### Vagueness and Formal Fuzzy Logic: Some Criticisms

DOI: http://dx.doi.org/10.12775/LLP.2017.031

#### Abstract

In the common man reasoning the presence of vague predicates is pervasive and under the name “fuzzy logic in narrow sense” or “formal fuzzy logic” there are a series of attempts to formalize such a kind of phenomenon. This paper is devoted to discussing the limits of these attempts both from a technical point of view and with respect the original and principal task: to define a mathematical model of the vagueness. For example, one argues that, since vagueness is necessarily connected with the intuition of the continuum, we have to look at the order-based topology of the interval [0,1] and not at the discrete topology of the set {0,1}. In accordance, in switching from classical logic to a logic for the vague predicates, we cannot avoid the use of the basic notions of real analysis as, for example, the ones of “approximation“, “convergence“, “continuity“. In accordance, instead of defining the compactness of the logical consequence operator and of the deduction operator in terms of finiteness, we have to define it in terms of continuity. Also, the effectiveness of the deduction apparatus has to be defined by using the tools of constructive real analysis and not the one of recursive arithmetic. This means that decidability and semi-decidability have to be defined by involving effective limit processes and not by finite steps stopping processes.

#### Keywords

#### Full Text:

PDF#### References

Belohlavek, R., 2002, “Fuzzy equational logic”, Arch. Math. Log. 41, 1: 83–90. DOI: 10.1007/s001530200006

Belohlavek, R., 2015, “Pavelka-style fuzzy logic in retrospect and prospect”, Fuzzy Sets and Systems 281: 61–72. DOI: 10.1016/j.fss.2015.07.007

Belohlavek, R., J.W. Dauben, G.J. Klir, 2017, Fuzzy Logic and Mathematics: A Historical Perspective, Oxford University Press (to appear). DOI: 10.1093/oso/9780190200015.001.0001

Belohlavek, R., and V. Vychodil, 2005, Fuzzy Equational Logic, Springer,Berlin, 2005. DOI: 10.1007/11376422_3

Belohlavek, R., and V. Vychodil, 2015, “A logic of graded attributes”, Arch. Math. Log. 4, 7–8: 785–802. DOI: 10.1007/s00153-015-0440-0

Belohlavek, R., G.J. Klir, H.W. Lewis III, E.C. Way, 2009, “Concepts and fuzzy sets: Misunderstandings, misconceptions, and oversights”, International Journal of Approximate Reasoning 51, 1: 23–34. DOI: 10.1016/j.ijar.2009.06.012

Biacino, L.G., G. Gerla, 2002, “Fuzzy logic, continuity and effectiveness”, Archive for Mathematical Logic 41, 7: 643–667. DOI: 10.1007/s001530100128

Carotenuto, G., G. Gerla, 2013, “Bilattices for deductions in multi-valued logic”, International Journal of Approximate Reasoning 54, 8: 1066–1086. DOI: 10.1016/j.ijar.2013.04.004

Cintula, P., and P. Hájek, 2010, “Triangular norm based predicate fuzzy logics”, Fuzzy Sets and Systems 161, 3: 311–346. DOI: 10.1016/j.fss.2009.09.006

Entemann, C.W., 2002, “Fuzzy logic: Misconceptions and clarifications”, Artificial Intelligence Review 17: 65–84.

Genito, D., G. Gerla, 2014, “Connecting bilattice theory with multivalued logic”, Logic and Logical Philosophy 23, 1: 15-45. DOI: 10.12775/LLP.2013.036

Gerla, G., 1982, “Sharpness relation and decidable fuzzy sets”, IEEE Trans. on Automatic Control, AC-27, 5: 1113. DOI: 10.1109/TAC.1982.1103075

Gerla, G., 2000, Fuzzy logic: Mathematical tools for approximate reasoning, Trends in Logic, Kluwer Ac. Press. DOI: 10.1007/978-94-015-9660-2

Gerla, G., 2005, “Fuzzy logic programming and fuzzy control”, Studia Logica 79, 2: 231–254. DOI: 10.1007/s11225-005-2977-0

Gerla, G., 2006, “Effectiveness and multi-valued logics”, Journal of Symbolic Logic 71, 1: 137–162. DOI: 10.2178/jsl/1140641166

Gerla, G., 2007, “Multi-valued logics, effectiveness and domains”, pages 336–347 in Computation and Logic in the Real World, Lecture Notes in Computer Science 4497, Springer. DOI: 10.1007/978-3-540-73001-9_35

Gerla, G., 2016,“Comments on some theories of fuzzy computation”, International Journal of General Systems 45, 4: 372–392. DOI: 10.1080/03081079.2015.1076403

Goguen, J.A., 1968/1969, “The logic of inexact concepts”, Synthese 19, 3–4: 325–373. DOI: 10.1007/BF00485654

Gottwald, S., 2008, “Mathematical fuzzy logic”, The Bulletin of Symbolic Logic 14, 2: 210–239. DOI: 10.2178/bsl/1208442828

Hájek, P., 1995, “Fuzzy logic and arithmetical hierarchy”, Fuzzy Sets and Systems 73, 3: 359–363. DOI: 10.1016/0165-0114(94)00299-M

Hájek, P., 1998, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht. DOI: 10.1007/978-94-011-5300-3

Hájek, P., 1999, “Ten questions and one problem on fuzzy logic”, Annals of Pure and Applied Logic 96, 1–3: 157–165. DOI: 10.1016/S0168-0072(98)00037-2

Hájek, P., 2006, “What is mathematical fuzzy logic?”, Fuzzy Sets and Systems 157, 5: 597–603. DOI: 10.1016/j.fss.2005.10.004

Hájek, P., and V. Novák, 2003, “The Sorites paradox and fuzzy logic”, International Journal of General Systems 32, 4: 373–383. DOI: 10.1080/0308107031000152522

Kamp, H., 1975, “Two theories of adjectives”, pages 123–155 in E. Keenan (ed.), Formal Semantics of Natural Languages, Cambridge University Press. DOI: 10.1017/CBO9780511897696.011

Montagna, F., 2001, “Three complexity problems in quantified fuzzy logic”, Studia Logica 68, 1: 143–152. DOI: 10.1023/A:1011958407631

Novák, V., 1990a, “On the syntactico-semantical completeness of first-order fuzzy logic. Part I: Syntax and Semantics”, Kybernetika 26: 47–66.

Novák, V., 1990b, “On the syntactico-semantical completeness of first-order fuzzy logic. Part II: Main results”, Kybernetika 26: 134–154.

Novák, V., 2005, “Are fuzzy sets a reasonable tool for modeling vague phenomena?”, Fuzzy Sets and Systems 156, 3: 341–348. DOI: 10.1016/j.fss.2005.05.029

Parikh, R., 1991, “A test for fuzzy logic”, Sigact News 22, 3: 49–50. DOI: 10.1145/126537.126542

Pavelka, J., 1979a, “On fuzzy logic I: Many-valued rules of inference”, Zeitschr. F. math. Logik und Grundlagen d. Math. 25, 3–6: 45–52. DOI: 10.1002/malq.19790250304

Pavelka, J., 1979b, “On fuzzy logic II: Enriched residuated lattices and semantics of propositional calculus”, Zeitschr. F. math. Logik und Grundlagen d. Math. 25, 7–12: 119–134. DOI: 10.1002/malq.19790250706

Pavelka, J., 1979c, “On fuzzy logic III: Semantical completeness of some many-valued propositional calculi”, Zeitschr. F. math. Logik und Grundlagen d. Math. 25, 25–29: 447-464. DOI: 10.1002/malq.19790252510

Pelletier, F.J., 2000, “Review of Metamathematics of Fuzzy Logic by P. Hájek”, The Bulletin of Symbolic Logic 6: 342–346.

Pelletier, F.J., 2004, “On some alleged misconceptions about fuzzy logic”, Artificial Intelligence Review 22, 1: 71–82. DOI: 10.1023/B:AIRE.0000044308.48654.c1

Rogers, H.,1967, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.

Sauerland, U., 2011, “Vagueness in language: The case against fuzzy logic revisited”, pages 185–198 in P. Cintula, C. Fermüller, L. Godo, and P. Hájek (eds.), Understanding Vagueness: Logical, Philosophical, and Linguistic Perspectives, Studies in Logic 36, College Publications, London, UK.

Sorensen, R., 2001, Vagueness and Contradiction, Clarendon Press, Oxford.

Tabacchi, M.E., and S. Termini, 2017a, “Back to ‘reasoning’”, pages 471–478 in M.B. Ferraro et al. (eds.), Soft Methods for Data Science, Advances in Intelligent Systems and Computing 456, Springer, Switzerland. DOI: 10.1007/978-3-319-42972-4_58

Tabacchi, M.E., and S. Termini, 2017b, “Fuzziness as an experimental science: An homage to Claudio Moraga”, pages 41–54 in R. Seising and H. Allende-Cid (eds.), Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing, Studies in Fuzziness and Soft Computing, Springer, Switzerland. DOI: 10.1007/978-3-319-48317-7_4

Termini, S., 2002, “On some vagaries of vagueness and information”, Annals of Mathematics and Artificial Intelligence 35: 343–355.

Trillas, E., 2006, “On the use of words and fuzzy sets”, Information Sciences 176, 11: 1463–1487. DOI: 10.1016/j.ins.2005.03.008

Turunen, E., 1999, Mathematics Behind Fuzzy Logic, Springer, Heidelberg.

Vojtáš, P., 2001, “Fuzzy logic programming”, Fuzzy Sets and Systems 124: 361–370.

Zadeh, L.A., 1965, “Fuzzy sets”, Information and Control 8, 3: 338–353. DOI: 10.1016/S0019-9958(65)90241-X URL: https://people.eecs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-1965.pdf

Print ISSN: 1425-3305

Online ISSN: 2300-9802