53 found
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  1.  18
    Proof Analysis: A Contribution to Hilbert's Last Problem.Sara Negri & Jan von Plato - 2014 - Cambridge University Press.
    Machine generated contents note: Prologue: Hilbert's Last Problem; 1. Introduction; Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction; 3. Axiomatic systems; 4. Order and lattice theory; 5. Theories with existence axioms; Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus; 7. Linear order; Part III. Proof Systems for Geometric Theories: 8. Geometric theories; 9. Classical and intuitionistic axiomatics; 10. Proof analysis in elementary geometry; Part IV. Proof Systems for Nonclassical (...)
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  2.  15
    Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - Cambridge University Press.
    Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic (...)
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  3.  13
    Natural Deduction with General Elimination Rules.Jan von Plato - 2001 - 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 derivation of the sequent Γ⇒C, no inactive weakening (...)
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  4.  32
    Gentzen's Proof Systems: Byproducts in a Work of Genius.von Plato Jan - 2012 - Bulletin of Symbolic Logic 18 (3):313-367.
    Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of (...)
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  5.  19
    Gentzen's Proof of Normalization for Natural Deduction.von Plato Jan - 2008 - Bulletin of Symbolic Logic 14 (2):240-257.
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  6. Sequent Calculus in Natural Deduction Style.Sara Negri & Jan von Plato - 2001 - 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 a subformula of the conclusion. Therefore (...)
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  7. In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs.Jan Von Plato - 2007 - Bulletin of Symbolic Logic 13 (2):189-225.
    The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is (...)
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  8. The Method of Arbitrary Functions.Jan Von Plato - 1983 - British Journal for the Philosophy of Science 34 (1):37-47.
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  9.  11
    A Proof of Gentzen's Hauptsatz Without Multicut.Jan von Plato - 2001 - Archive for Mathematical Logic 40 (1):9-18.
    Gentzen's original proof of the Hauptsatz used a rule of multicut in the case that the right premiss of cut was derived by contraction. Cut elimination is here proved without multicut, by transforming suitably the derivation of the premiss of the contraction.
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  10.  27
    Gentzen's Proof of Normalization for Natural Deduction.Jan von Plato & G. Gentzen - 2008 - Bulletin of Symbolic Logic 14 (2):240 - 257.
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  11.  3
    The Axioms of Constructive Geometry.Jan von Plato - 1995 - Annals of Pure and Applied Logic 76 (2):169-200.
    Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to (...)
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  12.  7
    Combinatorial Analysis of Proofs in Projective and Affine Geometry.Jan von Plato - 2010 - Annals of Pure and Applied Logic 162 (2):144-161.
    The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates . This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those (...)
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  13.  1
    Il Silenzio Delle Sirene: La Matematica Greca Antica.Jan von Plato - 2013 - History and Philosophy of Logic 34 (4):381 - 392.
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  14.  2
    Proof-Theoretical Analysis of Order Relations.Sara Negri, Jan von Plato & Thierry Coquand - 2004 - Archive for Mathematical Logic 43 (3):297-309.
    A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.
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  15.  7
    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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  16.  3
    Normal Derivability in Modal Logic.Jan von Plato - 2005 - Mathematical Logic Quarterly 51 (6):632-638.
    The standard rule of necessitation in systems of natural deduction for the modal logic S4 concludes □A from A whenever all assumptions A depends on are modal formulas. This condition prevents the composability and normalization of derivations, and therefore modifications of the rule have been suggested. It is shown that both properties hold if, instead of changing the rule of necessitation, all elimination rules are formulated in the manner of disjunction elimination, i.e. with an arbitrary consequence.
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  17. The Logic of Euclidean Construction Procedures.P. Mäenpää & Jan von Plato - 1990 - Acta Philosophica Fennica 39:275-293.
     
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  18.  17
    Skolem's Discovery of Gödel-Dummett Logic.Jan von Plato - 2003 - Studia Logica 73 (1):153 - 157.
    Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.
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  19.  20
    The Significance of the Ergodic Decomposition of Stationary Measures for the Interpretation of Probability.Jan von Plato - 1982 - Synthese 53 (3):419 - 432.
    De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.
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  20.  2
    Skolem's Discovery of Gödel-Dummett Logic.Jan von Plato - 2003 - Studia Logica 73 (1):153-157.
    Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.
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  21.  40
    Rereading Gentzen.Jan Von Plato - 2003 - Synthese 137 (1-2):195 - 209.
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  22.  37
    Formalization of Hilbert's Geometry of Incidence and Parallelism.Jan von Plato - 1997 - Synthese 110 (1):127-141.
    Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.
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  23.  1
    Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective.Jan von Plato - 1997 - Philosophical Quarterly 47 (186):122-125.
  24. Normal Form and Existence Property for Derivations in Heyting Arithmetic.Jan von Plato - 2006 - Acta Philosophica Fennica 78:159.
     
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  25.  37
    Probability and Determinism.Jan Von Plato - 1982 - Philosophy of Science 49 (1):51-66.
    This paper discusses different interpretations of probability in relation to determinism. It is argued that both objective and subjective views on probability can be compatible with deterministic as well as indeterministic situations. The possibility of a conceptual independence between probability and determinism is argued to hold on a general level. The subsequent philosophical analysis of recent advances in classical statistical mechanics (ergodic theory) is of independent interest, but also adds weight to the claim that it is possible to justify an (...)
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  26.  18
    Generality and Existence: Quantificational Logic in Historical Perspective.von Plato Jan - 2014 - Bulletin of Symbolic Logic 20 (4):417-448,.
    Frege explained the notion of generality by stating that each its instance is a fact, and added only later the crucial observation that a generality can be inferred from an arbitrary instance. The reception of Frege's quantifiers was a fifty-year struggle over a conceptual priority: truth or provability. With the former as the basic notion, generality had to be faced as an infinite collection of facts, whereas with the latter, generality was based on a uniformity with a finitary sense: the (...)
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  27.  22
    Probabilistic Causality From a Dynamical Point of View.Jan von Plato - 1990 - Topoi 9 (2):101-108.
  28.  22
    From Axiomatic Logic to Natural Deduction.Jan von Plato - 2014 - 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 deduction, rather than any single (...)
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  29.  25
    Reductive Relations in Interpretations of Probability.von Plato Jan - 1981 - Synthese 48 (1):61 - 75.
  30.  9
    For Oiva Ketonen's 85th Birthday.Sara Negri & Jan von Plato - 1998 - Bulletin of Symbolic Logic 4 (4).
  31.  25
    De Finetti's Earliest Works on the Foundations of Probability.Jan von Plato - 1989 - Erkenntnis 31 (2-3):263 - 282.
    Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.
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  32.  8
    Gentzen's Logic.Jan von Plato - 2009 - In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. pp. 667-721.
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  33.  11
    Normal Derivability in Classical Natural Deduction.Jan Von Plato & Annika Siders - forthcoming - Review of Symbolic Logic.
  34.  6
    Proof Theory of Classical and Intuitionistic Logic.Jan von Plato - 2008 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.
    This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second reason for the importance of intuitionism (...)
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  35.  15
    On Partial Exchangeability as a Generalization of Symmetry Principles.Jan von Plato - 1981 - Erkenntnis 16 (1):53 - 59.
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  36.  11
    Illustrations of Method in Ptolemaic Astronomy.Jan Von Plato - 1994 - Grazer Philosophische Studien 49:63-75.
    Mathematical Astronomy as the most developed branch of ancient exact sciences has been widely discussed - especially epistemological issues e.g. concerning astronomy as a prime example of the distinction between instrumentalist and realist understanding of theories. In contrast to these the very methodology of ancient astronomy has received little attention. Following the work of Jaakko Hintikka and Unto Remes Aristarchus' method of determining the distance of the Sun is sketched and Ptolemy's solar model is discussed in detail.
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  37.  11
    The Generalization of de Finetti's Representation Theorem to Stationary Probabilities.Jan von Plato - 1982 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982:137 - 144.
    de Finetti's representation theorem of exchangeable probabilities as unique mixtures of Bernoullian probabilities is a special case of a result known as the ergodic decomposition theorem. It says that stationary probability measures are unique mixtures of ergodic measures. Stationarity implies convergence of relative frequencies, and ergodicity the uniqueness of limits. Ergodicity therefore captures exactly the idea of objective probability as a limit of relative frequency (up to a set of measure zero), without the unnecessary restriction to probabilistically independent events as (...)
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  38.  9
    Review: Vincent F. Hendricks, Stig Andur Pedersen, Klaus Frovin Jørgensen, Proof Theory, History and Philosophical Significance. [REVIEW]Jan von Plato - 2002 - Bulletin of Symbolic Logic 8 (3):431-432.
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  39.  7
    Kurt Gödel: Essays for His Centennial.Jan von Plato - 2011 - History and Philosophy of Logic 32 (4):402 - 404.
    History and Philosophy of Logic, Volume 32, Issue 4, Page 402-404, November 2011.
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  40.  3
    David Hilbert's Lectures on the Foundations of Geometry 1891–1902. Edited by Hallett Michael and Majer Ulrich, Hilbert's David Lectures on the Foundations of Mathematics and Physics, 1891–1933, Vol. 1. Springer, Berlin, Heidelberg and New York, 2004, Xviii+ 661 Pp. [REVIEW]Jan von Plato - 2006 - Bulletin of Symbolic Logic 12 (3):492-494.
  41.  3
    Gentzen Writes in the Published Version of His Doctoral Thesis Untersuchun-Gen Über Das Logische Schliessen (Investigations Into Logical Reasoning) That He Was Able to Prove the Normalization Theorem Only for Intuitionistic Natural Deduction, but Not for Classical. To Cover the Latter, He Developed Classical Sequent Calculus and Proved a Corresponding Theorem, the Famous Cut Elim.Jan von Plato - 2008 - Bulletin of Symbolic Logic 14 (2).
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  42.  3
    Proof Theory, History and Philosophical Significance, Edited by Hendricks Vincent F., Pedersen Stig Andur, and Jørgensen Klaus Frovin, Synthese Library, Vol. 292, Kluwer Academic Publishers, Dordrecht, Boston, and London, 2000, Xii+ 244 Pp.—. [REVIEW]Jan von Plato - 2002 - Bulletin of Symbolic Logic 8 (3):431-432.
  43.  2
    Review: Dirk van Dalen, Mystic, Geometer, and Intuitionist. The Life of L. E. J. Brouwer. Volume 1. The Dawning Revolution. [REVIEW]Jan von Plato - 2001 - Bulletin of Symbolic Logic 7 (1):62-65.
  44.  3
    Probabilistic Causality, Randomization and Mixtures.Jan von Plato - 1986 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:432-437.
    A formulation of probabilistic causality is given in terms of the theory of abstract dynamical systems. Causal factors are identified as invariants of motion of a system. Repetition of an experiment leads to the notion of stationarity, and causal factors yield a decomposition of the stationary probability law of the experiment into ergodic components. In these, statistical behaviour is uniform. Control of identified causal factors leads to a corresponding statistical law for the events, which is offered as a notion of (...)
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  45.  1
    Ptolemy's Almagest by Ptolemy; G. J. Toomer; Preceptum Canonis Ptolomei by David Pingree.Jan von Plato - 2001 - Isis: A Journal of the History of Science 92:149-150.
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  46. Articles In Proof Theory, History And Philosophical Significance. [REVIEW]von Plato Jan - 2002 - Bulletin of Symbolic Logic 8 (3):431-431.
     
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  47. David Hilbert’s Lectures on the Foundations of Geometry 1891—1902. [REVIEW]Jan von Plato - 2006 - Bulletin of Symbolic Logic 12 (3):492-493.
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  48. David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933. Edited byEwald William andSieg Wilfried.Springer,Berlin, Heidelberg and New York,2013, Xxv + 1062 Pp. [REVIEW]Jan von Plato - 2014 - Bulletin of Symbolic Logic 20 (3):363-365.
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  49. Illustrations of Method in Ptolemaic Astronomy.Jan Von Plato - 1994 - Grazer Philosophische Studien 49:63-75.
    Mathematical Astronomy as the most developed branch of ancient exact sciences has been widely discussed - especially epistemological issues e.g. concerning astronomy as a prime example of the distinction between instrumentalist and realist understanding of theories. In contrast to these the very methodology of ancient astronomy has received little attention. Following the work of Jaakko Hintikka and Unto Remes Aristarchus' method of determining the distance of the Sun is sketched and Ptolemy's solar model is discussed in detail.
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  50. K. Gödel Collected Works Iv—V. [REVIEW]Jan von Plato - 2004 - Bulletin of Symbolic Logic 10 (4):558-562.
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