A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus


In the XII Latin American Symposium on Mathematical Logic we presented a work introducing a Hilbert-style propositional calculus called four-valued Monteiro propositional calculus. This calculus, denoted by M4, is introduced in terms of the binary connectives (implication), (weak implication), (conjunction) and the unary ones (negation) and (modal operator). In this paper, it is proved that M4 belongs to the class of standard systems of implicative extensional propositional calculi as defined by Rasiowa (1974). Furthermore, we show that the definitions of four-valued modal algebra and M4 -algebra are equivalent and, in addition, obtain the completeness theorem for M4. We also introduce the notion of modal distributive lattices with implication and show that these algebras are more convenient than four-valued modal algebras for the study of four-valued Monteiro propositional calculus from an algebraic point of view. This follows from the fact that the implication is one of its basic binary operations.

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Figallo, A. , Bianco, E. and Ziliani, A. (2014) A New Algebraic Version of Monteiro’s Four-Valued Propositional Calculus. Open Journal of Philosophy, 4, 319-331. doi: 10.4236/ojpp.2014.43036.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Bianco, E. (2004). Four-Valued Monteiro Propositional Calculus. XII Latin American Symposium on Mathematical Logic. Abstracts of Contributed Papers, San José, 3.
[2] Bianco, E. (2008). Una contribución al estudio de las álgebras de De Morgan modales 4-valuadas. Ms. Thesis, Baha Blanca: Universidad Nacional del Sur.
[3] Birkhoff, G. (1967). Lattice Theory (3rd ed.). Providence: American Mathematical Society, Col Pub.
[4] Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge: Cambridge University Press.
[5] Burris, S., & Sankappanavar, H. P. (1981). A Course in Universal Algebra. Graduate Texts in Mathematics 78. Berlin: Springer. http://dx.doi.org/10.1007/978-1-4613-8130-3
[6] Carnielli, W. A., & Marcos, J. (2002). A Taxonomy of C-Systems. In W. A. Carnielli, M. E. Coniglio, & I. M. L. D’ottaviano (Eds.), Paraconsistency—The Logical Way to the Inconsistent, Volume 228 of Lecture Notes in Pure and Applied Mathematics (pp. 1-94). New York: Marcel Dekker.
[7] Coniglio, M. E., & Figallo, M. (2013). Hilbert-Style Presentations of Two Logics Associated to Tetravalent Modal Algebras. Studia Logica.
[8] Figallo, A., & Landini, P. (1995). On Generalized I—Algebras and Modal 4-Valued Algebras. Rep Math Logic Impact Factor, 29, 3-18.
[9] Figallo, A., & Ziliani, A. (1991). Symmetric Tetra-Valued Modal Algebras. Discreta Notas de la Sociedad de Matem tica de Chile, 10, 133-141.
[10] Figallo, A. V. (1992). On the Congruences in Four-Valued Modal Algebras. Portugaliae Mathematica, 49, 249-261.
[11] Figallo, A. V. (1994). Tópicos sobre álgebras modales 4-valuadas. Proceedings of the 9th Latin American Symposium on Mathematical Logic, 38, 145-157.
[12] Font, J. M., & Rius, M. (1990). A Four-Valued Modal Logic Arising from Monteiros’s Last Algebras. Proceedings of the 20th International Symposium on Multiple-Valued Logic, IEEE Computer Society Press, 85-92.
[13] Font, J. M., & Rius, M. (2000). An Abstract Algebraic Logic Approach to Tetravalent Modal Logics. The Journal of Symbolic Logic, 65, 481-518. http://dx.doi.org/10.2307/2586552
[14] Kalman, J. A. (1958). Lattices with Involution. Transactions of the American Mathematical Society, 87, 485-491.
[15] Lemmon, E. J., & Scott, D. (1977). An Introduction to Modal Logic. In K. Segerberg (Ed.), The Lemmon Notes (Vol. 11). American Philosophical Quarterly Monograph Series. Oxford: Basil Blackwell.
[16] Loureiro, I. (1980). Homomorphism Kernels of a Tetravalent Modal Algebra. Portugaliae Mathematica, 39, 371-379.
[17] Loureiro, I. (1982). Axiomatisation et propriétés des algèbres modales tétravalentes. Comptes Rendus de l’Académie des Sciences, 295, 555-557.
[18] Loureiro, I. (1983a). Algebras Modais Tetravalentes. Ph. D. Thesis, Lisbon: Faculdade de Ciências de Lisboa.
[19] Loureiro, I. (1983b). Prime Spectrum of a Tetravalent Modal Algebras. Notre Dame Journal of Formal Logic, 24, 389-394.
[20] Loureiro, I. (1983c). Finitely Generated Free Tetravalent Modal Algebras. Discrete Mathematics, 46, 41-48.
[21] Loureiro, I. (1984). Finite Tetravalent Modal Algebras. Revista de la Unión Matemática Argentina, 31, 187-191.
[22] Loureiro, I. (1985). Principal Congruences of Tetravalent Modal Algebras. Notre Dame Journal of Formal Logic, 26, 76-80.
[23] MacLane, S. (1971). Categories for the Working Mathematician. Berlin: Springer.
[24] Marona, R. (1964). A Characterisation of Morgan Lattices. Notas de Lógica Matemática, 18, 1-3.
[25] Monteiro, A. (1960). Matrices de Morgan caractéristiques pour le calcul propositionnel classique. Anais da Academia Brasileira de Ciências, 52, 1-7.
[26] Rasiowa, H. (1974). An Algebraic Approach to Non-Classical Logics. Warzawa & Amsterdam: North-Holland Publishing Company.
[27] Scroggs, S. J. (1951). Extensions of the Lewis System S5. The Journal of Symbolic Logic, 16, 112-120.

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