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).