47 found
Order:
  1. Structural Proof Theory.Sara Negri, Jan von Plato & Aarne Ranta - 2001 - New York: Cambridge University Press. Edited by Jan Von Plato.
    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 (...)
    Direct download  
     
    Export citation  
     
    Bookmark   119 citations  
  2.  62
    Proof Analysis: A Contribution to Hilbert's Last Problem.Sara Negri & Jan von Plato - 2011 - Cambridge and New York: Cambridge University Press. Edited by Jan Von Plato.
    This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim (...)
  3.  85
    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 (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   55 citations  
  4.  49
    (1 other version)Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective.Lawrence Sklar & Jan von Plato - 1994 - Journal of Philosophy 91 (11):622.
  5. The method of arbitrary functions.Jan von Plato - 1983 - British Journal for the Philosophy of Science 34 (1):37-47.
  6. Gentzen's proof systems: byproducts in a work of genius.Jan von Plato - 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 (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   16 citations  
  7.  57
    David Hilbert's lectures on the foundations of geometry 1891–1902. edited by Michael Hallett and Ulrich Majer, David Hilbert's Lectures on the Foundations of Mathematics and Physics, 1891–1933, vol. 1. Springer, Berlin, Heidelberg and New York, 2004, xviii + 661 pp.Jan von Plato - 2006 - Bulletin of Symbolic Logic 12 (3):492-494.
  8.  85
    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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  9.  28
    A Problem of Normal Form in Natural Deduction.Jan von Plato - 2000 - 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.
    Direct download  
     
    Export citation  
     
    Bookmark   15 citations  
  10.  98
    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 (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   15 citations  
  11.  60
    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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   14 citations  
  12.  73
    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.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  13. 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 (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  14.  35
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  15.  33
    From Gentzen to Jaskowski and Back: Algorithmic Translation of Derivations Between the Two Main Systems of Natural Deduction.Jan Von Plato - 2017 - Bulletin of the Section of Logic 46 (1/2).
    The way from linearly written derivations in natural deduction, introduced by Jaskowski and often used in textbooks, is a straightforward root-first translation. The other direction, instead, is tricky, because of the partially ordered assumption formulas in a tree that can get closed by the end of a derivation. An algorithm is defined that operates alternatively from the leaves and root of a derivation and solves the problem.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  16.  67
    Translations from natural deduction to sequent calculus.Jan von Plato - 2003 - 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 explains the cuts in (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  17.  64
    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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  18.  59
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  19. Normal form and existence property for derivations in heyting arithmetic.Jan von Plato - 2006 - Acta Philosophica Fennica 78:159.
     
    Export citation  
     
    Bookmark   7 citations  
  20. The logic of Euclidean construction procedures.P. Mäenpää & Jan von Plato - 1990 - Acta Philosophica Fennica 39:275-293.
     
    Export citation  
     
    Bookmark   6 citations  
  21.  73
    Kurt gödel’s first steps in logic: Formal proofs in arithmetic and set theory through a system of natural deduction.Jan von Plato - 2018 - Bulletin of Symbolic Logic 24 (3):319-335.
    What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  22.  35
    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.
    Direct download  
     
    Export citation  
     
    Bookmark   6 citations  
  23.  54
    Reductive relations in interpretations of probability.Jan Von Plato - 1981 - Synthese 48 (1):61 - 75.
  24. 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 (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  25.  79
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  26.  39
    Gentzen's Logic.Jan von Plato - 2009 - In Dov Gabbay (ed.), The Handbook of the History of Logic. Elsevier. pp. 667-721.
  27.  56
    Logic Lectures. Gödel's Basic Logic Course at Notre Dame.Jan von Plato - 2018 - History and Philosophy of Logic 39 (4):396-401.
    Biographies of Kurt Gödel tell that he gave in 1939 an introductory lecture course on logic at the University of Notre Dame. In the early 1930s, Gödel took part and presented many of hi...
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  28.  47
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  29.  45
    Proof theory of classical and intuitionistic logic.Jan von Plato - 2009 - In Leila Haaparanta (ed.), The development of modern logic. New York: 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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  30.  74
    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.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  31.  77
    Rereading Gentzen.Jan Von Plato - 2003 - Synthese 137 (1-2):195 - 209.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  32.  38
    Il silenzio delle sirene: La matematica greca antica.Jan von Plato - 2013 - History and Philosophy of Logic 34 (4):381 - 392.
    Fabio Acerbi, Il silenzio delle sirene: La matematica greca antica. Rome: Carocci editore, 2010. 445 pp. € 44. ISBN 978-88-430-5579-1.Fabio Acerbi's recent book Il silenzio delle sirene: La matemat...
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  33.  62
    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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  34.  35
    Illustrations of Method in Ptolemaic Astronomy.Jan Von Plato - 1994 - Grazer Philosophische Studien 49 (1):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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  35.  38
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  36.  62
    On Partial Exchangeability as a Generalization of Symmetry Principles.Jan Von Plato - 1981 - Erkenntnis 16 (1):53 - 59.
  37.  19
    Ptolemy's Almagest. Ptolemy, G. J. ToomerPreceptum Canonis Ptolomei. David Pingree.Jan von Plato - 2001 - Isis 92 (1):149-150.
  38.  55
    Probabilistic causality from a dynamical point of view.Jan von Plato - 1990 - Topoi 9 (2):101-108.
  39.  31
    (1 other version)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 (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  40.  27
    Portrait of Young Gödel: Education, First Steps in Logic, the Problem of Completeness.Jan von Plato - 2024 - Springer Nature Switzerland.
    In the summer of 1928, Kurt Gödel (1906–1978) embarked on his logical journey that would bring him world fame in a mere three years. By early 1929, he had solved an outstanding problem in logic, namely the question of the completeness of the axioms and rules of quantificational logic. He then went on to extend the result to the axiom system of arithmetic but found, instead of completeness, his famous incompleteness theorem that got published in 1931. It belongs to the (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  41.  26
    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 (...)
    Direct download  
     
    Export citation  
     
    Bookmark  
  42.  28
    (1 other version)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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  43. Articles In Proof Theory, History And Philosophical Significance. [REVIEW]Jan von Plato - 2002 - Bulletin of Symbolic Logic 8 (3):431-431.
  44.  12
    David Hilbert’s lectures on the foundations of arithmetic and logic 1917–1933. Edited by William Ewald and Wilfried Sieg. Springer, Berlin, Heidelberg and New York, 2013, xxv + 1062 pp. [REVIEW]Jan von Plato - 2014 - Bulletin of Symbolic Logic 20 (3):363-365.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  45.  20
    Dirk van Dalen. Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1. The dawning revolution. Clarendon Press, Oxford University Press, Oxford, New York, etc., 1999, xv + 440 pp. [REVIEW]Jan von Plato - 2001 - Bulletin of Symbolic Logic 7 (1):62-65.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  46. K. Gödel Collected Works Iv—v. [REVIEW]Jan von Plato - 2004 - Bulletin of Symbolic Logic 10 (4):558-562.
  47.  9
    Review of Kurt Gödel Collected Works IV–V: Correspondence. [REVIEW]Jan von Plato - 2004 - Bulletin of Symbolic Logic 10 (4):558-563.