Definitions Derived from Neutrosophics

by Florentin Smarandache

Abstract:  Twenty-seven new definitions are presented, derived from

neutrosophic set, neutrosophic probability, and neutrosophic statistics.

Each one is independent, short, with references and cross references

like in a dictionary style.

Keywords:  Fuzzy set, fuzzy logic; neutrosophic logic;

Neutrosophic set, intuitionistic set, paraconsistent set, faillibilist

set, paradoxist set, tautological set, nihilist set, dialetheist set,

trivialist set;

Classical probability and statistics, imprecise probability;

Neutrosophic probability and statistics, intuitionistic probability

and statistics, paraconsistent probability and statistics, faillibilist

probability and statistics, paradoxist probability and statistics,

tautological probability and statistics, nihilist probability and

statistics, dialetheist probability and statistics, trivialist

probability and statistics.

2000 MSC:  03E99, 03-99, 03B99, 60A99, 62A01, 62-99.

Introduction:

As an addenda to , , [5-7] we display the below unusual

extension of definitions resulted from neutrosophics in the Set

Theory and Probability.  Some of  them are listed in the Dictionary

of Computing .  Further development of these definitions

(including properties, applications, etc.) is in our research plan.

1. Definitions of New Sets

====================================================

1.1. Neutrosophic Set:

<logic, mathematics> A set which generalizes many existing classes

of sets, especially the fuzzy set.

Let U be a universe of discourse, and M a set included in U.

An element x from U is noted, with respect to the set M, as x(T,I,F),

and belongs to M in the following way: it is T% in the set

(membership appurtenance), I% indeterminate (unknown if it is in the

set), and F% not in the set (non-membership);

here T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

Therefore: -0 # inf(T) + inf(I) + inf(F) # sup(T) + sup(I) + sup(F) # 3+.

Generalization of {classical set}, {fuzzy set}, {intuitionistic set},

{paraconsistent set}, {faillibilist set}, {paradoxist set},

{tautological set}, {nihilist set}, {dialetheist set}, {trivialist}.

Related to {neutrosophic logic}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

1.2. Intuitionistic Set:

<logic, mathematics> A set which provides incomplete

information on its elements.

A class of {neutrosophic set} in which every element x is

incompletely known, i.e. x(T,I,F) such that

sup(T)+sup(I)+sup(F)<1;

here T,I,F are real standard or non-standard subsets, included in

the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

Contrast with {paraconsistent set}.

Related to {intuitionistic logic}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

1.3. Paraconsistent Set:

<logic, mathematics> A set which provides paraconsistent information

on its elements.

A class of {neutrosophic set} in which every element x(T,I,F) has the

property that sup(T)+sup(I)+sup(F)>1;

here T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth, indeterminacy,

and falsity percentages respectively.

Contrast with {intuitionistic set}.

Related to {paraconsistent logic}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

1.4. Faillibilist Set:

<logic, mathematics> A set whose elements are uncertain.

A class of {neutrosophic set} in which every element x has a

percentage of indeterminacy, i.e. x(T,I,F) such that inf(I)>0;

here T,I,F are real standard or non-standard subsets, included

in the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

Related to {faillibilism}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

<logic, mathematics> A set which contains and doesn't contain

itself at the same time.

A class of {neutrosophic set} in which every element x(T,I,F) has

the form x(1,I,1), i.e. belongs 100% to the set and doesn't

belong 100% to the set simultaneously;

here T,I,F are real standard or non-standard subsets, included in

the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

1.6. Tautological Set:

<logic, mathematics> A set whose elements are absolutely

determined in all possible worlds.

A class of {neutrosophic set} in which every element x has the

form x(1+,-0,-0), i.e. absolutely belongs to the set;

here T,I,F are real standard or non-standard subsets, included

in the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

Contrast with {nihilist set} and {nihilism}.

Related to {tautologism}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

1.7. Nihilist Set:

<logic, mathematics> A set whose elements absolutely

don’t belong to the set in all possible worlds.

A class of {neutrosophic set} in which every element x has the

form x(-0,-0,1+), i.e. absolutely doesn’t belongs to the set;

here T,I,F are real standard or non-standard subsets, included

in the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

The empty set is a particular set of {nihilist set}.

Contrast with {tautological set}.

Related to {nihilism}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

(http://fs.unm.edu/FirstNeutConf.htm,

====================================================

1.8. Dialetheist Set:

<logic, mathematics> /di:-al-u-theist/ A set which contains at

least one element which also belongs to its complement.

A class of {neutrosophic set} which models a situation

where the intersection of some disjoint sets is not empty.

There is at least one element x(T,I,F) of the dialetheist set

M which belongs at the same time to M and to the set C(M),

which is the complement of M;

here T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and  falsity percentages respectively.

Contrast with {trivialist set}.

Related to {dialetheism}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

1.9. Trivialist Set:

<logic, mathematics>  A set all of whose  elements also belong

to its complement.

A class of {neutrosophic set} which models a situation

where the intersection of any disjoint sets is not empty.

Every element x(T,I,F) of the trivialist set M belongs at the

same time to M and to the set C(M), which is the

complement of M;

here T,I,F are real standard or non-standard subsets,

included in the non-standard unit interval ]-0, 1+[, representing

truth, indeterminacy, and falsity percentages respectively.

Contrast with {dialetheist set}.

Related to {trivialism}.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

# 2. Definitions of New Probabilities and Statistics

====================================================

2.1. Neutrosophic Probability:

<probability> The probability that an event occurs is (T, I, F),

where T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively.

Therefore: -0 # inf(T) + inf(I) + inf(F) # sup(T) + sup(I) + sup(F) # 3+.

Generalization of {classical probability} and {imprecise probability},

{intuitionistic probability}, {paraconsistent probability}, {faillibilist

{nihilistic probability}, {dialetheist probability}, {trivialist probability}.

Related with {neutrosophic set} and {neutrosophic logic}.

The analysis of neutrosophic events is called Neutrosophic Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.2. Intuitionistic Probability:

<probability> The probability that an event occurs is (T, I, F),

where T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively,

and n_sup = sup(T)+sup(I)+sup(F) < 1,

i.e. the probability is incompletely calculated.

Contrast with {paraconsistent probability}.

Related to {intuitionistic set} and {intuitionistic logic}.

The analysis of intuitionistic events is called Intuitionistic Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.3. Paraconsistent Probability:

<probability> The probability that an event occurs is (T, I, F),

where T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively,

and n_sup = sup(T)+sup(I)+sup(F) > 1,

i.e. contradictory information from various sources.

Contrast with {intuitionistic probability}.

Related to {paraconsistent set} and {paraconsistent logic}.

The analysis of paraconsistent events is called

Paraconsistent Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.4. Faillibilist Probability:

<probability> The probability that an event occurs is (T, I, F),

where T,I,F are real standard or non-standard subsets, included in the

non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and falsity percentages respectively,

and inf(I) > 0,

i.e. there is some percentage of indeterminacy in calculation.

Related to {faillibilist set} and {faillibilism}.

The analysis of faillibilist events is called Faillibilist Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

<probability> The probability that an event occurs is (1, I, 1),

where I is a standard or non-standard subset, included in the

non-standard unit interval ]-0, 1+[, representing indeterminacy.

may occur and may not occur simultaneously).

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.6. Tautological Probability:

<probability> The probability that an event occurs is more than one,

i.e. (1+, -0, -0).

Tautological probability is used for universally sure events (in all

possible worlds, i.e. do not depend on time, space, subjectivity, etc.).

Contrast with {nihilistic probability} and {nihilism}.

Related to {tautological set} and {tautologism}.

The analysis of tautological events is called Tautological Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.7. Nihilist Probability:

<probability> The probability that an event occurs is less than zero,

i.e. (-0, -0, 1+).

Nihilist probability is used for universally impossible events (in all

possible worlds, i.e. do not depend on time, space, subjectivity, etc.).

Contrast with {tautological probability} and {tautologism}.

Related to {nihilist set} and {nihilism}.

The analysis of nihilist events is called Nihilist Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.8. Dialetheist Probability:

<probability> /di:-al-u-theist/ A probability space where at least

one event and its complement are not disjoint.

A class of {neutrosophic probability} which models a situation

where the intersection of some disjoint events is not empty.

Here, similarly, the probability of an event to occur is (T, I, F),

where T,I,F are real standard or non-standard subsets, included

in the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and  falsity percentages respectively.

Contrast with {trivialist probability}.

Related to {dialetheist set} and {dialetheism}.

The analysis of dialetheist events is called Dialetheist Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

2.9. Trivialist Probability:

<probability> A probability space where every event and its

complement are not disjoint.

A class of {neutrosophic probability}which models a situation

where the intersection of any disjoint events is not empty.

Here, similarly, the probability of an event to occur is (T, I, F),

where T,I,F are real standard or non-standard subsets, included

in the non-standard unit interval ]-0, 1+[, representing truth,

indeterminacy, and  falsity percentages respectively.

Contrast with {dialetheist probability}.

Related to {trivialist set} and {trivialism}.

The analysis of trivialist events is called Trivialist Statistics.

{ ref. Florentin Smarandache, "A Unifying Field in Logics.

Neutrosophy: Neutrosophic Probability, Set, and Logic",

American Research Press, Rehoboth, 1999;

====================================================

General References:

1.  Jean Dezert, Open Questions on Neutrosophic Inference, Multiple-Valued Logic Journal, 2001 (to appear).

2.  Denis Howe, On-Line Dictionary of Computing,

3.  Charles Le, Preamble to Neutrosophy and Neutrosophic Logic, Multiple-Valued Logic Journal, 2001 (to appear).

4.  Florentin Smarandache, organizer, First International Conference on Neutrosophy, Neutrosophic Probability, Set, and Logic, University of New Mexico, 1-3 December 2001.

5.  Florentin Smarandache, Neutrosophy, a New Branch of Philosophy, Multiple-Valued Logic Journal, 2001 (to appear).

6.  Florentin Smarandache, Neutrosophic Set, Probability and Statistics, Multiple-Valued Logic Journal, 2001 (to appear).

7.  Florentin Smarandache, A Unifying Field in Logics, Neutrosophic Logic, Multiple-Valued Logic Journal, 2001 (to appear).