Abstract
Non-standard logics departs from traditional logic mostly in extended views, on one hand syntactically related to logical operators, and on the other hand semantically related to truth values. Typical for these approaches is the remaining traditional view on ’sets and relations’ and on terms based on signatures. Thus the cornerstones of the languages remain standard, and so does mostly the view on knowledge representation and reasoning using traditional substitution theories and unification styles. In previous papers we have dealt with particular problems such as generalizing terms and substitution, extending our views on sets and relations, and demonstrated the use of these non-standard language elements in various applications such as for fuzzy logic, generalized convergence spaces, rough sets and Kleene algebras. In this paper we provide an overview and summarized picture of what indeed happens when we drop the requirement for using traditional sets with relations and terms with equational settings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, London (1975)
Barr, M., Wells, C.: Toposes, Triples and Theories. Springer, Heidelberg (1985)
Butzmann, H.P., Kneis, G.: On Čech-Stone compactifications of pseudo-topological spaces. Math. Nachr. 128, 259–264 (1986)
Eklund, P., Gähler, W.: Generalized Cauchy spaces. Math. Nachr. 147, 219–233 (1990)
Eklund, P., Gähler, W.: Fuzzy Filter Functors and Convergence. In: Rodabaugh, S.E., Klement, E.P., Höhle, U. (eds.) Applications of category theory to fuzzy subsets. Theory and Decision Library B, pp. 109–136. Kluwer, Dordrecht (1992)
Eklund, P., Gähler, W.: Completions and Compactifications by Means of Monads. In: Lowen, R., Roubens, M. (eds.) Fuzzy Logic, State of the Art, pp. 39–56. Kluwer, Dordrecht (1993)
Eklund, P., Gähler, W.: Partially ordered monads and powerset Kleene algebras. In: IPMU 2004. Proc. 10th Information Processing and Management of Uncertainty in Knowledge Based Systems Conference (2004)
Eklund, P., Galán, M.A.: The rough powerset monad. In: Proc. of the 37th International Symposium on Multiple Valued Logics, ISMVL-2007, Oslo (Norway) (Accepted)
Eklund, P., Galán, M.A., Gähler, W., Medina, J., Ojeda Aciego, M., Valverde, A.: A note on partially ordered generalized terms. In: Proc. of Fourth Conference of the European Society for Fuzzy Logic and Technology and Rencontres Francophones sur la Logique Floue et ses applications (Joint EUSFLAT-LFA 2005), pp. 793–796 (2005)
Eklund, P., Galán, M.A., Medina, J., Ojeda Aciego, M., Valverde, A.: A categorical approach to unification of generalised terms. Electronic Notes in Theoretical Computer Science, vol. 66(5) (2002), http://www.elsevier.nl/locate/entcs/volume66.html
Eklund, P., Galán, M.A., Medina, J., Ojeda-Aciego, M., Valverde, A.: Similarities between powersets of terms, Fuzzy Sets and Systems, 144, 213–225 (2004) Godo, L., Sandri, S. (eds.) Possibilistic Logic and Related Issues (2004)
Eklund, P., Galán, M.A.: Monads can be rough. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 77–84. Springer, Heidelberg (2006)
Gähler, W.: A topological approach to structure theory. Math. Nachr. 100, 93–144 (1981)
Gähler, W.: Monads and convergence. In: Proc. Conference Generalized Functions, Convergences Structures, and Their Applications, Dubrovnik (Yugoslavia), pp. 29–46. Plenum Press, New York (1988)
Gähler, W.: Completion theory. In: Mathematisches Forschungsinstitut Oberwolfach, p. 8, Tagungsbericht 48/1991 (1991)
Gähler, W.: General Topology – The monadic case, examples, applications. Acta Math. Hungar. 88, 279–290 (2000)
Gähler, W.: Extension structures and completions in topology and algebra, Seminarberichte aus dem Fachbereich Mathematik, Band 70, FernUniversität in Hagen (2001)
Gähler, W., Eklund, P.: Extension structures and compactifications. In: CatMAT 2000. Categorical Methods in Algebra and Topology, pp. 181–205 (2000)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)
Keller, H.H.: Die Limesuniformisierbarkeit der Limesräume. Math. Ann. 176, 334–341 (1968)
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956)
Kowalsky, H.-J.: Limesräume und Komplettierung. Math. Nachr. 12, 301–340 (1954)
Manes, E.G.: Algebraic Theories. Springer, Heidelberg (1976)
Meseguer, J.: General logics. In: Ebbinghaus, H.-D. et al. (eds.) Logic Colloquium ’87, pp. 275–329. Elsevier/North-Holland, Amsterdam (1989)
Pawlak, Z.: Rough sets. Int. J. Computer and Information Sciences 5, 341–356 (1982)
Rydeheard, D.E., Burstall, R.M.: A categorical unification algorithm. In: Poigné, A., Pitt, D.H., Rydeheard, D.E., Abramsky, S. (eds.) Category Theory and Computer Programming. LNCS, vol. 240, pp. 493–505. Springer, Heidelberg (1986)
Salomaa, A.: Two complete axiom systems for the algebra of regular events. J. ACM 13, 158–169 (1966)
Tarski, A.: On the calculus of relations. J. Symbolic Logic 6, 65–106 (1941)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eklund, P., Galán, M.A. (2007). On Logic with Fuzzy and Rough Powerset Monads. In: Kryszkiewicz, M., Peters, J.F., Rybinski, H., Skowron, A. (eds) Rough Sets and Intelligent Systems Paradigms. RSEISP 2007. Lecture Notes in Computer Science(), vol 4585. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73451-2_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-73451-2_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73450-5
Online ISBN: 978-3-540-73451-2
eBook Packages: Computer ScienceComputer Science (R0)