrito 28 (2):317-329.
    A common misconception among logicians is to think that intuitionism is necessarily tied-up with single conclusion calculi. Single conclusion calculi can be used to model intuitionism and they are convenient, but by no means are they necessary. This has been shown by such influential textbook authors as Kleene, Takeuti and Dummett, to cite only three. If single conclusions are not necessary, how do we guarantee that only intuitionistic derivations are allowed? Traditionally one insists on restrictions on particular rules: implication right, (...)
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  • (1 other version)From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s.Paolo Mancosu (ed.) - 1997 - Oxford, England: Oxford University Press USA.
    From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...)
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  • Dual-Intuitionistic Logic.Igor Urbas - 1996 - Notre Dame Journal of Formal Logic 37 (3):440-451.
    The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a (...)
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  • Applications of Kripke models to Heyting-Brouwer logic.Cecylia Rauszer - 1977 - Studia Logica 36 (1-2):61 - 71.
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  • Structure in mathematics and logic: A categorical perspective.S. Awodey - 1996 - Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...)
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  • Introduction to Higher Order Categorical Logic.Joachim Lambek & Philip J. Scott - 1986 - Cambridge University Press.
    In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language (...)
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  • From absolute to local mathematics.J. L. Bell - 1986 - Synthese 69 (3):409 - 426.
    In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.
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  • (1 other version)Constructible falsity.David Nelson - 1949 - Journal of Symbolic Logic 14 (1):16-26.
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  • Dualising Intuitionictic Negation.Graham Priest - 2009 - Principia: An International Journal of Epistemology 13 (2):165-184.
    One of Da Costa’s motives when he constructed the paraconsistent logic C! was to dualise the negation of intuitionistic logic. In this paper I explore a different way of going about this task. A logic is defined by taking the Kripke semantics for intuitionistic logic, and dualising the truth conditions for negation. Various properties of the logic are established, including its relation to C!. Tableau and natural deduction systems for the logic are produced, as are appropriate algebraic structures. The paper (...)
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  • The Logic of Contradiction.Nicolas D. Goodman - 1981 - Mathematical Logic Quarterly 27 (8-10):119-126.
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  • The ambiguous Relationship between Pragma-Dialectics and Logic.Ralph H. Johnson - 2006 - In F. H. van Eemeren, Peter Houtlosser, Haft-van Rees & A. M. (eds.), Considering pragma-dialectics: a festschrift for Frans H. van Eemeren on the occasion of his 60th birthday. Mahwah, N.J.: L. Erlbaum Associates. pp. 127.
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  • Complement-Topoi and Dual Intuitionistic Logic.Luis Estrada-González - 2010 - Australasian Journal of Logic 9:26-44.
    Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus. In this paper I present a sequent calculus for complement-topos logic, which throws some light on the problem of giving a dualization for LJ.
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  • Empirical Negation.Michael De - 2013 - Acta Analytica 28 (1):49-69.
    An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.
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  • The theory of rejected propositions. II.Jerzy Słupecki, Grzegorz Bryll & Urszula Wybraniec-Skardowska - 1972 - Studia Logica 30 (1):97 - 145.
    This paper is a continuation of Part I under the same title. Its Chapter III contains results given in the following publications: U. Wybraniec-Skardowska, Teoria zdań odrzuconych (Theory of Rejected Sentences), (doctoral dissertation under the supervision of Jerzy Słupecki, published as a monograph), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i Monografie, Nr 22 (1969), 5-131. G. Bryll, Związki logiczne pomiędzy zdaniami nauk empirycznych (Logical relations between sentences of empirical sciences). Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Studia i (...)
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  • Duality and Inferential Semantics.James Trafford - 2015 - Axiomathes 25 (4):495-513.
    It is well known that classical inferentialist semantics runs into problems regarding abnormal valuations. It is equally well known that the issues can be resolved if we construct the inference relation in a multiple-conclusion sequent calculus. The latter has been prominently developed in recent work by Restall, with the guiding interpretation that the valid sequent says that the simultaneous assertion of all of Γ with the denial of all of Δ is incoherent. However, such structures face significant interpretive challenges, and (...)
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  • On pure refutation formulations of sentential logics.Tomasz Skura - 1990 - Bulletin of the Section of Logic 19 (3):102-107.
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  • Theory of rejected propositions. I.Jerzy Słupecki, Grzegorz Bryll & Urszula Wybraniec-Skardowska - 1971 - Studia Logica 29 (1):75 - 123.
    The idea of rejection of some sentences on the basis of others comes from Aristotle, as Jan Łukasiewicz states in his studies on Aristotle's syllogistic [1939, 1951], concerning rejection of the false syllogistic form and those on certain calculus of propositions. Short historical remarks on the origin and development of the notion of a rejected sentence, introduced into logic by Jan Łukasiewicz, are contained in the Introduction of this paper. This paper is to a considerable extent a summary of papers (...)
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  • (1 other version)Temporal and atemporal truth in intuitionistic mathematics.Enrico Martino & Gabriele Usberti - 1994 - Topoi 13 (2):83-92.
    In section 1 we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of theclassical meaning of the logical constants, and consequently is incompatible with the antirealism of orthodox intuitionism. In section 2 we show how what we call the potential intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability (...)
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  • [Omnibus Review].Robert Goldblatt - 1986 - Journal of Symbolic Logic 51 (1):225-227.
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  • On Refutation Rules.Tomasz Skura - 2011 - Logica Universalis 5 (2):249-254.
    The goal of this paper is to generalize specific techniques connected with refutation rules involving certain normal forms. In particular, a method of axiomatizing both a logic L and its complement −L is introduced.
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  • Errata Corrige to “Pragmatic and dialogic interpretation of bi-intuitionism. Part I”.Gianluigi Bellin, Massimiliano Carrara, Daniele Chiffi & Alessandro Menti - 2016 - Logic and Logical Philosophy 25 (2).
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  • The trilaticce of constructive truth values.Yaroslav Shramko, J. Michael Dunn & Tatsutoshi Takenaka - 2001 - Journal of Logic and Computation 11 (1):761--788.
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  • Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic.Linda Postniece - unknown
    Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cut-free’ sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in (...)
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