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Reconciling First-Order Logic to Algebra

  • Walter Carnielli
  • Hugo Luiz Mariano
  • Mariana Matulovic
Chapter
Part of the Trends in Logic book series (TREN, volume 47)

Abstract

We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Walter Carnielli
    • 1
  • Hugo Luiz Mariano
    • 2
  • Mariana Matulovic
    • 3
  1. 1.Department of Philosophy and Centre for Logic, Epistemology and the History of ScienceUniversity of Campinas-UnicampCampinasBrazil
  2. 2.Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil
  3. 3.Faculty of Science and EngineeringSão Paulo State University (UNESP)TupãBrazil

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