Introduction to Fuzzy Logic. James K. Peckol

1 A Brief Introduction and History

      THINGS TO LOOK FOR…

1.1 Introduction

      We open this text with a challenge and a foundation. Whether crisp or fuzzy, whether involving animals, humans, or machines, philosophers, scientists, and educators have studied, debated, and analyzed terms such as think, ponder, logic, reason, philosophize, or learn for centuries. Yet today, our understanding of these processes still has the opportunity to grow. Given such a history, what do we know?

      Let us start with learning. Learning is a process that starts (at least) immediately after birth and continues, often unobtrusively, through the remaining years of life. Recent research, however, has found that learning may actually begin months earlier. Nevertheless, the term itself generally evokes childhood memories of old books, pedagogical teachers, and stuffy classrooms on warm spring afternoons when we would rather be outside playing. If we pause and reflect for a moment, we realize that learning is not limited to that proffered by the pendants of previous days but is a natural part of our daily existence. Each time that we encounter a fresh idea, make a new discovery, or solve one of life's many challenges, we are learning; we are growing and enriching our perceived model of our world or potentially what lies in space beyond.

      Understanding the concepts of learning and reasoning is playing an increasingly significant role in the modern high‐tech design, development, and implementation of perceptrons, neural networks, artificial intelligence, machine learning, and the primary topic of this text, i.e., fuzzy systems. We mentioned two terms: “crisp” and “fuzzy.” We now introduce and explore thought and reasoning in such systems.

1.2 Models of Human Reasoning

      We now move from animals and humans to raise the question that has perplexed for eons. Can a computing machine be designed to think, to reason, to learn, to create, that is, to self‐modify? Can such machines learn and operate like a human being? To be able to design and implement such tools and machines, we must first fully understand their intended applications and what these terms actually mean in such contexts.

      Throughout history, as an outgrowth of the work of researchers in both the information processing and the epistemological schools, many different models of human reasoning have been explored, proposed, and tried. In each instance, the hypothetical model tries to capture the dynamic nature, inexactness, or intuitive nature of the underlying process.

      Frequently, the heuristic character of human reasoning is quantified numerically. This is seen in Lotfi Zadeh's characterization of notions such as young or old on a mathematical scale or Ted Shortliffe's measures of belief and disbelief. Herb Simon has countered that people do not reason numerically. Perhaps they do not. However, mathematics is a reasonable first‐order approach when attempting to capture the essence of the intuitive inexactness humans so readily accept but which computers have difficulty accommodating. Three or four simple words illustrate the essence of the two philosophies:

      Let's begin our study by looking at some of the early works. This work is rooted mainly in the studies, writings, and teachings of early Greek philosophers including Socrates, Plato, Aristotle, Parmenides, and Heraclitus. Philosophy in the early days often included mathematics and related reasoning.

      1.2.1 The Early Foundation

      Socrates was one of the early classic Greek philosophers and is often regarded as the founding father of Western philosophy. He is particularly noted for his creation of the Socratic Method in which the teacher repeatedly poses clarifying questions until the student grasps and understands the concept(s) being taught.

      Plato, a student of Socrates, formed the first institution of higher learning in the Western world. Reflecting the Socratic Method, he wrote dialogues in which the participants discuss, analyze, and dissect a topic from various perspectives. He is also considered the developer of the concept of forms in which an ideal world of forms exists in contrast to a false world of phenomena.

      Aristotle, mentored by Plato, brought symbolic logic and the notion of scientific thinking to Western philosophy. In doing so, he contradicted Plato's ideal world of forms with a more pragmatic view and contributed advances in the branch of philosophy known as metaphysics. Among Aristotle's fundamental assertions was that it is impossible to both be something and not be the same thing at the same time. Today, that assertion falls apart in the field of fuzzy logic.

Parmenides was a pre‐Socratic philosopher known as the Philosopher of Changeless Being. He believed that there are two views of reality. One is the way of truth or fact and the other is that change is impossible and existence is unchanging. Over the course of his career, he also had a significant influence on a young Plato.

      Heraclitus views contradicted those of Parmenides with his insistence on ever‐present change as fundamental to the universe. Such a view was reflected in his saying: “No man ever steps in the same river twice.” His beliefs continued to identify him as one of the founders of the branch of metaphysics referred to as ontology, which deals with the nature of being.

      From these early philosophers and their different views on reality, learning, and reasoning, we have the following classic laws of thought:

      1.2.1.1 Three Laws of Thought

      The laws of thought are stipulated to be the rules by which rational discourse can be considered to be based. Thus, they are rules that apply, without exception, to any subject matter of thought. The three laws are as follows: