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3 results

  • BETWEEN PROOF CONSTRUCTION AND SAT-SOLVING

  • Part of
  • ALEKSY SCHUBERT, PAWEŁ URZYCZYN, KONRAD ZDANOWSKI
  • Journal:
    The Journal of Symbolic Logic , First View
    Published online by Cambridge University Press:
    08 August 2025, pp. 1-22
    • Article
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  • The classical satisfiability problem (SAT) is used as a natural and general tool to express and solve combinatorial problems that are in NP. We postulate that provability for implicational intuitionistic propositional logic (IIPC) can serve as a similar natural tool to express problems in Pspace. We demonstrate it by proving two essential results concerning the system. One is a natural reduction from full IPC (with all connectives) to implicational formulas of order three. Another result is a convenient interpretation in terms of simple alternating automata. Additionally, we distinguish some natural subclasses of IIPC corresponding to the complexity classes NP and co-NP.

The Road to Paradox
  • A Guide to Syntax, Truth and Modality
  • Volker Halbach, Graham E. Leigh
  • Published online:
    17 October 2024
    Print publication:
    31 October 2024
  • Truth, provability, necessity, and other concepts are fundamental to many branches of philosophy, mathematics, computer science, and linguistics. Their study has led to some of the most celebrated achievements in logic, such as Gödel's incompleteness theorems, Tarski's theorem on the undefinability of truth, and numerous accounts of the paradoxes associated with these concepts. This book provides a clear and direct introduction to the theory of paradoxes and the Gödel incompleteness theorems. It offers new analyses of the ideas of self-reference, circularity, and the semantic paradoxes, and helps readers to see both how paradoxes arise and what their common features are. It will be valuable for students and researchers with a minimal background in logic and will equip them to understand and discuss a wide variety of topics in philosophical logic.

  • 4 - Gödel and Formalism Independence

  • Juliette Kennedy, University of Helsinki
  • Book:
    Gödel, Tarski and the Lure of Natural Language
    Published online:
    03 December 2020
    Print publication:
    17 December 2020, pp 61-98

  • Summary

    Gödel’s suggestion in his 1946 Princeton Bicentennial Lecture, to extend what he characterises as the formalism independence of the concept of computabilityto definability and provability, are discussed and implenented. The implementation for the concept of definabilitytakes the form of extended constructibility. Possible implementations having to do with provability are also discussed, mainly in connection with informal proof systems involving Gödel’s so-called large cardinal program.