Abstract
During a presidential forum in the 2008 U.S. presidential campaign, the moderator, Pastor Rick Warren, wanted Sen. John McCain and then-Sen. Barack Obama to define rich with a specific number. Warren wanted to know at what specific income level a person goes from being not rich to rich. The problem with this question is that there is no specific income at which a person makes the leap from being not rich to being rich. This is because rich is a fuzzy set, not a crisp set, with different incomes having different degrees of membership in the rich fuzzy set. Similarly, middle class and poor are fuzzy sets. Fuzzy logic is needed to properly ask and answer Warren's question about quantitatively defining rich. Similarly, fuzzy logic is needed to properly ask and answer queries about quantitatively defining imprecise linguistic terms in politics and public policy like middle class, poor, low inflation, medium inflation, and high inflation. Type-one or interval type-two fuzzy logic can be used for quantitatively defining imprecise linguistic terms. This chapter shows how to use type-one fuzzy logic and interval type-two fuzzy logic for this purpose, as well as the advantages and disadvantages of each. Imprecise terms in natural languages should be considered to have qualitative definitions, quantitative definitions, crisp quantitative definitions, fuzzy quantitative definitions, type-one fuzzy quantitative definitions, and interval type-two fuzzy quantitative definitions.
Top2. Types Of Imprecision And Uncertainty
There are various types of uncertainty and imprecision. However, they can be classified under two broad categories: type one uncertainty and type two uncertainty (Gupta, 1988a; Gupta, 1988b; Gupta, 1991; Gupta, 1992; Solo & Gupta, 2007; Gupta & Solo, 2010; Gupta & Solo, 2015).
Key Terms in this Chapter
Fuzzy Logic: Fuzzy logic is a field created by Lotfi A. Zadeh for information arising from computational perception and cognition that is uncertain, imprecise, vague, partially true, or without sharp boundaries.
Type-Two Fuzzy Set: A type-two fuzzy set allows the inclusion of uncertainty in the parameters of a membership function. The membership function of a type-two fuzzy set is in itself a fuzzy set. A type-two fuzzy set is three-dimensional where the third dimension indicates the degree of membership of the two-dimensional membership function at each point in its two-dimensional domain. A type-two fuzzy set has a footprint of uncertainty with an upper membership function and lower membership function.
Fuzzy Quantitative Definition: Fuzzy quantitative definition is a term coined by Ashu M. G. Solo to refer to a quantitative definition of an imprecise word using a fuzzy set instead of a quantitative definition of an imprecise word using a crisp set ( crisp quantitative definition ) or linguistic definition without numerical parameters ( qualitative definition ).
Interval Type-Two Fuzzy Quantitative Definition: Interval type-two fuzzy quantitative definition is a term coined by Ashu M. G. Solo to refer to a fuzzy quantitative definition of an imprecise word using an interval type-two fuzzy set instead of a quantitative definition of an imprecise word using a type-one fuzzy set ( type-one fuzzy quantitative definition ), quantitative definition of an imprecise word using a crisp set ( crisp quantitative definition ), or linguistic definition without numerical parameters ( qualitative definition ).
Fuzzy Quantitative Definition: Type-one fuzzy quantitative definition is a term coined by Ashu M. G. Solo to refer to a fuzzy quantitative definition of an imprecise word using a type-one fuzzy set instead of a quantitative definition of an imprecise word using an interval type-two fuzzy set ( interval type-two fuzzy definition ), quantitative definition of an imprecise word using a crisp set ( crisp quantitative definition ), or linguistic definition without numerical parameters ( qualitative definition ).
Type-One Fuzzy Set: In a type-one fuzzy set, elements can have degrees of membership.
Crisp Quantitative Definition: Crisp quantitative definition is a term coined by Ashu M. G. Solo to refer to a quantitative definition of an imprecise word using a crisp set instead of a quantitative definition of an imprecise word using a fuzzy set ( fuzzy quantitative definition ) or linguistic definition without numerical parameters ( qualitative definition ).
Footprint of Uncertainty: A footprint of uncertainty indicates the upper and lower bounds in the two-dimensional domain of a type-two fuzzy set. A footprint of uncertainty in a type-two fuzzy set is a region bounded by an upper membership function and lower membership function.
Fuzzy Sets: In a fuzzy set , elements can have degrees of membership. The concept of a fuzzy set was developed by Lotfi A. Zadeh.
Quantitative Definition: Quantitative definition is a term coined by Ashu M. G. Solo to refer to a definition of an imprecise word using a crisp set or fuzzy set instead of a linguistic definition of an imprecise word without numerical parameters ( qualitative definition ).
John McCain: John McCain was a U.S. senator and the Republican presidential nominee in 2008.
Type Two Uncertainty: Type two uncertainty is a term coined by Madan M. Gupta for information or phenomena that arise from human perception and cognitive processes or from cognitive information in general. This subject has received relatively little attention. Perception and cognition through biological sensors (eyes, ears, nose, etc.), perception of pain, and other similar biological events throughout our nervous system and neural networks deserve special attention. The perception and cognition phenomena associated with these processes are characterized by many great uncertainties and cannot be described by conventional statistical theory. A person can linguistically express perceptions experienced through the senses, but these perceptions cannot be described using conventional statistical theory. Fuzzy logic has proven to be a very promising tool for dealing with type two uncertainty.
Barack Obama: Barack Obama was the president of the United States and the Democratic presidential nominee in 2008.
Qualitative Definition: Qualitative definition is a term coined by Ashu M. G. Solo to refer to a linguistic definition of an imprecise word without numerical parameters, such as is found in dictionaries, instead of a definition of an imprecise word using a crisp set or fuzzy set ( quantitative definition ).
Interval Type-Two Fuzzy Set: An interval type-two fuzzy set is a type-two fuzzy set in which the third dimension is constant meaning the degree of membership is constant for the two-dimensional membership function at each point in its two-dimensional domain. Therefore, the third dimension is ignored.
Type One Uncertainty: Type one uncertainty is a term coined by Madan M. Gupta for information that arises from the random behavior of physical systems. The pervasiveness of this type of uncertainty can be witnessed in random vibrations of a machine, random fluctuations of electrons in a magnetic field, diffusion of gases in a thermal field, random electrical activities of cardiac muscles, uncertain fluctuations in the weather pattern, and turbulent blood flow through a damaged cardiac valve. Type one uncertainty has been studied for centuries. Complex statistical mathematics has evolved for the characterization and analysis of such random phenomena. Stochastic theory is effective in dealing with type one uncertainty.