Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off- Logic, Probability, and Statistics
This work provides a theoretical extension for handling uncertain or imprecise data in fields like logic and statistics, but it appears incremental as it builds on existing neutrosophic concepts.
The paper tackles the limitation of traditional fuzzy and probability frameworks by introducing neutrosophic overset, underset, and offset, which allow values outside the [0,1] interval to model real-world scenarios like overtime work or negative contributions.
Neutrosophic Over-/Under-/Off-Set and -Logic were defined by the author in 1995 and published for the first time in 2007. We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is over 1}, Neutrosophic Underset {when some neutrosophic component is below 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component over 1 and other neutrosophic component below 0}. This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components. For example, person working overtime deserves a membership degree over 1, while a person producing more damage than benefit to a company deserves a membership below 0. Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were extended to respectively Neutrosophic Over-/Under-/Off-Logic, -Measure, -Probability, -Statistics etc. [Smarandache, 2007].