Extended Nonstandard Neutrosophic Logic, Set, and Probability based on Extended Nonstandard Analysis

We extend for the second time the Nonstandard Analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad - all these in order to close the newly extended nonstandard space under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations. Then, we extend the Nonstandard Neutrosophic Logic, Nonstandard Neutrosophic Set, and Nonstandard Probability on this Extended Nonstandard Analysis space - that we prove it is a nonstandard neutrosophic lattice of first type (endowed with a nonstandard neutrosophic partial order) as well as a nonstandard neutrosophic lattice of second type (as algebraic structure, endowed with two binary neutrosophic laws, inf_N and sup_N). Many theorems, new terms introduced, and examples of nonstandard neutrosophic operations are given.


Short Introduction
In order to more accurately situate and fit the neutrosophic logic into the framework of extended nonstandard analysis, we present the nonstandard neutrosophic inequalities, nonstandard neutrosophic equality, nonstandard neutrosophic infimum and supremum, nonstandard neutrosophic intervals, including the cases when the neutrosophic logic standard and nonstandard components T, I, F get values outside of the classical unit interval [0, 1], and a brief evolution of neutrosophic operators.

Theoretical Reason for the Nonstandard Form of Neutrosophic Logic
The only reason I have added the nonstandard form to neutrosophic logic (and similarly to neutrosophic set and probability) was in order to make a distinction between Relative Truth (which is truth in some Worlds, according to Leibniz) and Absolute Truth (which is truth in all possible Words, according to Leibniz as well) that occur in philosophy. [15,16] An infinitesimal [or infinitesimal number] (  ) is a number  such that | | 1 / n   , for any non-null positive integer n. An infinitesimal is close to zero, and so small that it cannot be measured.

Introduction to Nonstandard Analysis
The infinitesimal is a number smaller, in absolute value, than anything positive nonzero.
Infinitesimals are used in calculus.
The set of hyperreals (or non-standard reals), denoted as R * , is the extension of set of the real numbers, denoted as R, and it comprises the infinitesimals and the infinites, that may be represented on the hyperreal number line The set of hyperreals satisfies the transfer principle, which states that the statements of first order in R are valid in R* as well.
A monad (halo) of an element a ∊ R * , denoted by μ(a), is a subset of numbers infinitesimally close to a.

First Extension of Nonstandard Analysis
Let's denote by R+ * the set of positive nonzero hyperreal numbers.

Second Extension of Nonstandard Analysis
For necessity of doing calculations that will be used in nonstandard neutrosophic logic in order to calculate the nonstandard neutrosophic logic operators (conjunction, disjunction, negation, implication, equivalence) and in order to have the Nonstandard Real MoBiNad Set closed under arithmetic operations, we extend now for the time: the left monad to the Left Monad Closed to the Right, the right monad to the Right Monad Closed to the Left; and the Pierced Binad to the Unpierced Binad, defined as follows: (10)

Right Monad Closed to the Left
Pierced Binad The element {a} has been included into the left monad, right monad, and pierced binad respectively.

Nonstandard Neutrosophic Function
In order to be able to define equalities and inequalities in the sets of monads, and in the sets of binads, we construct a nonstandard neutrosophic function that approximates the monads and binads to tiny open (or half open and half closed respectively) standard real intervals as below. It is called 'neutrosophic' since it deals with indeterminacy: unclear, vague monads and binads, and the function approximates them with some tiny real subsets.
Taking an arbitrary infinitesimal ε1 > 0, and writinga = a-ε1, a + = a+ε1, anda + = a  ε1, (13) or taking an arbitrary infinitesimal ε2 ≥ 0, and writing (14) we meant actually picking up a representative from each class of the monads and of the binads respectively. Representations of the monads and binads by intervals is not quite accurate from a classical point of view, but it is an approximation that helps in finding a partial order and computing nonstandard arithmetic operations on the elements of the nonstandard set NRMB.
Let ε be a generic positive infinitesimal, while a be a generic standard real number.
Let P(R) be the power set of the real number set R.
For any a ∊ R, the set of real numbers, one has: μN( (a + ) ) =N (a, a + ε),  (22) in order to set it as real interval too.
The notations of monad's and binad's diacritics above (not laterally) the number a as are the best, since they also are designed to avoid confusion for the case when the real number a is negative.
For example, if a = -2, then the corresponding monads and binads are respectively represented as:

Neutrosophic Strict Inequalities
We recall the neutrosophic strict inequality which is needed for the inequalities of nonstandard numbers.
Let α, β be elements in a partially ordered set M.
We have defined the neutrosophic strict inequality and read as "α is neutrosophically greater than β" if α in general is greater than β, or α is approximately greater than β, or subject to some indeterminacy (unknown or unclear ordering relationship between α and β) or subject to some contradiction (situation when α is smaller than or equal to β) α is greater than β.
It means that in most of the cases, on the set M, α is greater than β.
And similarly for the opposite neutrosophic strict inequality α <N β.

Neutrosophic Equality
We have defined the neutrosophic inequality and read as "α is neutrosophically equal to β" if α in general is equal to β, or α is approximately equal to β, or subject to some indeterminacy (unknown or unclear ordering relationship between α and β) or subject to some contradiction (situation when α is not equal to β) α is equal to β.
It means that in most of the cases, on the set M, α is equal to β.

Neutrosophic (Non-Strict) Inequalities
Combining the neutrosophic strict inequalities with neutrosophic equality, we get the ≥N and ≤N neutrosophic inequalities.
Let α, β be elements in a partially ordered set M.
The neutrosophic (non-strict) inequality and read as "α is neutrosophically greater than or equal to β" if α in general is greater than or equal to β, or α is approximately greater than or equal to β, or subject to some indeterminacy (unknown or unclear ordering relationship between α and β) or subject to some contradiction (situation when α is smaller than β) α is greater than or equal to β.
It means that in most of the cases, on the set M, α is greater than or equal to β.

Neutrosophic Infimum and Neutrosophic Supremum
As an extension of the classical infimum and classical supremum, and using the neutrosophic inequalities and neutrosophic equalities, we define the neutrosophic infimum ( denoted as infN ) and the neutrosophic supremum ( denoted as supN ).

Neutrosophic Infimum.
Let (S, <N) be a set that is neutrosophically partially ordered, and M a subset of S. The neutrosophic infimum of M, denoted as infN(M) is the neutrosophically greatest element in S that is neutrosophically less than or equal to all elements of M: Neutrosophic Supremum. Let (S, <N) be a set that is neutrosophically partially ordered, and M a subset of S. The neutrosophic supremum of M, denoted as supN(M) is the neutrosophically smallest element in S that is neutrosophically greater than or equal to all elements of M.

Definition of Nonstandard Real MoBiNad Set
Let ℝ be the set of standard real numbers, ℝ * the set of hyper-reals (or non-standard reals) which consists of infinitesimals and infinites.
(36) Therefore: where ( − ℝ) is the set of all real left monads,

Etymology of MoBiNad
MoBiNad comes from monad + binad, introduced now for the first time.

Definition of Nonstandard Complex MoBiNad Set
The Nonstandard Complex MoBiNad Set, introduced here for the first time, is defined as:

Definition of Nonstandard Neutrosophic Real MoBiNad Set
The Nonstandard Neutrosophic Real MoBiNad Set, introduced now for the first time, is defined as:

Definition of Nonstandard Neutrosophic Complex MoBiNad Set
The Nonstandard Neutrosophic Complex MoBiNad Set, introduced now for the first time, is defined as:

Properties of the Nonstandard Neutrosophic Real Mobinad Set
Since in nonstandard neutrosophic logic we use only the nonstandard neutrosophic real mobinad set, we study some properties of it.

Theorem 1
The nonstandard real mobinad set ( , ≤ ), endowed with the nonstandard neutrosophic inequality is a lattice of first type [as partially ordered set (poset)].

Proof
If = , one has: and there is no neutrosophic ordering relationship between and -+ , nor between a and 0 a  (that is why ≤ on is a partial ordering set).
Any two-element set { , } ⊂ has a neutrosophic nonstandard infimum (meet, or greatest lower bound) that we denote by inf , and a neutrosophic nonstandard supremum (joint, or least upper bound) that we denote by sup , where both For the non-ordered elements and -+ : And similarly for non-ordered elements and -0 + : Dealing with monads and binads which neutrosophically are real subsets with indeterminate borders, and similarly = [ , ] can be treated as a subset, we may compute inf N and sup N of each of them.
Therefore, ( , ≤ ) is a nonstandard real mobinad lattice of first type (as partially ordered set).

Theorem 2
Any finite non-empty subset of ( ,≤ ) is also a sublattice of first type.

Proof
It is a consequence of any classical lattice of first order (as partially ordered set).

Theorem 3
( , ≤ ) is not bounded neither to the left nor to the right, since it does not have a minimum (bottom, or least element), nor a maximum (top, or greatest element).

Theorem 4
( , inf , sup ), where inf and sup are two binary operations, dual to each other, defined before, is a lattice of second type (as an algebraic structure).

Proof
We have to show that the two laws inf and sup are commutative, associative, and verify the absorption laws.

Commutativity Laws
Their proofs are straightforward.

Associativity Laws
and where we have extended the binary operation inf to a trinary operation inf . and where similarly we have extended the binary operation sup to a trinary operation sup . (80)

Proof
The axioms of idempotency follow directly from the axioms of absorption proved above.
Thus, we have proved that (NRMB, infN, supN) is a lattice of second type (as algebraic structure).
The sublattice Identity Laws are verified below.

Definition of Nonstandard Real MoBiNad Unit Interval
This is an extension of the previous definition (1998) of nonstandard unit interval Associated to the first published definitions of neutrosophic set, logic, and probability was used.

Theorem 7
The Nonstandard Real MoBiNad Unit Interval ] − 0, 1 + [ is a partially ordered set (poset) with respect to ≤ , and any of its two elements have an inf N and sup N whence ] − 0, 1 + [ is a nonstandard neutrosophic lattice of first type (as poset).

Theorem 8
The Nonstandard Real MoBiNad Unit Interval ] − 0, 1 + [ , endowed with two binary operations inf and sup , is also a nonstandard neutrosophic lattice of second type (as an algebraic structure).

Definition of Extended General Neutrosophic Logic
We extend and present in a clearer way our 1995 definition (published in 1998) of neutrosophic logic.
Let be a universe of discourse of propositions, and ∈ a generic proposition.
A General Neutrosophic Logic is a multivalued logic in which each proposition has a degree of truth ( ), a degree of indeterminacy ( ), and a degree of falsehood ( ), where , , are standard or nonstandard real mobinad subsets of the nonstandard real mobinat unit interval

Definition of Standard Neutrosophic Logic
If in the above definition of general neutrosophic logic all neutrosophic components, T, I, F, are standard real subsets, included in or equal to the standard real unit interval, , , we have a standard neutrosophic logic.

Definition of Extended Nonstandard Neutrosophic Logic
If in the above definition of general neutrosophic logic at least one of the neutrosophic components T, I, F is a nonstandard real mobinad subset, neutrosophically included in or equal to the nonstandard real mobinad unit interval ] − 0, 1 + [ , we have an extended nonstandard neutrosophic logic.

Definition of Extended General Neutrosophic Set
We extend and present in a clearer way our 1995 definition of neutrosophic set.
Let be a universe of discourse of elements, and ∈ a subset.
A Neutrosophic Set is a set such that each element from has a degree of membership ( ), a degree of indeterminacy ( ), and a degree of nonmembership ( ), where , , are standard or nonstandard real mobinad subsets, neutrosophically included in or equal to the nonstandard real mobinat unit interval ] − 0, 1 + [ ,

Definition of Standard Neutrosophic Set
If in the above general definition of neutrosophic set all neutrosophic components, T, I, F, are standard real subsets included in or equal to the classical real unit interval, we have a standard neutrosophic set.

Definition of Extended Nonstandard Neutrosophic Set
If in the above general definition of neutrosophic set at least one of the neutrosophic components T, I, F is a nonstandard real mobinad subsets, neutrosophically included in or equal to we have a nonstandard neutrosophic set.

Definition of Extended General Neutrosophic Probability
We extend and present in a clearer way our 1995 definition of neutrosophic probability.
Let be a universe of discourse of events, and ∈ be an event.

Definition of Standard Neutrosophic Probability
If in the above general definition of neutrosophic probability all neutrosophic components, T, I, F, are standard real subsets, included in or equal to the standard unit interval, we have a standard neutrosophic probability.

Definition of Extended Nonstandard Neutrosophic Probability
If in the above general definition of neutrosophic probability at least one of the neutrosophic components T, I, F is a nonstandard real mobinad subsets, neutrosophically included in or equal to ] − 0, 1 + [ , we have a nonstandard neutrosophic probability.

Operations on the Nonstandard Real MoBiNad Set ( )
For all nonstandard (addition, subtraction, multiplication, division, and power) operations, where *N is any neutrosophic arithmetic operations with neutrosophic numbers (+N, -N, × ,÷ ,  ^N), while the corresponding ⊛ is an arithmetic operation with real subsets.
So, we approximate the nonstandard operations by standard operations of real subsets.
We sink the nonstandard neutrosophic real mobinad operations into the standard real subset operations, then we resurface the last ones back to the nonstandard neutrosophic real mobinad set.
Let 1 and 2 be two non-null positive infinitesimals. We present below some particular cases, all others should be deduced analogously.

Nonstandard Addition
First Method Adding two left monads, one also gets a left monad.

Nonstandard Subtraction
First Method  Subtracting two left monads, one obtains an unpierced binad (that's why the unpierced binad had to be introduced).
Since 1 > 0 and 2 > 0, For or/and negative numbers, it's similar but it's needed to compute the and of the products of intervals.
Dividing two left monads, one obtains an unpierced binad.
For or/and negative numbers, it's similar but it's needed to compute the and of the products of intervals.
Multiplying a positive left monad closed to the right, with a positive unpierced binad, one obtains an unpierced binad.
Raising a right monad closed to the left to a power equal to a left monad closed to the right, for both monads above 1, the result is an unpierced binad.

Consequence
In general, when doing arithmetic operations on nonstandard real monads and binads, the result may be a different type of monad or binad.
That's why is was imperious to extend the monads to closed monads, and the pierced binad to unpierced binad, in order to have the whole nonstandard neutrosophic real mobinad set closed under arithmetic operations.
be the left monads, left monads closed to the right, right monads, right monads closed to the left, and binads, and binads nor prierced of the elements (standard real numbers) a and b respectively. Since all monads and binads are real subsets, we may treat the single real numbers NRMB 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, then α >N β (partially).

ii)
If the subset α has many of its elements below all elements of the subset β, then α <N β (partially).

iii)
If the subset α has many of its elements equal with elements of the subset β, then α =N β (partially).
If the subset α verifies i) and iii) with respect to subset β, then α ≥N β.
If the subset α verifies ii) and iii) with respect to subset β, then α ≤N β.
If the subset α verifies i) and ii) with respect to subset β, then there is no neutrosophic order (inequality) between α and β.
{ For example, between a and (a + ) there is no neutrosophic order, similarly between a and . } Similarly, if the subset α verifies i), ii) and iii) with respect to subset β, then there is no neutrosophic order (inequality) between α and β.

Open Neutrosophic Research
The quantity or measure of "many of its elements" of the above i), ii), or iii) conditions depends on each neutrosophic application and on its neutrosophic experts.
An approach would be to employ the Neutrosophic Measure [21,22], that handles indeterminacy, which may be adjusted and used in these cases.
In general, we do not try in purpose to validate or invalidate an existing scientific result, but to investigate how an existing scientific result behaves in a new environment (that may contain indeterminacy), or in a new application, or in a new interpretation.

Nonstandard Neutrosophic Inequalities
For the neutrosophic nonstandard inequalities, we propose based on the previous six neutrosophic equalities, the following: since the standard real interval (a -ε, a) is below a, and a is below the standard real interval (a, a + ε) by using the approximation provided by the nonstandard neutrosophic function μ, where x is of course a (nonzero) positive infinitesimal (the above double neutrosophic inequality actually becomes a double classical standard real inequality for each fixed positive infinitesimal).
The converse double neutrosophic inequality is also neutrosophically true:

(a) ≤N (a)  (a + ) ≤N (a + ) (145)
whence the left side of the inequality's middle term coincides with the inequality first term, while the right side of the inequality middle term coincides with the third inequality term.
Using the nonstandard general notation one has: If a > b, which is a (standard) classical real inequality, then

Nonstandard Neutrosophic Equalities
Let a, b be standard real numbers; if a = b that is a (classical) standard equality, then:

Nonstandard Hesitant Sets
Are while other elements may be standard real numbers, infinitesimals, or also monads or binads (of any type).
If the neutrosophic components T, I, F are nonstandard hesitant sets, then one has a Nonstandard Hesitant Neutrosophic Logic / Set / Probability.

Nonstandard Neutrosophic Strict Interval Inclusion
On the nonstandard real set NRMB,

Nonstandard Neutrosophic (Non-Strict) Set Inclusion
The nonstandard set A is neutrosophically not-strictly included in the nonstandard set B, N AB  , iff: ,

Nonstandard Neutrosophic Set Equality
The nonstandard sets A and B are neutrosophically equal, and →F, →N, →P representing respectively the fuzzy implication, neutrosophic implication, and plithogenic implication.
I agree that my beginning neutrosophic operators (when I applied the same fuzzy t-norm, or the same fuzzy t-conorm, to all neutrosophic components T, I, F) were less accurate than others developed later by the neutrosophic community researchers. This was pointed out since 2002 by Ashbacher [9] and confirmed in 2008 by Rivieccio [10]. They observed that if on T1 and T2 one applies a fuzzy t-norm, on their opposites F1 and F2 one needs to apply the fuzzy t-conorm (the opposite of fuzzy t-norm), and reciprocally.
About inferring I1 and I2, some researchers combined them in the same directions as T1 and T2.
Then: Even more, recently, in an extension of neutrosophic set to plithogenic set [11] (which is a set whose each element is characterized by many attribute values), the degrees of contradiction c( , ) between the neutrosophic components T, I, F have been defined (in order to facilitate the design of the aggregation operators), as follows:

Conclusion
In the history of mathematics, critics on nonstandard analysis, in general, have been made by Paul Halmos, Errett Bishop, Alain Connes and others.
That's why we have extended in 1998 for the first time the monads to pierced binad, and then in 2019 for the second time we extended the left monad to left monad closed to the right, the right monad to right monad closed to the left, and the pierced binad to unpierced binad. These were necessary in order to construct a general nonstandard neutrosophic real mobinad space, which is closed under the nonstandard neutrosophic arithmetic operations (such as addition, subtraction, multiplication, division, and power) which are needed in order to be able to define the nonstandard neutrosophic operators (such as conjunction, disjunction, negation, implication, equivalence) on this space, andto transform the newly constructed nonstandard neutrosophic real mobinad space into a lattice of first order (as partially ordered nonstandard set, under the neutrosophic inequality ) and a lattice of second type [as algebraic structure, endowed with two binary laws: neutrosophic infimum (infN) and neutrosophic supremum (supN)].