[Paper Review] Plithogeny, Plithogenic Set, Logic, Probability, and Statistics

Florentín Smarandache|arXiv (Cornell University)|Aug 12, 2018

Advanced Mathematical Theories1 references41 citations

TL;DR

The paper introduces plithogenic sets, logic, probability, and statistics as generalizations of classical and neutrosophic frameworks, with definitions, aggregation operators, theorems, and applications.

ABSTRACT

In this book we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), plithogenic logic (as generalization of classical, fuzzy, intuitionistic fuzzy, and neutrosophic logics), plithogenic probability (as generalization of classical, imprecise, and neutrosophic probabilities), and plithogenic statistics (as generalization of classical, and neutrosophic statistics). Plithogenic Set is a set whose elements are characterized by one or more attributes, and each attribute may have many values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, logic, probability and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion, equality are influenced by the attribute values contradiction (dissimilarity) degrees. Formal definitions of plithogenic set, logic, probability, statistics are presented into the book, followed by plithogenic aggregation operators, various theorems related to them, and afterwards examples and applications of these new concepts in our everyday life.

Motivation & Objective

• Present a generalized framework called plithogenic set for elements characterized by multiple attributes with multiple values.
• Define plithogenic logic, probability, and statistics as extensions of classical and neutrosophic concepts.
• Introduce aggregation operators, including linear combinations of t-norms/t-conorms, governed by attribute-value contradiction degrees.
• Provide formal definitions, theorems, and examples/applications of plithogenic concepts in everyday life.

Proposed method

• Define plithogenic set with elements characterized by attributes and values with corresponding degrees of appurtenance d(x,v).
• Introduce a dominant attribute value and a dissimilarity/contradiction degree between attribute values and the dominant value.
• Formulate plithogenic intersection/union as linear combinations of fuzzy t-norms and t-conorms.
• Incorporate attribute-value contradiction into definitions of plithogenic complement, inclusion, and equality.
• Present formal definitions, aggregation operators, and theorems for plithogenic set, logic, probability, and statistics.
• Provide examples and applications illustrating these concepts.

Experimental results

Research questions

• RQ1 How can plithogenic sets generalize crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets?
• RQ2 What are the formal definitions and properties of plithogenic logic, probability, and statistics?
• RQ3 How do contrariety (dissimilarity) degrees between attribute values and the dominant value affect aggregation operators?
• RQ4 What are the key aggregation operators and theorems for plithogenic systems?
• RQ5 How can plithogenic concepts be applied to real-world problems?

Key findings

• Plithogenic sets generalize multiple classical frameworks by incorporating attribute-value degrees of appurtenance and a dominant value with contradiction degrees.
• Plithogenic intersection and union are constructed as linear combinations of t-norms and t-conorms.
• Complement, inclusion, and equality are influenced by the contradiction degrees between attribute values and the dominant value.
• The framework provides formal definitions, aggregation operators, theorems, and example applications across logic, probability, and statistics.