The sorites paradox and fuzzy logic

Abstract

The sorites paradox (interpreted as the paradox of small natural numbers) is analyzed using mathematical fuzzy logic. In the first part, we present an extension of BL-fuzzy logic by a new unary connective At of almost true and the crisp Peano arithmetic extended by a fuzzy predicate of feasibility. Then we give examples of possible semantics of At and examples of semantics of feasible numbers. In the second part, we present an analysis of the sorites paradox within fuzzy logic with evaluated syntax and show that under a very natural assumption we obtain a consistent fuzzy theory. Thus, sorites is not paradoxical at all.

Keywords:

• Sorites paradox
• Basic fuzzy logic
• Fuzzy logic in narrow sense
• Peano arithmetic

Acknowledgements

Partial support of grant No. A1030004/00 of the Grant Agency of the Academy of Sciences of the Czech Republic is acknowledged by the first author.

Notes

^†[email protected].

^‡Note that the notion of a large number is discussed in Hájek (1998) 3.3.21 for Łukasiewicz logic and 4.1.27 for product logic. Note also the analysis of the liar paradox in Hájek et al. (2000).

^¶Note that by no means can we say that n ⁰ is the first large number; it is only the number which we surely know to be large.

Prof. RNDr. Petr Hájek, DrSc. graduated in mathematics from Charles University in Prague. He obtained a PhD and DSc. (doctor of sciences) in mathematical logic in the Czechoslovak Academy of Sciences. In 1998, he obtained full professorship in mathematics (awarded by the president of the Czech Republic). He worked in the Mathematical Institute of the Czechoslovak Academy of Sciences—now the Academy of Sciences of the Czech Republic—and then became director of the Institute of Computer Science of AS CR, where he is now a senior scientist. He is author or coauthor of 6 monographs and more than 200 scientific papers in the field of mathematical logic. He is a member of the Association for Symbolic Logic and of the editorial boards of the Archive for Mathematical Logic, Fundamenta Informaticae, Soft Computing, and the Czechoslovak Mathematical Journal.

Prof. Ing. Vilém Novák, DrSc. graduated from Mining University, Ostrava (Czech Republic) as a system engineer. He obtained a PhD in mathematical logic in Charles University, Prague, and DSc. (doctor of sciences) in computer science at the Polish Academy of Sciences, Warsaw. He obtained a full professorship at Masaryk University, Brno (awarded by the president of the Czech Republic). He worked in the Mining Institute of the Czechoslovak Academy of Sciences and since 1995 he has worked in the University of Ostrava in the Department of Mathematics and at the same time is the director of the Institute for Research and Applications of Fuzzy Modeling. His research interests are fuzzy logic, modeling of natural language semantics, fuzzy control and modeling. He is author or coauthor of 4 monographs and more than 140 scientific papers. He is a member of the European Society for Fuzzy Logic and Technology (EUSFLAT) and also a member of the editorial board of the International Journal of Fuzzy Sets and Systems, the Journal of Soft Computing and the Journal of Applied Non-Classical Logics.