×

zbMATH — the first resource for mathematics

Algebraic models of deviant modal operators based on De Morgan and Kleene lattices. (English) Zbl 1242.03088
Summary: An algebraic model of a kind of modal extension of De Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a De Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and De Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)–(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.

MSC:
03G25 Other algebras related to logic
03B45 Modal logic (including the logic of norms)
03E72 Theory of fuzzy sets, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baaz, M., Infinite-valued Gödel logics with 0-1 projections and relativizations, (), 23-33 · Zbl 0862.03015
[2] Banerjee, M.; Chakraborty, M., Rough algebra, Bulletin of the Polish Academy of sciences – mathematics, 41, 193-297, (1993) · Zbl 0795.03035
[3] Banerjee, M.; Chakraborty, M., Rough sets through algebraic logic, Fundamenta informaticae, 28, 211-221, (1996) · Zbl 0864.03041
[4] Becchio, D., Sur LES definitions des algebres trivalentes de łukasiewicz donnees par A, Monteiro. logique et analyse, 16, 339-344, (1973) · Zbl 0307.02040
[5] Bialynicki-Birula, A., Remarks on quasi-Boolean algebras, Bulletin de l’académie polonaise des sciences, classe III, 5, 615-619, (1957) · Zbl 0086.01002
[6] Bialynicki-Birula, A.; Rasiowa, H., On the representation of quasi-Boolean algebras, Bulletin de l’académie polonaise des sciences, classe III, 5, 259-261, (1957) · Zbl 0082.01403
[7] G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publication, third ed., vol. XXV, American Mathematical Society, Providence, Rhode Island, 1967 (first ed., 1940, second (revisited) ed., 1948).
[8] Birkhoff, G.; von Neumann, J., The logic of quantum mechanics, Annals of mathematics, 37, 823-843, (1936) · JFM 62.1061.04
[9] Bonikowski, Z., A certain conception of the calculus of rough sets, Notre dame journal of formal logic, 33, 412-421, (1992) · Zbl 0762.04001
[10] ()
[11] Cattaneo, G., Brouwer – zadeh posets for unsharp quantum mechanics, International journal of theoretical physics, 31, 1573-1597, (1992) · Zbl 0789.03047
[12] Cattaneo, G., Fuzzy quantum logic II: the logics of unsharp quantum mechanics, International journal of theoretical physics, 32, 1709-1734, (1993) · Zbl 0812.03030
[13] Cattaneo, G., Generalized rough sets (preclusivity fuzzy-intuitionistic BZ lattices), Studia logica, 58, 47-77, (1997) · Zbl 0864.03040
[14] Cattaneo, G., A unified framework for the algebra of unsharp quantum mechanics, International journal of theoretical physics, 36, 3085-3117, (1997) · Zbl 0898.03022
[15] G. Cattaneo, Abstract approximation spaces for rough theories, in: [89], 1998, pp. 59-98. · Zbl 0927.68087
[16] Cattaneo, G.; Ciucci, D., Heyting wajsberg algebras as an abstract environment linking fuzzy and rough sets, Lecture notes in artificial intelligence, vol. 2475, (2002), Springer-Verlag Berlin, pp. 77-84 · Zbl 1013.03073
[17] G. Cattaneo, D. Ciucci, Generalized negations and intuitionistic fuzzy sets, A criticism to a widely used terminology, in: Proceedings of International Conference in Fuzzy Logic and Technology (EUSFLAT03). University of Applied Sciences of Zittau-Goerlitz, Zittau, 2003, pp. 147-152.
[18] G. Cattaneo, D. Ciucci, Intuitionistic fuzzy sets or orthopair fuzzy sets? in: Proceedings of International Conference in Fuzzy Logic and Technology (EUSFLAT03). University of Applied Sciences of Zittau-Goerlitz, Zittau, 2003, pp. 153-158.
[19] Cattaneo, G.; Ciucci, D., Algebraic structures for rough sets, (), 218-264
[20] Cattaneo, G.; Ciucci, D., Basic intuitionistic principles in fuzzy set theories and its extensions (a terminological debate on atanassov IFS), Fuzzy sets and systems, 157, 3198-3219, (2006) · Zbl 1112.03050
[21] Cattaneo, G.; Ciucci, D., Some methodological remarks about categorical equivalence in the abstract approach to roughness. part I, Lecture notes in artificial intelligence, vol. 4062, (2006), Springer-Verlag Berlin, pp. 277-283 · Zbl 1196.03071
[22] Cattaneo, G.; Ciucci, D., A hierarchical lattice closure approach to abstract approximation spaces, Lecture notes in artificial intelligence, vol. 5009, (2008), Springer-Verlag Berlin, pp. 363-370
[23] Cattaneo, G.; Ciucci, D., Lattice with interior and closure operators and abstract approximation spaces, (), 67-116 · Zbl 1248.06005
[24] Cattaneo, G.; Ciucci, D., Posets and lattices, (), 2190-2208
[25] Cattaneo, G.; Ciucci, D.; Giuntini, R.; Konig, M., Algebraic structures related to many valued logical systems. part I: Heyting wajsberg algebras, Fundamenta informaticae, 63, 331-355, (2004) · Zbl 1090.03035
[26] Cattaneo, G.; Ciucci, D.; Giuntini, R.; Konig, M., Algebraic structures related to many valued logical systems. part II: equivalence among some widespread structures, Fundamenta informaticae, 63, 357-373, (2004) · Zbl 1092.03035
[27] Cattaneo, G.; Dalla Chiara, M.; Giuntini, R.; Paoli, F., Quantum logic and nonclassical logics, (), 127-226 · Zbl 1273.03171
[28] Cattaneo, G.; Dalla Chiara, M.L.; Giuntini, R., Fuzzy intuitionistic quantum logics, Studia logica, 52, 419-442, (1993) · Zbl 0791.03036
[29] G. Cattaneo, Dalla M.L. Chiara, R. Giuntini, Some algebraic structures for many-valued logics, Tatra Mountains Mathematical Publication 15 (1998) 173-196 (Special Issue: Quantum Structures II, Dedicated to Gudrun Kalmbach). · Zbl 0939.03077
[30] Cattaneo, G.; Giuntini, R., Some results on BZ structures from Hilbertian unsharp quantum physics, Foundations of physics, 25, 1147-1183, (1995)
[31] Cattaneo, G.; Giuntini, R.; Pilla, R., BZMV^{dm} and Stonian MV algebras (applications to fuzzy sets and rough approximations), Fuzzy sets and systems, 108, 201-222, (1999) · Zbl 0948.06008
[32] Cattaneo, G.; Manià, A., Abstract orthogonality and orthocomplementation, Proceedings of the Cambridge philosophical society, 76, 115-132, (1974) · Zbl 0295.06007
[33] G. Cattaneo, G. Marino, Partially ordered structures of fuzzy projections in Hilbert spaces, in: A. Di Nola, A. Ventre, (Eds.), Proceedings of the 1st Napoli Meeting on Mathematics of Fuzzy Systems, Napoli, 1984, pp. 59-71.
[34] Cattaneo, G.; Marino, G., Non-usual orthocomplementations on partially ordered sets and fuzziness, Fuzzy sets and systems, 25, 107-123, (1988) · Zbl 0631.06005
[35] Cattaneo, G.; Nisticò, Semantical structures for fuzzy logics: an introductory approach, (), 33-50 · Zbl 0619.03018
[36] Cattaneo, G.; Nisticò, G., Brouwer – zadeh posets and three valued łukasiewicz posets, Fuzzy sets and systems, 33, 165-190, (1989) · Zbl 0682.03036
[37] Chang, C.C., Algebraic analysis of many valued logics, Transactions of the American mathematical society, 88, 467-490, (1958) · Zbl 0084.00704
[38] Chellas, B.F., Modal logic, an introduction, (1988), Cambridge University Press Cambridge, MA, first published 1980 · Zbl 0431.03009
[39] A. Ciabattoni, G. Metcalfe, F. Montagna, Adding modalities to MTL and its extensions, in: Proceedings of the Linz Symposium, 2005.
[40] Cignoli, R., Boolean elements in łukasiewicz algebras. I, Proceedings of the Japan Academy, 41, 670-675, (1965) · Zbl 0168.00601
[41] Cignoli, R., Injective De Morgan and Kleene algebras, Proceedings of the American mathematical society, 47, 269-278, (1975) · Zbl 0301.06009
[42] Cignoli, R.; Monteiro, A., Boolean elements in łukasiewicz algebras. II, Proceedings of the Japan Academy, 41, 676-680, (1965) · Zbl 0168.00602
[43] D. Ciucci, D. Dubois, Truth-functionality, rough sets and three-valued logics, in: Proceedings ISMVL, 2010, pp. 98-103.
[44] Dubois, D., On ignorance and contradiction considered as truth-values, Logic journal of the IGPL, 16, 195-216, (2008) · Zbl 1139.03013
[45] Dubois, D.; Prade, H., Conditional objects as nonmonotonic consequence relationships, IEEE transaction of systems, man, and cybernetics, 24, 12, 1724-1740, (1994) · Zbl 1371.03041
[46] Dunn, J.M., Relevance logic and entailment, (), 117-224 · Zbl 0875.03051
[47] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017
[48] Esteva, F.; Godo, L.; Hájek, P.; Navara, M., Residuated fuzzy logics with an involutive negation, Archive for mathematical logic, 39, 103-124, (2000) · Zbl 0965.03035
[49] J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, in: Proceedings of FUZZ-IEEE 1995, 1995, pp. 2077-2082.
[50] Frink, O., New algebras of logic, American mathematical monthly, 45, 210-219, (1938) · JFM 64.0030.03
[51] Gehrke, M.; Walker, E., On the structure of rough sets, Bulletin Polish Academy of science (mathematics), 40, 235-245, (1992) · Zbl 0778.04002
[52] Gentilhomme, M.Y., LES ensembles flous en linguistique, Cahiers de linguistique theoretique et applique, bucarest, 47, 47-65, (1968)
[53] Goldblatt, R., Mathematical modal logic: a view of its evolution, Journal of applied logic, 1, 309-392, (2003) · Zbl 1041.03015
[54] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030
[55] Hájek, P., On very true, Fuzzy sets and systems, 124, 329-333, (2001) · Zbl 0997.03028
[56] Halmos, P.R., Algebraic logic, I. monadic Boolean algebras, Composition Mathematica, 12, 217-249, (1955) · Zbl 0087.24505
[57] Halmos, P.R., The basic concepts of algebraic logic, American mathematical monthly, 53, 363-387, (1956) · Zbl 0070.24506
[58] Halmos, P.R., Algebraic logic, (1962), Chelsea Pub. Co. New York · Zbl 0101.01101
[59] Halpern, J.Y.; Fagin, R.; Moses, Y.; Vardi, M., Reasoning about knowledge, (2003), MIT Press, Revised paperback edition · Zbl 1060.03008
[60] Hardegree, G.M., The conditional in abstract and concrete quantum logic, (), 49-108
[61] Hardegree, G.M., Material implication in orthomodular (and Boolean) lattices, Notre dame journal of modal logic, 22, 163-182, (1981) · Zbl 0438.03060
[62] Hintikka, J., Knowledge and belief, (1962), Cornell University Press
[63] Hughes, G.; Cresswell, M., A companion to modal logic, (1984), Methuen London · Zbl 0625.03005
[64] McKinsey, J.C.C., A solution to decision problem for the Lewis systems S2 ans S4 with an application to topology, The journal of symbolic logic, 6, 117-134, (1941) · JFM 67.0974.01
[65] McKinsey, J.C.C.; Tarski, A., The algebra of topology, Annals of mathematics, 45, 141-191, (1944) · Zbl 0060.06206
[66] McKinsey, J.C.C.; Tarski, A., On closed elements in closure algebras, Annals of mathematics, 47, 122-162, (1946) · Zbl 0060.06207
[67] McKinsey, J.C.C.; Tarski, A., Some theorems about the sentential calculi of Lewis and Heyting, Journal of symbolic logic, 13, 1-15, (1948) · Zbl 0037.29409
[68] Moisil, G.C., Recherches sur l’algebres de la logiques, annales scientifiques de l’université de jassy, 22, 1-117, (1935)
[69] Moisil, G.C., Recherches sur LES logiques non-chrysippiennes, annales scientifiques de l’université de jassy, 26, 431-466, (1940) · Zbl 0025.00409
[70] Moisil, G.C., Notes sur LES logiques non-chrysippiennes, annales scientifiques de l’université de jassy, 27, 86-98, (1941) · Zbl 0025.29401
[71] G.C. Moisil, Essais sur les logiques non Chrysippiennes. Edition de l’Académie de la République Socialiste de Roumanie, Bucharest, 1972.
[72] G.C. Moisil, Exemples de propositions a plusieurs valeurs (texte inédit), Chapter 1.7, in: [71], 1972.
[73] G.C. Moisil, Les ensembles flous et la logique à trois valeurs (texte inédit), Chapter 1.5. in: [71], 1972.
[74] A. Monteiro, Algebras monádicas. Atas do Segundo Colóquio Brasileiro de Matemática (33-52), appeared also as Notas de Logica Matematica, n. 7, Instituto de Matematica, Universitad Nacional del Sur, Bahia Blanca - Argentina, 1960 (1974).
[75] Monteiro, A., Matrices De Morgan caractéristiques pour le calcul propositionel classique, Anais da academia brasiliera de ciencias, 32, 1-7, (1960) · Zbl 0094.00605
[76] Monteiro, A., Sur la definition des algebres de łukasiewicz trivalentes, Bulletin mathématique de la société de sciences mathématiques et physiques de la République populaire roumaine, 7, 1-2, (1963) · Zbl 0143.00605
[77] Monteiro, A., Construction des algèbres de łukasiewicz trivalentes dens LES algèbres de Boole monadiques - I, Mathematica japonicae, 1-23, (1967) · Zbl 0165.30903
[78] Monteiro, A., Sur LES algèbres de Heyting symétriques, Portugaliae Mathematica, 39, 1-237, (1980) · Zbl 0582.06012
[79] A. Monteiro, Unpublished papers, I. Vol. 40 of Notas de Logica Matematica. NMABB-Conicet, Universidad Nacional del Sur, Bahia Blanca - Argentina, 1996.
[80] A. Monteiro, L. Monteiro, Algèbres de Stone libres, in: [79], 1968.
[81] Monteiro, L., Sur LES algebres de łukasiewicz injectives, Proceedings of the Japan Academy, 41, 578-581, (1965) · Zbl 0143.00607
[82] Nielsen, M.A.; Chuang, I.L., Quantum computation and quantum information, (2000), Cambridge University Press Cambridge · Zbl 1049.81015
[83] Orlowska, E., A logic of indiscernibility relations, Lncs, vol. 208, (1985), Springer-Verlag Berlin, pp. 177-186
[84] Orlowska, E., Kripke semantics for knowledge representation logics, Studia logica, 49, 255-272, (1990) · Zbl 0726.03023
[85] ()
[86] P. Pagliani, Rough set theory and logic-algebraic structures, in: [85], 1998, pp. 109-190.
[87] Pawlak, Z., Rough sets, International journal of computer and information sciences, 11, 341-356, (1982) · Zbl 0501.68053
[88] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Information sciences, 177, 3-27, (2007) · Zbl 1142.68549
[89] ()
[90] Rasiowa, H., An algebraic approach to non-classical logics, (1974), North Holland Amsterdam · Zbl 0299.02069
[91] Rasiowa, H.; Sikorski, R., The mathematics of metamathematics, () · Zbl 0122.24311
[92] M.B. Smyth, Topology, in: S. Abramsky, D. Gabbay, T. Maibaum, (Eds.), Handbook of Logic in Computer Science, Oxford University Press, 1992, pp. 641-761.
[93] S. Surma, Logical Works, Polish Academy of Sciences, Wroclaw, 1977.
[94] Tarski, A.; Thompson, F.B., Some general properties of cylindric algebras, Bulletin of the American mathematical society, 58, 65, (1952), Abstracts of the Meeting in Pasadena, December 1, 1951
[95] Tsao-Chen, T., Algebraic postulates and a geometrical interpretation for the Lewis calculus of strict implication, Bulletin of the American mathematical society, 44, 737-744, (1938) · Zbl 0019.38504
[96] Varadarajan, V.S., Geometry of quantum physics. I, (1968), Van Nostrand Princeton · Zbl 0155.56802
[97] M. Wajsberg, Aksjomatyzacja trówartościowego rachunkuzdań [Axiomatization of the three-valued propositional calculus]. Comptes Rendus des Séances de la Societé des Sciences et des Lettres de Varsovie 24 (1931) 126-148 (English Translation in [93]).
[98] Wajsberg, M., Beiträge zum metaaussagenkalkül I, Monashefte fur Mathematik un physik, 42, 221-242, (1935), English Translation in [93] · Zbl 0013.09701
[99] Walker, E., Stone algebras, conditional events, and three valued logic, IEEE transaction of systems, man, and cybernetics, 24, 12, 1699-1707, (1994) · Zbl 1371.03034
[100] Zadeh, L.A., A fuzzy-set-theoretic interpretation of linguistic hedges, Journal of cybernetics, 2, 4-34, (1972)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.