** **

** 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 [1], [3], [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 [2]. 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;

(http://www.gallup.unm.edu/~smarandache/NeutrosophicSet.pdf,

http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**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://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.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://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.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://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.htm__)
}

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

**1.5.
Paradoxist Set**:

<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.

Related to {paradoxism}.

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

Neutrosophy: Neutrosophic Probability, Set, and
Logic",

American Research Press, Rehoboth, 1999;

(__http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.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://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.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://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,__

__ http://www.gallup.unm.edu/~smarandache/neut-ad.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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

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

**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

probability}, {paradoxist probability}, {tautological
probability},

{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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**2.5.
Paradoxist Probability**:

<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.

Paradoxist probability is used for paradoxal events (i.e.
which

may occur and may not occur simultaneously).

Related to {paradoxist set} and {paradoxism}.

The analysis of paradoxist events is called **Paradoxist Statistics**.

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

Neutrosophy:
Neutrosophic Probability, Set, and Logic",

American Research
Press, Rehoboth, 1999;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

**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;

(http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm,

http://www.gallup.unm.edu/~smarandache/neut-ad.htm) }

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

** **

** **

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*,

http://foldoc.doc.ic.ac.uk/foldoc/

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).