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  1. (2 other versions)Multiple-Conclusion Logic.Ronald Harrop - 1981 - Journal of Symbolic Logic 46 (1):161-163.
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  • Multiple Conclusion Logic.D. J. Shoesmith & Timothy John Smiley - 1978 - Cambridge, England / New York London Melbourne: Cambridge University Press. Edited by T. J. Smiley.
    Multiple -conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations. This is a sustained study of the subject and is certain to stimulate further research. Part I reworks (...)
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  • Concepts of Proof in Mathematics, Philosophy, and Computer Science.Peter Schuster & Dieter Probst (eds.) - 2016 - Boston: De Gruyter.
    A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be (...)
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  • Eliminating disjunctions by disjunction elimination.Davide Rinaldi, Peter Schuster & Daniel Wessel - 2017 - Bulletin of Symbolic Logic 23 (2):181-200.
    Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive (...)
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  • Finite Methods in Mathematical Practice.Peter Schuster & Laura Crosilla - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. Boston: De Gruyter. pp. 351-410.
    In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so--called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid on (...)
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  • Algebraische und logistische untersuchungen über freie verbände.Paul Lorenzen - 1951 - Journal of Symbolic Logic 16 (2):81-106.
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  • Multiple Conclusion Logic.N. Tennant - 1980 - Philosophical Quarterly 30 (121):379-382.
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  • (1 other version)Cut Elimination in the Presence of Axioms.Sara Negri & Jan Von Plato - 1998 - Bulletin of Symbolic Logic 4 (4):418-435.
    A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...)
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  • On engendering an illusion of understanding.Dana Scott - 1971 - Journal of Philosophy 68 (21):787-807.
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  • Über das Verhältnis zwischen intuitionistischer und klassischer Arithmetik.Gerhard Gentzen - 1974 - Archive for Mathematical Logic 16 (3-4):119-132.
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  • (1 other version)Dynamical algebraic structures, pointfree topological spaces and Hilbert's program.Henri Lombardi - 2006 - Annals of Pure and Applied Logic 137 (1-3):256-290.
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  • (1 other version)Structures algébriques dynamiques, espaces topologiques sans points et programme de Hilbert.Henri Lombardi - 2006 - Annals of Pure and Applied Logic 137 (1-3):256-290.
    A possible relevant meaning of Hilbert’s program is the following one: “give a constructive semantic for classical mathematics”. More precisely, give a systematic interpretation of classical abstract proofs about abstract objects, as constructive proofs about constructive versions of these objects.If this program is fulfilled we are able “at the end of the tale” to extract constructive proofs of concrete results from classical abstract proofs of these results.Dynamical algebraic structures or geometric theories seem to be a good tool for doing this (...)
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  • (1 other version)For Oiva Ketonen's 85th birthday.Sara Negri & Jan von Plato - 1998 - Bulletin of Symbolic Logic 4 (4):418-435.
    A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...)
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  • Remarks on the Scott–Lindenbaum Theorem.Gillman Payette & Peter K. Schotch - 2014 - Studia Logica 102 (5):1003-1020.
    In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted (...)
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  • Inductively generated formal topologies.Thierry Coquand, Giovanni Sambin, Jan Smith & Silvio Valentini - 2003 - Annals of Pure and Applied Logic 124 (1-3):71-106.
    Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in (...)
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  • (1 other version)Relecture constructive de la théorie d'Artin-Schreier.Henri Lombardi - 1998 - Annals of Pure and Applied Logic 91 (1):59-92.
    RésuméNous introduisons la notion de structure algébrique dynamique, inspirée de l'évaluation dynamique et de la théorie des modèles. Nous montrons comment cette notion constructive permet une relecture de la théorie d'Artin-Schreier, avec la modification capitale que le résultat final est alors établi de manière constructive. Nous pensons que ce que nous avons réalisé ici sur un cas d'école peut être généralisé à des parties significatives de l'algèbre classique, et est donc une contribution à la réalisation du programme de Hilbert pour (...)
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  • Über Axiomensysteme beliebiger Satzsysteme.P. Hertz - 1929 - Annalen der Philosophie Und Philosophischen Kritik 8 (1):178-204.
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  • Formal topologies on the set of first-order formulae.Thierry Coquand, Sara Sadocco, Giovanni Sambin & Jan Smith - 2000 - Journal of Symbolic Logic 65 (3):1183-1192.
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  • Jan von Plato and Sara Negri, Structural Proof Theory. [REVIEW]Harold T. Hodes - 2006 - Philosophical Review 115 (2):255-258.
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  • Kreisel's unwinding of Artin's proof.Charles Delzell - 1996 - In Piergiorgio Odifreddi (ed.), Kreiseliana: About and Around Georg Kreisel. A K Peters. pp. 113--246.
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  • Dynamical method in algebra: effective Nullstellensätze.Michel Coste, Henri Lombardi & Marie-Françoise Roy - 2001 - Annals of Pure and Applied Logic 111 (3):203-256.
    We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorn's lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that (...)
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  • On preserving.Gillman Payette & Peter K. Schotch - 2007 - Logica Universalis 1 (2):295-310.
    . This paper examines the underpinnings of the preservationist approach to characterizing inference relations. Starting with a critique of the ‘truth-preservation’ semantic paradigm, we discuss the merits of characterizing an inference relation in terms of preserving consistency. Finally we turn our attention to the generalization of consistency introduced in the early work of Jennings and Schotch, namely the concept of level.
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  • (1 other version)A constructive rereading of the Artin-Schreier theory.Henri Lombardi - 1998 - Annals of Pure and Applied Logic 91 (1):59-92.
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