It is possible to define a Heyting and a Łukasiewicz implication over a Lukasiewicz–Moisil algebra of order n ≤ 5 (ŁM-algebra, for short); however, for n > 5 we can only define a Heyting implication. At the same time, the Heyting implication on a chain is a Gödel implication, and it is also known as trivial Hilbert implication. This fact allow us to axiomatize this implicative reduct of ŁM-algebras by the notion of n-valued Hilbert algebras.

In this talk, we will present the propositional calculus for the {→, ∨,△, 1}-reduct of ŁM-algebra of order 3 which will be demonstrated Soundness and Completeness Theorem respect this algebraic structures. For this purpose, we will use the Brikhoff theorem for this new class of algebras. Thereafter, the first order calculus is studied and investigated using algebraic and Henkin’s tools in order to show adequacy theorem with respect of suitable algebraic-like structures.

On the other hand, we will discust about the possibility of the study of some first order Logics of Formal Inconsistency (for short, LFI) by adapting the above results in the context of Swap structures and Fidel structures for some non-algebraizable LFI via representation theorems presentated in [3, 4].

References

[1] V.Boicescu and A.Filipoiu and G.Georgescu and S.Rudeanu, Łukasiewicz - Moisil Algebras, Annals of Discrete Mathematics 49, North - Holland, 1991.

[2] S. Celani and D. Montangie, Hilbert algebras with supremum, Algebra Universalis 67(2012), no. 3, 237–255.

[3] M. E. Coniglio, A. Figallo-Orellano and A. C. Golzio, Non-deterministic algebraization of logics by swap structures, to appear in Logic Journal of the IGPL.

[4] M.E. Coniglio, A. Figallo-Orellano, A model-theoretic analysis of Fidel-structures for mbC, to appear in The volume ”Priest on Dialetheism and Paraconsistency”.

[5] A. Diego, Sur les algèbres de Hilbert, Collection de Logique Mathématique, Sér. A, Hermann, 21(1966).

[6] Itala M. L. D’Ottaviano, Sobre uma Teoria de Modelos Trivalente, PhD thesis in Mathematics, IMECC, State University of Campinas, Brazil, 1982.

[7] A. V. Figallo, G. Ramón and S. Saad, iH-Propositional calculus, Bull. Sect. Logic Univ. Ldz 35 (2006), no. 4, 157–162.

[8] Luiz Monteiro, Algèbres de Hilbert n−valentes, Portugaliae Math. 36(1977), 159–174.

[9] Ivo Thomas, Finite limitations on Dummett’s LC, Notre Dame Journal of Formal Logic, 3 (1962), 170 – 174.