Skip to main content
SpringerLink
Log in

Information Gaps as Communication Needs: A New Semantic Foundation for Some Non-Classical Logics

  • Published:
Journal of Logic, Language and Information Aims and scope Submit manuscript

Abstract

Semantics connected to some information based metaphor are well-known in logic literature: a paradigmatic example is Kripke semantic for Intuitionistic Logic. In this paper we start from the concrete problem of providing suitable logic-algebraic models for the calculus of attribute dependencies in Formal Contexts with information gaps and we obtain an intuitive model based on the notion of ’’passage of information‘‘ showing that Kleene algebras, semi-simple Nelson algebras, three-valued Łukasiewicz algebras and Post algebras of order three are, in a sense, ’’naturally‘‘ and directly connected to partially defined information systems. In this way wecan provide for these logic-algebraic structures a ’’raison d‘êetre‘‘ different from the original motivations concerning, for instance, computability theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Allwein, G. and Dunn, J.M., 1993, "Kripke models for linear logic," JSL 58(2), 514–544.

    Google Scholar 

  • Balbes, R. and Dwinger, Ph., 1974, Distributive Lattices, Columbia, Missouri: University of Missouri Press.

    Google Scholar 

  • Boicescu, V., Filipoiu, A., Georgescu, G., and Rudeanu, S., 1991, Łukasiewicz–Moisil Algebra, Amsterdam: North-Holland.

    Google Scholar 

  • Buneman, P., Jung, A., and Ohori, A., 1991, "Using powerdomains to generalize relational databases," Theoretical Comp. Science, 25–55.

  • Burmeister, P., 1989, "Merkmalsimplikationen bei unsicherem Wissen," Technische Hochschule Darmstadt, Preprint No. 1218.

  • Burmeister, P., 1993, "Formal concept analysis with a three-valued conceptual logic," Technische Hochschule Darmstadt (preliminary report).

  • Cattaneo, G. and Nisticó, G., 1988, "Brouwer–Zadeh posets and three-valued Łukasiewicz posets," Journal of Fuzzy Sets and Systems 25, 165–190.

    Google Scholar 

  • Cignoli, R., 1986, "The class of Kleene algebras satisfying an interpolation property and Nelson algebras," Algebra Universalis 23, 262–292.

    Google Scholar 

  • Dahn, B., 1974, "Kripke-style semantics for some many-valued propositional calculi," Bull. Polish Acad. of Sciences. Math. 32(2), 747–758.

    Google Scholar 

  • Davey, B.A. and Duffus, D., 1982, "Exponentiation and duality," pp. 43–95 in Ordered Sets, I. Rival, ed., NATO ASI Series 83, Dordrecht: Reidel.

    Google Scholar 

  • Dunn, J.M., 1966, "The algebra of intensional logics," PhD Dissertation, University of Pittsburg, 1966.

  • Epstein, G. and Horn, A., 1974, "Chain based lattices," Journal of Mathematics 55(1), 65–84.

    Google Scholar 

  • Fitting, M., 1991, "Kleene’s logic generalized," Journal of Logic and Computation 1(6), 797–810.

    Google Scholar 

  • Girard, J.Y., 1987, "Linear logic," Theoretical Computer Science 50, 1–102.

    Google Scholar 

  • Halpern, J.Y. and Moses, Y., 1985, "A guide to the modal logic of knowledge and belief: preliminary report," pp. 480–490 in Proc. of the 9th IJCAI.

  • Iturrioz, L., 1990, "Logiques multivaluées. Un outil souple et pluriel," pp. 113–138 in Modéles logiques et systèmes d’intelligence artificielle, L. Iturrioz and A. Dussauchoy, eds., Hermes.

  • Kent, R.E., 1993, "Rough concept analysis," pp. 245–253 in Proc. Intern. Conference on Rough Sets and Knowledge Discovery, Banff, Canada.

  • Kleene, S.C., 1952, Introduction to Metamathematics, Amsterdam: North-Holland.

    Google Scholar 

  • Kreisel, G., 1965a, "Mathematical logic," pp. 95–195 in Lectures in Modern Mathematics, L.S. Saaty, ed., New York.

  • Kreisel, G., 1965b, "Informal rigour and completeness proofs" (and related panel discussion), pp. 138–180 in Problems in the Philosophy of Mathematics, Proc. Intern. Coll. Phil. of Science, London, Vol. I, I. Lakatos, ed., Amsterdam.

  • Kozen, D., 1981, "On the duality of dynamic algebras and Kripke models," LNCS 125, 1–11.

    Google Scholar 

  • Kripke, S.A., 1963a, "Semantical analysis of Modal Logic I," Zeitschr. Math. und Grund. Math. 9, 67–96.

    Google Scholar 

  • Kripke, S.A., 1963b, "Semantical analysis of intuituitionistic logic," pp. 92–129 in Formal Systems and Recursive Functions, J.N. Crossley and M.A.E. Dummett, eds., Amsterdam: North-Holland.

    Google Scholar 

  • Libkin, L., 1991, "A relational algebra for complex objects based on partial information," pp. 30–43 in Proc. 3rd Symp. on Fundamentals of Database Knowledge Base Systems. Lecture Notes in Comp. Science, Vol. 495.

    Google Scholar 

  • Maximova, L. and Vakarelov, D., 1974, "Semantics for ω+-valued predicate calculi," Bull. Polish Acad. of Sciences, Math. Astr. Phys. 8, 765–771.

    Google Scholar 

  • Miglioli, P.A., Moscato, U., Ornaghi, M., and Usberti, U., 1989, "A constructivism based on classical truth," Notre Dame Journal of Formal Logic 30(1), 67–90.

    Google Scholar 

  • Monteiro, A., 1963, "Construction des algebres de Nelson finies," Bull. Polish Acad. of Sciences, Math. 11, 359–362.

    Google Scholar 

  • Monteiro, A., 1967, "Construction des algebres de Łukasiewicz trivalentes dans les algebres de Boole monadiques," I. Math. Japonicae 12, 1–23.

    Google Scholar 

  • Nelson, D., 1949, "Constructible falsity," JSL 14, 16–26.

    Google Scholar 

  • Orłowska, E., 1988, "Kripke models with relative accessibility and their applications to inferences fromincomplete information," Mathem. Problems in Comput. Theory. Banach Center Publication 21, 329–339.

    Google Scholar 

  • Orłowska, E., 1989, "Logic for reasoning about knowledge," Zeitschr. f. math. Log. und Grund. d. Math. 35, 559–572.

    Google Scholar 

  • Orłowska, E. and Pawlak, Z., 1984, "Representation of non deterministic information," Theoretical Computer Science 29, 27–39.

    Google Scholar 

  • Pagliani, P., 1990, "Some remarks on special lattices and related constructive logics with strong negation," Notre Dame Journal of Formal Logic 31(4), 515–528.

    Google Scholar 

  • Pagliani, P., 1993, "From concept lattices to approximation spaces: Algebraic structures of some spaces of partial objects," Fundamenta Informaticae 18(1), 1–25.

    Google Scholar 

  • Pagliani, P., 1993, A Logic-Algebraic Approach to Formal Contexts with Partial Information. International Workshop on Order and Data Analysis (abstr. of the lectures). University of Massachusetts, Amherst.

    Google Scholar 

  • Pagliani, P., 1994, "A pure logic-algebraic analysis on rough top and rough bottom equalities," pp. 227–236 in Proc. of Rough Sets and Knowledge Discovery ’93, Banff, October 1993, W. Ziarko, ed., Springer-Verlag.

  • Pagliani, P., 1994, "Towards a logic of rough sets systems," in Soft Computing, Proceedings International Conference on Rough Sets and Soft Computing, S. Jose, 1994, T.Y. Lin and A.M. Wildberg, eds., The Society for Computer Simulation.

  • Pagliani, P., "Rough sets systems and logic algebraic structures," in Modelling Incomplete Information: Fundamentals and Applications, E. Orłowska, ed. (forthcoming).

  • Parikh, R., 1981, "Propositional dynamic logics of programs: A survey," LNCS 125, 101–144.

    Google Scholar 

  • Parikh, R., 1982, "Some applications of topology to program semantics." LNCS 131, 375–385.

    Google Scholar 

  • Pawlak, Z., 1982, "Rough sets: Algebraic and topological approach," Int. J. of Inf. and Comp. Science 11, 341–356.

    Google Scholar 

  • Pawlak, Z., 1991, Rough Sets: A Theoretical Approach to Reasoning about Data, Dordrecht: Kluwer.

    Google Scholar 

  • Pawlak, Z. and Rauszer, C., 1984, "Dependency of attributes in information systems," Bull. Polish Acad. of Sciences., Math. 33(9/10), 551–559.

    Google Scholar 

  • Rival, I. (ed.), 1982, Ordered Sets, NATO ASI Series 83, Dordrecht: Reidel.

    Google Scholar 

  • Sendlewski, A., 1990, "Nelson algebras through Heyting ones: I," Studia Logica 49(1), 105–126.

    Google Scholar 

  • Thomason, R., 1969, "A semantical study of constructible falsity," Zeitschr. Math. Logik und Grund. Math. 15, 247–257.

    Google Scholar 

  • Vakarelov, D., 1977, "Notes on N-lattices and constructive logics with strong negation," Studia Logica 36, 109–125.

    Google Scholar 

  • Vikers, S., 1989, Topology via Logic. Cambridge Tracts in Theoretical Computer Science, Vol. 5.

  • Wille, R., 1982, "Restructuring lattice theory," pp. 445–470 in Ordered Sets, I. Rival, ed., Dordrecht: Reidel.

    Google Scholar 

  • Wille, R., 1988, "Dependencies of many-valued attributes," pp. 581–586 in Classification and Related Methods in Data Analysis, H.H. Bock, ed., Amsterdam: North-Holland.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Group on Knowledge and Communication Models, Via Imperia, 6, 00161, Roma, Italy

    Piero Pagliani

Authors
  1. Piero Pagliani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pagliani, P. Information Gaps as Communication Needs: A New Semantic Foundation for Some Non-Classical Logics. Journal of Logic, Language and Information 6, 63–99 (1997). https://doi.org/10.1023/A:1008286102793

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008286102793

Keywords

Search

Navigation