# Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis

## Abstract

**:**

_{N}and sup

_{N}). Many theorems, new terms introduced, better notations for monads and binads, and examples of nonstandard neutrosophic operations are given.

## 1. Short Introduction

## 2. Theoretical Reason for the Nonstandard Form of Neutrosophic Logic

^{+}), or infinitesimally smaller than 0.8 (or

^{−}0.8). But these can easily be overcome by roughly using interval neutrosophic values, for example (0.80, 0.81) and (0.79, 0.80), respectively.

## 3. Why the Sum of Neutrosophic Components Is Up to 3

## 4. Neutrosophic Components outside the Unit Interval [0, 1]

^{−}0 <

_{N}0 and 1

^{+}>

_{N}1

## 5. Refined Neutrosophic Logic, Set, and Probability

_{1}, T

_{2}, …; I

_{1}, I

_{2}, …; F

_{1}, F

_{2}, …, which were deduced from our everyday life [16].

## 6. From Paradoxism Movement to Neutrosophy Branch of Philosophy and then to Neutrosophic Logic

^{+}(a tinny bigger than the Relative Truth’s value): 1

^{+}>

_{N}1, where >

_{N}is a neutrosophic inequality, meaning 1

^{+}is neutrosophically bigger than 1.

## 7. Introduction to Nonstandard Analysis

## 8. First Extension of Nonstandard Analysis

_{+}* the set of positive nonzero hyperreal numbers.

**Left Monad**{that we denote, for simplicity, by (

^{−}a) or only

^{−}a} is defined as:

**Right Monad**{that we denote, for simplicity, by (a

^{+}) or only by a

^{+}} is defined as:

**Pierced Binad**{that we denote, for simplicity, by (

^{−}a

^{+}) or only

^{−}a

^{+}} is defined as:

## 9. Second Extension of Nonstandard Analysis

**Left Monad Closed to the Right**

**Right Monad Closed to the Left**

**Unpierced Binad**

## 10. Nonstandard Neutrosophic Function

_{2}≥ 0, and writing

_{MB}.

_{N}((

^{−}a)) =

_{N}(a − ε, a),

_{N}((a

^{+})) =

_{N}(a, a + ε),

_{N}((

^{−}a

^{+})) =

_{N}(a − ε, a) ∪ (a, a + ε),

## 11. General Notations for Monads and Binads

^{−},

^{−0},

^{+},

^{+0},

^{−}

^{+},

^{−}

^{0+}} = {

^{0},

^{−},

^{−0},

^{+},

^{+0},

^{−}

^{+},

^{−}

^{0+}};

^{0}, or nothing above), a left monad (

^{−}), a left monad closed to the right (

^{−0}), a right monad (

^{+}), a right monad closed to the left (

^{0+}), a pierced binad (

^{−}

^{+}), or a unpierced binad (

^{−0+}), respectively.

^{−}2, then the corresponding monads and binads are respectively represented as:

**Classical and Neutrosophic Notations**

+, −, ×,÷, ^, *

_{N}, ≤

_{N}, >

_{N}, ≥

_{N}, ∧

_{N}, ∨

_{N}, →

_{N}, ↔

_{N}, ∩

_{N}, ∪

_{N}, ⊂

_{N}, ⊃

_{N}, ⊆

_{N}, ⊇

_{N}, =

_{N}, ∈

_{N}+

_{N}, −

_{N}, ×

_{N}, ÷

_{N}, ^

_{N}, *

_{N}

## 12. Neutrosophic Strict Inequalities

_{N}β

_{N}β

## 13. Neutrosophic Equality

_{N}β

## 14. Neutrosophic (Nonstrict) Inequalities

_{N}and ≤

_{N}neutrosophic inequalities.

_{N}β

_{N}β.

## 15. Neutrosophically Ordered Set

_{N}) is called a neutrosophically ordered set if

## 16. Neutrosophic Infimum and Neutrosophic Supremum

_{N}) and the neutrosophic supremum (denoted as sup

_{N}).

_{N}) be a set that is neutrosophically partially ordered, and M a subset of S.

_{N}(M) is the neutrosophically greatest element in S that is neutrosophically less than or equal to all elements of M:

_{N}) be a set that is neutrosophically partially ordered and M a subset of S.

_{N}(M) is the neutrosophically smallest element in S that is neutrosophically greater than or equal to all elements of M.

## 17. Definition of Nonstandard Real MoBiNad Set

- $\mu {(}^{-}\mathbb{R})$ is the set of all real left monads,
- $\mu {(}^{-}{\mathbb{R}}^{0})$ is the set of all real left monads closed to the right,
- $\mu \left({\mathbb{R}}^{+}\right)$ is the set of all real right monads,
- $\mu {(}^{0}{\mathbb{R}}^{+})$ is the set of all real right monads closed to the left,
- $\mu {(}^{-}{\mathbb{R}}^{+})$ is the set of all real pierced binads,
- and $\mu {(}^{-}{\mathbb{R}}^{0\text{}+})$ is the set of all real unpierced binads.

_{MB}is closed under addition, subtraction, multiplication, division (except division by $\stackrel{m}{a}$, with a = 0 and $m\in \left\{\text{}{,}^{-}{,}^{-\text{}0}{,}^{+}{,}^{0+}{,}^{-\text{}+}{,}^{-\text{}0\text{}+}\right\}$), and power

## 18. Etymology of MoBiNad

**mo**nad +

**bi**nad, introduced now for the first time.

## 19. Definition of Nonstandard Complex MoBiNad Set

## 20. Definition of Nonstandard Neutrosophic Real MoBiNad Set

## 21. Definition of Nonstandard Neutrosophic Complex MoBiNad Set

## 22. Properties of the Nonstandard Neutrosophic Real Mobinad Set

**Theorem**

**1.**

**Proof.**

^{−}${a}^{+}$,

^{−}${a}^{+}$:

^{−}${a}^{0\text{}+}$:

**Consequence**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

_{MB}includes the set of real number R = (−$\infty ,\text{}+\infty $) which is clearly unbounded to the left and right-hand sides. □

**Theorem**

**4.**

**Proof.**

**Proof.**

**Proof.**

**Proof.**

_{MB}, inf

_{N}, sup

_{N}) is a lattice of second type (as algebraic structure).

## 23. Definition of General Nonstandard Real MoBiNad Interval

^{−}${a}^{+}$ (there is no relation of order between $a$ and

^{−}${a}^{+}$);

**Theorem**

**5.**

**Proof.**

_{MB}. □

**Theorem**

**6.**

**Proof.**

^{−}$a$, and the nonstandard neutrosophic modinad Identity Meet Element (Top) is ${b}^{+}$, or

## 24. Definition of Nonstandard Real MoBiNad Unit Interval

_{MB}means: all monads and binads included in ${]}^{-}0,\text{}{1}^{+}[$, for example,

^{−}0.2), (

^{−}0.3

^{0}), (0.5

^{+}), (

^{−}0.7

^{+}), (

^{−}0.8

^{0+}) etc.

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 25. Definition of Extended General Neutrosophic Logic

## 26. Definition of Standard Neutrosophic Logic

## 27. Definition of Extended Nonstandard Neutrosophic Logic

**Theorem**

**9.**

**Proof.**

## 28. Definition of Extended General Neutrosophic Set

## 29. Definition of Standard Neutrosophic Set

## 30. Definition of Extended Nonstandard Neutrosophic Set

## 31. Definition of Extended General Neutrosophic Probability

## 32. Definition of Standard Neutrosophic Probability

## 33. Definition of Extended Nonstandard Neutrosophic Probability

## 34. Classical Operations with Real Sets

## 35. Operations on the Nonstandard Real MoBiNad Set ($N{R}_{MB}$)

_{N}NR

_{MB}, α *

_{N}β =

_{N}μ

_{N}(α) ⊛ μ

_{N}(β)

_{N}is any neutrosophic arithmetic operations with neutrosophic numbers (+

_{N}, −

_{N}, ${\times}_{N},\text{}{\xf7}_{N}$, ^

_{N}), while the corresponding $\u229b$ is an arithmetic operation with real subsets.

**Nonstandard Division**

**Nonstandard Multiplication**

**Nonstandard Power**

## 36. Conditions of Neutrosophic Nonstandard Inequalities

_{MB}be the Nonstandard Real MoBiNad. Let’s endow (NR

_{MB}, <

_{N}) with a neutrosophic inequality.

_{MB}is a set of subsets, and thus we deal with neutrosophic inequalities between subsets.

- (i)
- If the subset α has many of its elements above all elements of the subset β,
- (ii)
- then α >
_{N}β (partially). - (iii)
- If the subset α has many of its elements below all elements of the subset β,
- (iv)
- then α <
_{N}β (partially). - (v)
- If the subset α has many of its elements equal with elements of the subset β,
- (vi)
- then α =
_{N}β (partially).

_{N}β.

_{N}β.

^{−}a

^{+}) there is no neutrosophic order, similarly between a and $\stackrel{-0+}{a}$.

## 37. Open Neutrosophic Research

## 38. Nonstandard Neutrosophic Inequalities

^{−}a) <

_{N}, a <

_{N}(a

^{+})

^{+}) >

_{N}, a >

_{N}(

^{−}a)

^{−}a) ≤

_{N}(

^{−}a

^{+}) ≤

_{N}(a

^{+})

^{−}a

^{+}) = (

^{−}a)$\cup $(a

^{+}) and, geometrically, on the Real Number Line, the number a is in between the subsets

^{−}a = (a − ε, a) and a

^{+}= (a, a + ε), so

^{−}a) ≤

_{N}(

^{−}a) ∪ (a

^{+}) ≤

_{N}(a

^{+})

^{+}) ≥

_{N}(

^{−}a) ∪ (a

^{+}) ≥

_{N}(

^{−}a)

^{−}a

^{+}) >

_{N}b, (

^{−}a

^{+}) >

_{N}(

^{−}b), (

^{−}a

^{+}) >

_{N}(b

^{+}), (

^{−}a

^{+}) >

_{N}(

^{−}b

^{+}), etc.

^{−}a

^{+}), or between the elements a and

_{MB}is a neutrosophically partially order set.

_{MB}, then (NR

_{MB}, ≤

_{N}) is neutrosophically totally ordered.

**Theorem**

**10.**

## 39. Nonstandard Neutrosophic Equalities

^{−}a) =

_{N}(

^{−}b), (a

^{+}) =

_{N}(b

^{+}), (

^{−}a

^{+}) =

_{N}(

^{−}b

^{+}),

## 40. Nonstandard Neutrosophic Belongingness

_{MB}, we say that

_{1}, m

_{2}, m ∈ {,

^{−},

^{−0},

^{+},

^{+0},

^{−+},

^{−0+}}.

## 41. Nonstandard Hesitant Sets

## 42. Nonstandard Neutrosophic Strict Interval Inclusion

_{MB},

## 43. Nonstandard Neutrosophic (Nonstrict) Interval Inclusion

_{MB},

## 44. Nonstandard Neutrosophic Strict Set Inclusion

## 45. Nonstandard Neutrosophic (Nonstrict) Set Inclusion

## 46. Nonstandard Neutrosophic Set Equality

## 47. The Fuzzy, Neutrosophic, and Plithogenic Logical Connectives ∧, ∨, →

_{F}, ∧

_{N}, ∧

_{P}representing respectively the fuzzy conjunction, neutrosophic

conjunction, and plithogenic conjunction;

_{F}, ∨

_{N}, ∨

_{P}representing respectively the fuzzy disjunction, neutrosophic

disjunction, and plithogenic disjunction,

_{F}, →

_{N}, →

_{P}representing respectively the fuzzy implication, neutrosophic

implication, and plithogenic implication.

_{1}and T

_{2}one applies a fuzzy t-norm, for their opposites F

_{1}and F

_{2}, one needs to apply the fuzzy t-conorm (the opposite of fuzzy t-norm), and reciprocally.

_{1}and I

_{2}, some researchers combined them in the same directions as T

_{1}and T

_{2}.

_{1}, I

_{1}, F

_{1}) ∧

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, F

_{1}∨

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, F

_{1}∧

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) →

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}, I

_{1}, T

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}∨

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, T

_{1}∧

_{F}F

_{2}).

_{1}and I

_{2}in the same direction as F

_{1}and F

_{2}(since both I and F are negatively qualitative neutrosophic components, while F is qualitatively positive neutrosophic component), the most used one is as follows.

_{1}, I

_{1}, F

_{1}) ∧

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, F

_{1}∨

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, F

_{1}∧

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) →

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}, I

_{1}, T

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, T

_{1}∧

_{F}F

_{2}).

(or 50%, because they are only half opposite).

_{1}, I

_{1}, F

_{1}) ∧

_{P}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, 0.5(I

_{1}∧

_{F}I

_{2}) + 0.5(I

_{1}∨

_{F}I

_{2}), F

_{1}∨

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) ∨

_{P}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, 0.5(I

_{1}∨

_{F}I

_{2}) + 0.5(I

_{1}∧

_{F}I

_{2}), F

_{1}∧

_{F}F

_{2}),

_{1}, I

_{1}, F

_{1}) →

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}, I

_{1}, T

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}∨

_{F}T

_{2}, 0.5(I

_{1}∨

_{F}I

_{2}) + 0.5(I

_{1}∧

_{F}I

_{2}),

T

_{1}∧

_{F}F

_{2}).

## 48. Fuzzy t-norms and Fuzzy t-conorms

_{F}(Fuzzy t-norms), and ∨

_{F}(Fuzzy t-conorms) are as follows.

_{F}b = min{a, b};

_{F}b = ab;

_{F}b = max{a + b − 1, 0}.

_{F}b = max{a, b};

_{F}b = a + b – ab;

_{F}b = min{a + b, 1}

## 49. Nonstandard Neutrosophic Operators

_{1}, I

_{1}, F

_{1}) ∧

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, F

_{1}∨

_{F}F

_{2}) =

(inf

_{N}(T

_{1}, T

_{2}), sup

_{N}(I

_{1}, I

_{2}), sup

_{N}(F

_{1}, F

_{2}))

_{1}, I

_{1}, F

_{1}) ∧

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, F

_{1}∨

_{F}F

_{2}) =

(T

_{1}×

_{N}T

_{2}, I

_{1}+

_{N}I

_{2}−

_{N}I

_{1}×

_{N}I

_{2}, F

_{1}+

_{N}F

_{2}−

_{N}F

_{1}×

_{N}F

_{2})

_{1}, I

_{1}, F

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, F

_{1}∧

_{F}F

_{2}) =

(sup

_{N}(T

_{1}, T

_{2}), inf

_{N}(I

_{1}, I

_{2}), inf

_{N}(F

_{1}, F

_{2}))

_{1}, I

_{1}, F

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, F

_{1}∧

_{F}F

_{2}) =

(T

_{1}+

_{N}T

_{2}−

_{N}T

_{1}×

_{N}T

_{2}, I

_{1}×

_{N}I

_{2}, F

_{1}×

_{N}F

_{2})

_{1}, I

_{1}, F

_{1}) = (F

_{1}, I

_{1}, T

_{1})

_{1}, I

_{1}, F

_{1}) = (F

_{1}, (1

^{+}) -

_{N}I

_{1}, T

_{1})

_{1}, I

_{1}, F

_{1}) →

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}, I

_{1}, T

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, T

_{1}∧

_{F}F

_{2})

= (F

_{1}+

_{N}T

_{2}−

_{N}F

_{1}×

_{N}T

_{2}, I

_{1}×

_{N}I

_{2}, T

_{1}×

_{N}F

_{2})

_{1}, I

_{1}, F

_{1}) →

_{N}(T

_{2}, I

_{2}, F

_{2}) = (F

_{1}, (1

^{+}) −

_{N}I

_{1}, T

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2})

= (F

_{1}∨

_{F}T

_{2}, ((1

^{+}) −

_{N}I

_{1}) ∧

_{F}I

_{2}, T

_{1}∧

_{F}F

_{2}) = (F

_{1}+

_{N}T

_{2}−

_{N}F

_{1}×

_{N}T

_{2}, ((1

^{+}) −

_{N}I

_{1}) ×

_{N}I

_{2}, T

_{1}×

_{N}F

_{2})

_{1}(T

_{1}, I

_{1}, F

_{1}) and P

_{2}(T

_{2}, I

_{2}, F

_{2}) be two nonstandard neutrosophic logical propositions, whose nonstandard neutrosophic components are, respectively,

_{1}, I

_{1}, F

_{1}, T

_{2}, I

_{2}, F

_{2}∈

_{N}NR

_{MB}.

## 50. Numerical Examples of Nonstandard Neutrosophic Operators

## 51. Conclusions

_{N}) and a lattice of second type (as algebraic structure, endowed with two binary laws: neutrosophic infimum (infN) and neutrosophic supremum (sup

_{N})).

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Smarandache, F.
Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis. *Symmetry* **2019**, *11*, 515.
https://doi.org/10.3390/sym11040515

**AMA Style**

Smarandache F.
Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis. *Symmetry*. 2019; 11(4):515.
https://doi.org/10.3390/sym11040515

**Chicago/Turabian Style**

Smarandache, Florentin.
2019. "Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis" *Symmetry* 11, no. 4: 515.
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