# zbMATH — the first resource for mathematics

The sorites paradox and fuzzy logic. (English) Zbl 1044.03017
The authors describe two interesting, slightly different ways to use fuzzy logic to discuss and explain the sorites paradox. Both approaches refer to an extended version of the Peano arithmetic: the extension being provided by a new unary predicate “feasible”, subject to some rather natural conditions.
One way is to use Hájek’s fuzzy logic BL and to extend it by a unary connective “almost true” with two suitable axioms
The other way is to refer to a “Pavelka-style” extension of infinite-valued Łukasiewicz logic, which allows for a graded notion of provability.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03F30 First-order arithmetic and fragments 03A05 Philosophical and critical aspects of logic and foundations
##### Keywords:
mathematical fuzzy logic; sorites paradox; Peano arithmetic
Full Text:
##### References:
 [1] Goguen J.A. 1968–69 The logic of inexact concepts Synthese 19 325 373 · Zbl 0184.00903 [2] Hájek P. 1998 Metamathematics of Fuzzy Logic Kluwer Dordrecht [3] Hájek P. 2000 Function symbols in fuzzy predicate logic Proc. East West Fuzzy Colloquium 2000 (Zittau-Görlitz) 2 8 [4] Hájek P. 2001 On very true Fuzzy Sets Syst. 124 329 333 · Zbl 0997.03028 [5] Hájek P. Pudlák P. 1993 Metamathematics of First-Order Arithmetic Springer Heidelberg [6] Hájek P. Paris J. Shepherdson J. 2000 The liar paradox and fuzzy logic J. Symb. Logic 65 339 346 · Zbl 0945.03031 [7] Höhle U. 1995 Commutative residuated l-monoids Höhle U. Klement E.P. Non-Classical Logics and Their Applications to Fuzzy Subsets. A Handbook of the Mathematical Foundations of Fuzzy Set Theory Kluwer Dordrecht [8] Keefe R. 2000 Theories of Vagueness Cambridge University Press Cambridge [9] Mesiar R. Novák V. 1997 On Fitting Operations Proc. of VIIth IFSA World Congress Academia Prague 286 290 [10] Novák V. 1996 Paradigm, Formal Properties and Limits of Fuzzy Logic Int. J. General Syst. 24 377 405 · Zbl 0855.03010 [11] Novák V. Perfilieva I. 2000 Discovering the World With Fuzzy Logic Studies in Fuzziness and Soft Computing, Springer-Verlag Heidelberg [12] Novák V. Perfilieva I. Močkoř J. 1999 Mathematical Principles of Fuzzy Logic Kluwer Dordrecht [13] Parikh R. 1971 On existence and feasibility in arithmetic J. Symb. Logic 36 494 508 · Zbl 0243.02037 [14] Read S. 1995 Thinking About Logic Oxford University Press Oxford [15] Vopěnka P. 1979 Mathematics in the Alternative Set Theory Teubner Leipzig
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.