Neutrosophic Modal Logic

I introduce now for the first time the neutrosophic modal logic. The Neutrosophic Modal Logic includes the neutrosophic operators that express the modalities. It is an extension of neutrosophic predicate logic, and of neutrosophic propositional logic. In order for the paper to be self-contained, I also recall the etymology and definition of neutrosophy and of neutrosophic logic. Several examples are presented as well.

Neutrosophy is a new branch of philosophy, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.
Neutrosophy is a generalization of dialectics, because while in dialectics one deals with the dynamics of opposites {let's note them by <A> and <antiA>, where <A> is an item (i.e.idea, notion, proposition, theory etc.) and <anti> is the opposite of <A>}, in neutrosophy one deals with the dynamics of opposites and their neutral {i.e.<A>, <antiA>, and their neutralities <neutA>}.
For example, if two countries C1 and C2 go to war against each other (this is the dynamics of opposites <A> and <antiA>), some neutral countries (<neutA>) interfere into the war in one side or another.So, the neutrals, which were ignored in dialectics, have to be taking into consideration as in neutrosophy.
Neutrosophic logic is a generalization of the intuitionistic fuzzy logic.In neutrosophic logic, a proposition has a degree of truth (t), a degree of indeterminacy/neutrality (neither true, not false) (i), and a degree of Neutrosophic Modal Logic is a logic where some neutrosophic modalities have been included.There is a large variety of neutrosophic modal logics, as actually happens in classical modal logic too.Similarly, the neutrosophic accessibility relation and possible neutrosophic worlds have many interpretations, depending on each application.Several neutrosophic modal applications are also listed.

Let
be a neutrosophic proposition.One has the following types of neutrosophic modalities: A) Neutrosophic Alethic Modalities (related to truth) has three neutrosophic operators: i.
Neutrosophic Possibility: It is neutrosophically possible that .ii. Neutrosophic Necessity: It is neutrosophically necessary that .iii.Neutrosophic Impossibility: It is neutrosophically impossible that .

B) Neutrosophic Temporal Modalities (related to time)
It was the neutrosophic case that .It will neutrosophically be that .And similarly: It has always neutrosophically been that .It will always neutrosophically be that .

C) Neutrosophic Epistemic Modalities (related to knowledge):
It is neutrosophically known that .

D) Neutrosophic Doxastic Modalities (related to belief):
It is neutrosophically believed that .
It is neutrosophically permissible that .

Neutrosophic Alethic Modal Operators
The modalities used in classical (alethic) modal logic can be neutrosophicated by inserting the indeterminacy.I insert the degrees of possibility and degrees of necessity, as refinement of classical modal operators.

Neutrosophic Possibility Operator
The classical Possibility Modal Operator «◊ » meaning «It is possible that P» is extended to Neutrosophic Possibility Operator: ◊ meaning «It is (t, i, f)-possible that », using Neutrosophic Probability, where «(t, i, f)-possible» means t % possible (chance that occurs), i % indeterminate (indeterminate-chance that occurs), and f % impossible (chance that does not occur).

Neutrosophic Necessity Operator
The classical Necessity Modal Operator «□ » meaning «It is necessary that P» is extended to Neutrosophic Necessity Operator: □ meaning «It is (t, i, f)-necessary that », using again the Neutrosophic Probability, where similarly «(t, i, f)-necessity» means t % necessary (surety that occurs), i % indeterminate (indeterminate-surety that occurs), and f % unnecessary (unsafety that occurs).

Connection between Neutrosophic Possibility Operator and Neutrosophic Necessity Operator
In classical modal logic, a modal operator is equivalent to the negation of the other: In neutrosophic logic one has a class of neutrosophic negation operators.The most used one is: where t, i, f are real subsets of the interval [0, 1], and P − is the negation of P. Let's check what's happening in the neutrosophic modal logic, using the previous example.Therefore, denoting by ↔ the neutrosophic equivalence, one has:

Neutrosophic Modal Equivalences
Neutrosophic Modal Equivalences hold within a certain accuracy, depending on the definitions of neutrosophic possibility operator and neutrosophic necessity operator, as well as on the definition of the neutrosophic negation -employed by the experts depending on each application.Under these conditions, one may have the following neutrosophic modal equivalences: For example, other definitions for the neutrosophic modal operators may be: or etc., while or etc.

Neutrosophic Truth Threshold
In neutrosophic logic, first I have to introduce a neutrosophic truth threshold,

Applications
If the neutrosophic theory is the Neutrosophic Mereology, or Neutrosophic Gnosisology, or Neutrosophic Epistemology etc., the neutrosophic accesibility relation is defined as above.Instead of the classical ( , ′), which means that the world ′ is accesible from the world , I generalize it to: ℛ ( ) , , … , ; , which means that the neutrosophic world is accesible from the neutrosophic worlds , , … , all together.
. ⊆ × is a neutrosophic accesibility relation of the neutrosophic Kripke frame.Actually, one has a degree of accesibility, degree of indeterminacy, and a degree of nonaccesibility.

Neutrosophic (t, i, f)-Assignement
The Neutrosophic (t, i, f)-Assignement is a neutrosophic mapping where, for any neutrosophic proposition ∈ and for any neutrosophic world , one defines: which is the neutrosophical logical truth value of the neutrosophic proposition in the neutrosophic world .

Neutrosophic Deducibility
One says that the neutrosophic formula is neutrosophically deducible from the neutrosophic Kripke frame , the neutrosophic (t, i, f) -assignment , and the neutrosophic world , and one writes as: , , ⊨ .
Let's make the notation: ( ; , , ) that denotes the neutrosophic logical value that the formula takes with respect to the neutrosophic Kripke frame , the neutrosophic (t, i, f)-assignement , and the neutrosphic world .One defines by neutrosophic induction:

Application of the Neutrosophic Threshold
I have introduced the term of (t, i, f)-physical law, meaning that a physical law has a degree of truth (t), a degree of indeterminacy (i), and a degree of falsehood (f).A physical law is 100% true, 0% indeterminate, and 0% false in perfect (ideal) conditions only, maybe in laboratory.
But our actual world ( * ) is not perfect and not steady, but continously changing, varying, fluctuating.
For example, there are physicists that have proved a universal constant (c) is not quite universal (i.e.there are special conditions where it does not apply, or its value varies between ( − , + ), for > 0 that can be a tiny or even a bigger number).
Thus, one can say that a proposition is neutrosophically nomological necessary, if is neutrosophically true at all possible neutrosophic worlds that obey the (t, i, f)-physical laws of the actual neutrosophic world * .

Neutrosophic Mereology
Neutrosophic Mereology means the theory of the neutrosophic relations among the parts of a whole, and the neutrosophic relations between the parts and the whole.
A neutrosophic relation between two parts, and similarly a neutrosophic relation between a part and the whole, has a degree of connection (t), a degree of indeterminacy (i), and a degree of disconnection (f).

Neutrosophic Mereological Threshold
Neutrosophic Mereological Threshold is defined as: THM = (min(tM), max(iG), max(fM)), where tM is the set of all degrees of connection between the parts, and between the parts and the whole; iM is the set of all degrees of indeterminacy between the parts, and between the parts and the whole; fM is the set of all degrees of disconnection between the parts, and between the parts and the whole.
I have considered all degrees as single-valued numbers.

Neutrosophic Gnosisology
Neutrosophic Gnosisology is the theory of (t, i, f)-knowledge, because in many cases we are not able to completely (100%) find whole knowledge, but only a part of it (t %), another part remaining unknown (f %), and a third part indeterminate (unclear, vague, contradictory) (i %), where t, i, f are subsets of the interval [0, 1].

Neutrosophic Gnosisological Threshold
Neutrosophic Gnosisological Threshold is defined, similarly, as: where tG is the set of all degrees of knowledge of all theories, ideas, propositions etc., iG is the set of all degrees of indeterminate-knowledge of all theories, ideas, propositions etc., fG is the set of all degrees of non-knowledge of all theories, ideas, propositions etc.I have considered all degrees as single-valued numbers.

Neutrosophic Epistemology
Neutrosophic Epistemology, as part of the Neutrosophic Gnosisology, is the theory of (t, i, f)scientific knowledge.
Science is infinite.We know only a small part of it (t %), another big part is yet to be discovered (f %), and a third part indeterminate (unclear, vague, contradictory) (i %).

Neutrosophic Epistemological Threshold
Neutrosophic Epistemological Threshold is defined as: THE = (min(tE), max(iE), max(fE)), where tE is the set of all degrees of scientific knowledge of all scientific theories, ideas, propositions etc., iE is the set of all degrees of indeterminate scientific knowledge of all scientific theories, ideas, propositions etc., fE is the set of all degrees of non-scientific knowledge of all scientific theories, ideas, propositions etc.I have considered all degrees as single-valued numbers.

Discussion & Conclusions
I have introduced for the first time the Neutrosophic Modal Logic and the Refined Neutrosophic Modal Logic.
Symbolic Neutrosophic Logic can be connected to the neutrosophic modal logic too, where instead of numbers we may use labels, or instead of quantitative neutrosophic logic we may have a quantitative neutrosophic logic.As an extension, I may introduce Symbolic Neutrosophic Modal Logic and Refined Symbolic Neutrosophic Modal Logic, where the symbolic neutrosophic modal operators (and the symbolic neutrosophic accessibility relation) have qualitative values (labels) instead on numerical values (subsets of the interval [0, 1]).
falsehood (f), where t, i, f are subsets of the unit interval [0, 1].The Modal Logic, from classical Boolean logic, I now extend to Neutrosophic Modal Logic.I introduce therefore the Neutrosophic Modal Logic and the Refined Neutrosophic Modal Logic.Then I can extend them to Symbolic Neutrosophic Modal Logic and Refined Symbolic Neutrosophic Modal Logic.I use labels instead of numerical values.

I
can also extend the classical binary accesibility relation ℛ to a neutrosophic n-ary accesibility relation ℛ ( ) , for n integer ≥ 2.

Neutrosophic Semantics of the Neutrosophic Modal Logic is
formed by a neutrosophic frame , which is a non-empty neutrosophic set, whose elements are called possible

neutrosophic worlds, and
a neutrosophic binary relation ℛ , called neutrosophic accesibility relation, between the possible neutrosophic worlds.By notation, one has: 〈 , ℛ 〉.A neutrosophic world ′ that is neutrosophically accessible from the neutrosophic world is symbolized as: .