A Modern Introduction to Fuzzy Mathematics

1Introduction

Vagueness is a fundamental property of this world. Vague objects are real objects and exist in the real world. Fuzzy mathematics is mathematics of vagueness. The core of fuzzy mathematics is the idea that objects have a property to some degree.

1.1 What Is Vagueness?

When we say that something is vague, we mean that its properties and capacities are not sharply determined. In different words, a vague concept is one that is characterized by fuzzy boundaries (i.e. there are cases where it is not clear if an object has or does not have a specific property or capacity). Jiri Benovsky [27] put forth an objection to this idea by claiming that everybody who thinks that there are ordinary objects must accept that they are vague, whereas everybody must accept the existence of sharp boundaries to ordinary objects. This does not lead to a contradiction since the two claims do not concern the same “everybody”.

The Sorites Paradox (σόφισμα τοῦ σωρείτη), which was introduced by Eubulides of Miletus (Eὐβουλίδης ὁ Mιλήσιος),¹ is a typical example of an argument that demonstrates what fuzzy boundaries are. The term “σωρείτες” (sorites) derives from the Greek word σωρός (soros), which means “heap.” The paradox is about the number of grains of wheat that makes a heap. All agree that a single grain of wheat does not comprise a heap. The same applies for two grains of wheat as they do not comprise a heap, etc. However, there is a point where the number of grains becomes large enough ...