neutrosophic logic

neutrosophic logic

A logic, in which each

proposition

is estimated to have the degree of truth in

T

, the degree of indeterminacy (neither true nor false) in

I

, and the degree of false in

F

, is called

neutrosophic logic

, where

T

,

I

,

F

are standard or

non-standard real

subsets of the non-standard unit interval

]

-

0

,

1

+

[

.

T

,

I

,

F

are called

neutrosophic components

.

Now let’s explain the previous notations:

A number

ε

is said to be

infinitesimal

if and only if for all positive integers

n

one has

|

ε

|

<

1

n

. Let

ε

>

0

be a such infinitesimal number. The

hyper-real number set

is an

extension

of the

real number

set, which includes classes of

infinite

numbers and classes of infinitesimal numbers.

Generally, for any real number

a

one defines

a

-

which signifies a

monad

, i.e. a set of hyper-real numbers in

non-standard analysis

, as follows:

a

-

=

{

a

-

ε

:

ε

∈

R

*

,

ε

is infinitesimal

}

,

and similarly one defines

a

+

, which is also a monad, as:

a

+

=

{

a

+

ε

:

ε

∈

R

*

,

ε

is infinitesimal

}

.

A

binad

a

+

-

is a union of the above two monads, i.e.

a

+

-

=

-

a

∪

a

+

.

For example: The non-standard finite number

1

+

=

1

+

ε

, where

1

is its

standard part

and

ε

its

non-standard part

, and similarly the non-standard finite number

0

-

=

0

-

ε

, where

0

is its standard part and

ε

its

non-standard part

.

Similarly for

3

+

=

3

+

ε

, etc.

Note that

]

-

0

,

1

+

[

is called the

non-standard unit interval

.

http://www.gallup.unm.edu/

smarandache

/

Introduction

.pdf

More information on non-standard

intervals

is available.

The superior sum of the neutrosophic components is defined as:

n

s

⁢

u

⁢

p

=

s

u

p

(

T

)

+

s

u

p

(

I

)

+

s

u

p

(

F

)

∈

]

-

0

,

3

+

[

may be as high as 3 or

3

+

.

While the inferior sum of the neutrosophic components is defined as:

n

i

⁢

n

⁢

f

=

i

n

f

(

T

)

+

i

n

f

(

I

)

+

i

n

f

(

F

)

∈

]

-

0

,

3

+

[

may be as low as 0 or

0

-

.

Neutrosophic logic was introduced by Florentin Smarandache in 1995 as a

generalization

of

fuzzy logic

(especially of intuitionistic fuzzy logic) when

n

s

⁢

u

⁢

p

=

1

, of

intuitionistic logic

when

n

s

⁢

u

⁢

p

<

1

, and of

paraconsistent logic

when

n

s

⁢

u

⁢

p

>

1

.

The main distinctions between the neutrosophic logic (NL) and intuitionistic fuzzy logic (IFL) are the facts that (a) the sum of neutrosophic components (or of their superior limits when they are subsets) in NL is not necessarily 1 as in IFL but any number from

0

-

to

3

+

in order to allow the

characterization

of incomplete or paraconsistent information, and (b) in NL one uses a non-standard interval

]

-

0

,

1

+

[

in order to make a

difference

in philosophy between

absolute truth

, denoted by

1

+

, and

relative truth

, denoted by

1

, and similarly distinctions between

absolute falsehood

and

relative falsehood

or between

absolute indeterminacy

and

relative indeterminacy

respectively, while in IFL one has a standard interval

[

0

,

1

]

.

Examples

:

- One uses a subset of truth (or indeterminacy, or falsity), instead of a number, because in many cases we are not able to exactly determine the degrees of truth and of false but to approximate them.

- In technical applications, where there is no need for distinctions between absolute truth and relative truth, we can use standard subsets instead of non-standard subsets and respectively the unit interval

[

0

,

1

]

instead of the non-standard unit interval

]

-

0

,

1

+

[

.

- Let say the proposition

One month from today it will be raining

can be between 0.30-0.40 or 0.45-0.50 true (according to various analyzers), 0.10 or 0.20

indeterminate

(neither true nor false, but unknown, due to possible hidden

parameters

that might influence the raining), and 0.60 or between 0.66-0.70 false.

- The subsets are not necessary intervals, but any sets (

discrete

,

continuous

, open or closed or half-open/half-closed interval,

intersections

or unions of the previous sets, etc.) in accordance with the given proposition.

- A subset may also have only one

element

, which is the easiest particular case of this logic.

References

1

Florentin Smarandache,

A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy,

Neutrosophic Set

, Neutrosophic Probability and Statistics

, third edition, Xiquan, Phoenix, 2003.

http://www.gallup.unm.edu/ smarandache/eBook-Neutrosophics2.pdf

Also online.

2

F. Smarandache, J. Dezert, A. Buller, M. Khoshnevisan, S. Bhattacharya, S. Singh, F. Liu, Gh. C. Dinulescu-Campina, C. Lucas, C. Gershenson,

Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics

, The University of New Mexico, Gallup Campus, 1-3 December 2001.

\htmladdnormallink

Also online.http://arxiv.org/pdf/math.GM/0306384

Title

neutrosophic logic

Canonical

name

NeutrosophicLogic

Date of creation

2013-03-22 14:21:45

Last modified on

2013-03-22 14:21:45

Owner

para0doxa (5174)

Last modified by

para0doxa (5174)

Numerical id

18

Author

para0doxa (5174)

Entry type

Definition

Classification

msc 03B50

Classification

msc 03B53

Classification

msc 03B60

Related topic

FlorentinSmarandache

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