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Introduction to Fuzzy Logic

2.1 Introduction

In classical (Newtonian) mechanics, uncertainty was considered as undesirable and to be avoided by any means. In the late nineteenth century, researchers started to realize that no physical system exists without a certain amount of uncertainty. This is a phenomenon without which the description of a system or model is incomplete. A trend started then in science and engineering to incorporate uncertainty in system models. At this stage uncertainty was quantified with the help of probability theory, developed in the eighteenth century by Thomas Bayes (Price, 1763). The expression of uncertainty using probability theory was first challenged by Max Black (Black, 1937). He proposed a degree as a measure of vagueness. Vagueness can be used to describe a certain kind of uncertainty. For example, John is young. The proposition defined here is vague. He pointed out two main ideas: one is the nature and observability of vagueness and the other is the relevance of vagueness for logic. Black proposed vague sets defined by a membership curve. This was the first attempt to give a precise mathematical theory for sets where there is a membership curve.

There was another movement present in the philosophy, among logicians. The most basic assumptions of classical (or two-valued) propositional as well as first-order logic are the principles of bivalence and compositionality. The principle of bivalence is the assumption that each sentence is either true ...

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