Special Issue "New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 July 2019

Special Issue Editor

Guest Editor
Prof. Dr. Florentin Smarandache

Department of Mathematics and Sciences, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
Website | E-Mail
Interests: neutrosophic triplet structures, neutrosophic duplet structures, artificial intelligence, number theory, information fusion, statistics, algebraic structures

Special Issue Information

Dear Colleagues,

The fields of neutrosophic set, logic, measure, probability, and statistics have been developed and explored very much in the last few years due to their multiple practical applications.

The neutrosophic components truth (T), indeterminacy (I), and falsehood (F) are symmetric in form, since T is symmetric to its opposite F with respect to I, which acts as an axis of symmetry:  T----------I----------F

This Special Issue invites state-of-the-art papers on new topics related of neutrosophic theories and applications, such as

- Studies of corner cases of neutrosophic sets/probabilities/statistics/logics, such as

  • neutrosophic intuitionistic sets (which are different from intuitionistic fuzzy sets), neutrosophic paraconsistent sets, neutrosophic faillibilist sets, neutrosophic paradoxist sets, neutrosophic, pseudo-paradoxist sets, neutrosophic tautological sets, neutrosophic nihilist sets, neutrosophic dialetheist sets, and neutrosophic trivialist sets;
  • neutrosophic intuitionistic probability and statistics, neutrosophic paraconsistent probability and statistics, neutrosophic faillibilist probability and statistics, neutrosophic paradoxist probability and statistics, neutrosophic pseudo-paradoxist probability and statistics, neutrosophic tautological probability and statistics, neutrosophic nihilist probability and statistics, neutrosophic dialetheist probability and statistics, and neutrosophic trivialist probability and statistics;
  • neutrosophic paradoxist logic (or paradoxism), neutrosophic pseudo-paradoxist logic (or neutrosophic pseudo-paradoxism), and neutrosophic tautological logic (or neutrosophic tautologism):



  • Refined Neutrosophic Set components (T, I, and F) of many neutrosophic sub-components:

(T1, T2, ...; I1, I2, ...; F1, F2, ...);



  • Degrees of Dependence and Independence between Neutrosophic Components

T, I, and F are independent components, leaving room for incomplete information (when their superior sum < 1), paraconsistent and contradictory information (when the superior sum > 1), or complete information (sum of components = 1). 

For software engineering proposals, the classical unit interval [0, 1] is used.

For single-valued neutrosophic logic, the sum of the components is:

0 ≤ t+i+f ≤ 3 when all three components are independent;

0 ≤ t+i+f ≤ 2 when two components are dependent, while the third one is independent from them;

0 ≤ t+i+f ≤ 1 when all three components are dependent.

When three or two of the components T, I, and F are independent; one leaves room for incomplete information (sum < 1), paraconsistent and contradictory information (sum > 1), or complete information (sum = 1). 

If all three components, T, I, and F are dependent; then, similarly one leaves room for incomplete information (sum < 1) or complete information (sum = 1). 

In general, the sum of two components x and y that vary in the unitary interval [0, 1] is

0 ≤ x + y ≤ 2 - d°(x, y), in which d°(x, y) is the degree of dependence between x and y, while

d°(x, y) is the degree of independence between x and y.



- Neutrosophic Overset (when some neutrosophic component is > 1), since he observed that, for example, an employee working overtime deserves a degree of membership > 1, with respect to an employee that only works regular full-time and whose degree of membership = 1;

and to Neutrosophic Underset (when some neutrosophic component is < 0), since, for example, an employee causing more damage than benefit to his company deserves a degree of membership < 0, with respect to an employee that produces benefit to the company and has the degree of membership > 0;

and to Neutrosophic Offset (when some neutrosophic components are off the interval [0, 1], i.e., some neutrosophic component > 1 and some neutrosophic component < 0).

Then, similarly, neutrosophic logic/measure/probability/statistics, etc., were extended to, respectively, Neutrosophic Over-/Under-/Off- Logic, Measure, Probability, and Statistics, etc.





  • Neutrosophic Tripolar Set and Neutrosophic Multipolar Set

and, consequently, the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph

http://fs.unm.edu/eBook-Neutrosophics6.pdf (p. 93)


- N-norm and N-conorm



- Neutrosophic Measure and Neutrosophic Probability 
          (chance that an event occurs, indeterminate chance of occurrence,
           chance that the event does not occur)



  • Law of Included Multiple-Middle (as middle part of Neutrosophy)

(<A>;  <neut1A>, <neut2A>, …;  <antiA>)


- Neutrosophic Statistics (indeterminacy is introduced into classical statistics with respect to the sample/population, or with respect to the individuals that only partially belong to a sample/population, or with respect to the neutrosophic probability distributions)



- Neutrosophic Precalculus and Neutrosophic Calculus



Refined Neutrosophic Numbers (a+ b1I1 + b2I2 + … + bnIn), in which I1, I2, …, In are subindeterminacies of indeterminacy I;

(t,i,f)-neutrosophic graphs;

Thesis-Antithesis-Neutrothesis, and Neutrosynthesis;

Neutrosophic Axiomatic System;

Neutrosophic Dynamic Systems;

Symbolic Neutrosophic Logic;

(t, i, f)-Neutrosophic Structures;

I-Neutrosophic Structures;

Refined Literal Indeterminacy;

Quadruple Neutrosophic Algebraic Structures;

Multiplication Law of Subindeterminacies:



- Theory of Neutrosophic Evolution: Degrees of Evolution, Indeterminacy or Neutrality, and Involution


- Plithogeny as Generalization of Dialectics and Neutrosophy;

Plithogenic Set/Logic/Probability/Statistics (as generalization of fuzzy, intuitionistic fuzzy, neutrosophic set/logic/probability/statistics)



- Neutrosophic Psychology (neutropsyche, refined neutrosophic memory: conscious, aconscious, unconscious, neutropsychic personality, Eros/Aoristos/Thanatos, neutropsychic crisp personality)


- Neutrosophic Applications in:

artificial intelligence, information systems, computer science, cybernetics, theory methods, mathematical algebraic structures, applied mathematics, automation, control systems, big data, engineering, electrical, electronic, philosophy, social science, psychology, biology, engineering, operational research, management science, imaging science, photographic technology, instruments, instrumentation, physics, optics, economics, mechanics, neurosciences, radiology nuclear, interdisciplinary applications, multidisciplinary sciences, etc.

[See: Xindong Peng and Jingguo Dai, A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017, Artificial Intelligence Review, Springer, 18 August 2018;


Prof. Dr. Florentin Smarandache
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers

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