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Vol. 14(1), 2014

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Vol. 15(1), 2015
Reconciling first-order logic to algebra
Walter Carnielli(*), Hugo Luiz Mariano(1) and Mariana Matulovic(2)

(*) CLE and Department of Philosophy State University of Campinas
(1) Institute of Mathematics and Statistics University of São Paulo
2) Department of Philosophy Paulista State University "Julio de Mesquita Filho" Marília, Brazil

Date Posted: February 5, 2015

vol. 15, n. 1, 2015

   
Abstract:

Starting from the algebraic method of theorem-proving based on the translation of logic formulasinto polynomials over nite elds, and by adapting the case of rst-order formulas by employingcertain rings equipped with in nitary operations, this paper de nes the notion of M-ring, a kind of polynomial ring de ned for each rst-order structure, by means of generators and relations. The notion of M-ring allows us to operate with some kind of in nitary version of Boolean sums and products, in this way expressing algebraically rst-order logic with a new gist. We then show how this polynomial representation of rst-order sentences could be seen as a legitimate algebraic semantics for first-order logic, alternative to cylindric and polyadic algebras an with a higher degree of naturalness. We brie y discuss how the method and their generalizations could be successfully lifted to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.



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