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47 results

Page 1 of 3

  • RUSSELL AND GÖDEL

  • ALASDAIR URQUHART
  • Journal:
    Bulletin of Symbolic Logic / Volume 22 / Issue 4 / December 2016
    Published online by Cambridge University Press:
    30 December 2016, pp. 504-520
    Print publication:
    December 2016
    • Article
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  • This paper surveys the interactions between Russell and Gödel, both personal and intellectual. After a description of Russell’s influence on Gödel, it concludes with a discussion of Russell’s reaction to the incompleteness theorems.

  • Von Neumann, Gödel and Complexity Theory

  • Alasdair Urquhart
  • Journal:
    Bulletin of Symbolic Logic / Volume 16 / Issue 4 / December 2010
    Published online by Cambridge University Press:
    15 January 2014, pp. 516-530
    Print publication:
    December 2010

  • Around 1989, a striking letter written in March 1956 from Kurt Gödel to John von Neumann came to light. It poses some problems about the complexity of algorithms; in particular, it asks a question that can be seen as the first formulation of the P = ? NP question. This paper discusses some of the background to this letter, including von Neumann's own ideas on complexity theory. Von Neumann had already raised explicit questions about the complexity of Tarski's decision procedure for elementary algebra and geometry in a letter of 1949 to J. C. C. McKinsey. The paper concludes with a discussion of why theoretical computer science did not emerge as a separate discipline until the 1960s.

  • 3 - Proof Theory

  • from Part II - Logic
  • Edited by Yves Crama, Université de Liège, Belgium, Peter L. Hammer
  • Book:
    Boolean Models and Methods in Mathematics, Computer Science, and Engineering
    Published online:
    05 June 2013
    Print publication:
    28 June 2010, pp 79-98

  • Summary

    Introduction

    The literature contains a wide variety of proof systems for propositional logic. In this chapter, we outline the more important of such proof systems, beginning with an equational calculus, then describing a traditional axiomatic proof system in the style of Frege and Hilbert.We also describe the systems of sequent calculus and resolution that have played an important part in proof theory and automated theorem proving. The chapter concludes with a discussion of the problem of the complexity of propositional proofs, an important area in recent logical investigations. In the last section,we give a proof that any consensus proof of the pigeonhole formulas has exponential length.

    An Equational Calculus

    The earliest proof systems for propositional logic belong to the tradition of algebraic logic and represent proofs as sequences of equations between Boolean expressions. The proof systems of Boole, Venn, and Schröder are all of this type. In this section, we present such a system, and prove its completeness, by showing that all valid equations between Boolean expressions can be deduced formally.

    We start from the concept of Boolean expression defined in Chapter 1 of the monograph Crama and Hammer [9]. If ϕ and ψ are Boolean expressions, then we write ϕ[ψ/xi] for the expression resulting from ϕ by substituting ψ for all occurrences of the variable xi in ϕ. With this notational convention, we can state the formal rules for deduction in our equational calculus.

  • Proofs, Snakes and Ladders

  • Alasdair Urquhart
  • Journal:
    Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie / Volume 13 / Issue 4 / December 1974
    Published online by Cambridge University Press:
    09 June 2010, pp. 723-731

  • Anyone who has worked at proving theorems of intuitionistic logic in a natural deduction system must have been struck by the way in which many logical theorems “prove themselves.” That is, proofs of many formulas can be read off from the syntactical structure of the formulas themselves. This observation suggests that perhaps a strong structural identity may underly this relation between formulas and their proofs. A formula can be considered as a tree structure composed of its subformulas (Frege 1879) and by the normal form theorem (Gentzen 1934) every formula has a normalized proof consisting of its subformulas. Might we not identify an intuitionistic theorem with (one of) its proof(s) in normal form?

  • From Berkeley to Bourbaki*

  • Alasdair Urquhart
  • Journal:
    Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie / Volume 38 / Issue 3 / Summer 1999
    Published online by Cambridge University Press:
    13 April 2010, pp. 587-592

  • This has been a great century for logic and the foundations of mathematics. Ewald's excellent sourcebook is a welcome addition to the literature on the exciting developments of this and the past two centuries. The richness of the material on which Ewald is drawing is shown by the fact that he has assembled a broad and representative selection without once duplicating anything to be found in the famous sourcebooks of van Heijenoort and Benacerraf/Putnam.

  • Enumerating Types of Boolean Functions

  • Alasdair Urquhart
  • Journal:
    Bulletin of Symbolic Logic / Volume 15 / Issue 3 / September 2009
    Published online by Cambridge University Press:
    15 January 2014, pp. 273-299
    Print publication:
    September 2009

  • Abstract.

    The problem of enumerating the types of Boolean functions under the group of variable permutations and complementations was first stated by Jevons in the 1870s, but not solved in a satisfactory way until the work of Pólya in 1940. This paper explains the details of Pólya's solution, and also the history of the problem from the 1870s to the 1970s.

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