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26 

Neutrosophic Modal Logic 
 

Florentin Smarandache 
University of New Mexico 

705 Gurley Ave., Gallup, New Mexico 87301, USA 
Phone: +1 505-277-0111 
fsmarandache@gmail.com 

 
Abstract 
We 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. 

 
1. Introduction 
The paper extends the fuzzy modal logic (Girle, 2010; Hájek & Harmancová, 1993; & Liau 

& Pen Lin, 1992), fuzzy environment (Hur et. al., 2006) and neutrosophic sets, numbers and 
operators (Liu et. al., 2014; Liu & Shi, 2015; Liu & Tang, 2016; Liu & Wang, 2016; Liu & Li, 
2017; Liu & Tang, 2016; Liu et. al., 2016; Liu, 2016), together with the last developments of the 
neutrosophic environment {including (t,i,f)-neutrosophic algeb-raic structures, neutrosophic triplet 
structures, and neutrosophic overset / underset / offset} (Smarandache, 2016a; Smarandache & Ali, 
2016; Smarandache, 2016b) passing through the symbolic neutrosophic logic (Smarandache, 2015), 
ultimately to neutrosophic modal logic. 

This paper also answers Rivieccio’s question on neutrosophic modalities. 
All definitions, sections, and notions in-troduced in this paper were never done before, 

neither in my previous work nor in other researchers’. 
Therefore, we introduce now the Neutrosophic Modal Logic and the Refined Neutrosophic 

Modal Logic. Then we can extend them to Symbolic Neutrosophic Modal Logic and Refined 
Symbolic Neutrosophic Modal Logic, using labels instead of numerical values. 

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 par-ticular application. Several neutrosophic modal applications are 
also listed. 

Due to numerous applications of neutrosophic modal logic (see the examples throughout the 
paper), the introduction of the neutrosophic modal logic was needed. 

Neutrosophic Modal Logic is a logic where some neutrosophic modalities have been included. 
Let 𝒫 be a neutrosophic proposition. We have the following types of neutrosophic modalities: 
 
I. Neutrosophic Alethic Modalities (related to truth) has three neutros-ophic operators: 
Neutrosophic Possibility: It is neutros-ophically possible that 𝒫. 
Neutrosophic Necessity: It is neutros-ophically necessary that 𝒫. 
Neutrosophic Impossibility: It is neutros-ophically impossible that 𝒫. 
 
II. Neutrosophic Temporal Modalities (related to time) 
It was the neutrosophic case that 𝒫. 
It will neutrosophically be that 𝒫. 
And similarly: 
It has always neutrosophically been that 𝒫. 
It will always neutrosophically be that 𝒫. 
 
III. Neutrosophic Epistemic Modalities (related to knowledge): 
It is neutrosophically known that 𝒫. 
 



  F. Smarandache - Neutrosophic Modal Logic 

27 

IV. Neutrosophic Doxastic Modalities (related to belief): 
It is neutrosophically believed that 𝒫.  
 
V. Neutrosophic Deontic Modalities:  
It is neutrosophically obligatory that 𝒫. 
It is neutrosophically permissible that 𝒫. 
 
2. Neutrosophic Alethic Modal Operators 
The modalities used in classical (alethic) modal logic can be neutrosophicated by inserting 

the indeterminacy. 
We insert the degrees of possibility and degrees of necessity, as refinement of classical 

modal operators. 
 
2.1. Neutrosophic Possibility Operator 
The classical Possibility Modal Operator «◊ 𝑃» meaning «It is possible that P» is extended 

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

If 𝒫 𝑡 , 𝑖 , 𝑓  is a neutrosophic proposition, with 𝑡 , 𝑖 , 𝑓  subsets of [0, 1], then the 
neutrosophic truth-value of the neutrosophic possibility operator is: 

◊ 𝒫 = sup 𝑡 , inf 𝑖 , inf 𝑓 ,   (1) 

which means that if a proposition P is 𝑡  true, 𝑖  indeterminate, and 𝑓  false, then the value 

of the neutrosophic possibility operator ◊ 𝒫 is: sup 𝑡  possibility, inf 𝑖  indeterminate-
possibility, and inf 𝑓  impossibility. 

For example. 
Let P = «It will be snowing tomorrow». 
According to the meteorological center, the neutrosophic truth-value of 𝒫 is: 
𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}),  
i.e. [0.5, 0.6] true, (0.2, 0.4) indeterminate, and {0.3, 0.5} false. 
Then the neutrosophic possibility operator is: 
◊ 𝒫 = (sup[0.5, 0.6], inf(0.2, 0.4), inf{0.3, 0.5}) = (0.6, 0.2, 0.3), 
i.e. 0.6 possible, 0.2 indeterminate-possibility, and 0.3 impossible. 
 
2.2. Neutrosophic Necessity Operator 
The classical Necessity Modal Operator «□𝑃» meaning «It is necessary that P» is extended 

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

If 𝒫 𝑡 , 𝑖 , 𝑓  is a neutrosophic proposition, with 𝑡 , 𝑖 , 𝑓  subsets of [0, 1], then the 
neutrosophic truth value of the neutrosophic necessity operator is: 

□ 𝒫 = inf 𝑡 , sup 𝑖 , sup 𝑓 ,   (2) 

which means that if a proposition 𝒫 is 𝑡  true, 𝑖  indeterminate, and 𝑓  false, then the value 

of the neutrosophic necessity operator □ 𝒫 is: inf 𝑡  necessary, sup 𝑖  indeterminate-necessity, 
and sup 𝑓  unnecessary. 

Taking the previous example: 
𝒫 = «It will be snowing tomorrow», with 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}),  
then the neutrosophic necessity operator is: 



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□ 𝒫 = (inf[0.5, 0.6], sup(0.2, 0.4), sup{0.3, 0.5}) = (0.5, 0.4, 0.5), 
i.e. 0.5 necessary, 0.4 indeterminate-necessity, and 0.5 unnecessary. 
 
2.3. Other Possibility and Necessity Operators 
The previously defined neutrosophic pos-sibility and respectively neutrosophic necessity 

operators, for 𝒫 𝑡 , 𝑖 , 𝑓  a neutrosophic propos-ition, with 𝑡 , 𝑖 , 𝑓  subset-valued included in [0, 1], 
 ◊𝑁𝒫 = (sup(𝑡𝑝), inf(𝑖𝑝), inf(𝑓𝑝)), 
          □𝑁𝒫 = (inf(𝑡𝑝), sup(𝑖𝑝), sup(𝑓𝑝)),  
work quite well for subset-valued (including interval-valued) neutrosophic components, but 

they fail for single-valued neutrosophic components because one gets ◊𝑁𝒫 = □𝑁𝒫. 
Depending on the applications, more possibility and necessity operators may be defined. 
Their definitions may work, mostly based on max / min / min for possibility operator and 

min / max / max for necessity operator ( when dealing with single-valued neutrosophic components 
in [0, 1] ), or based on sup / inf / inf for possibility operator and inf / sup / sup for necessity operator 
(when dealing with interval-valued or more general with subset-valued of neutrosophic components 
included in [0, 1] ): 

For examples. 
Let 𝒫 𝑡 , 𝑖 , 𝑓  be a neutrosophic proposition, with 𝑡 , 𝑖 , 𝑓  single-valued of [0, 1], then 

the neutrosophic truth-value of the neutrosophic possibility operator is: 
◊ 𝒫 = ( max{𝑡 , 1-𝑓 }, min{𝑖 , 1-𝑖 }, min{𝑓 , 1- 𝑡 } ) 
or 
◊ 𝒫 = ( max{𝑡 , 1-𝑡 }, min{𝑖 , 1-𝑖 }, min{𝑓 , 1- 𝑓 } ) 
or  
◊ 𝒫 = (1- 𝑓 , 𝑖 , 𝑓 )  
{defined by Anas Al-Masarwah}. 
Let 𝒫 𝑡 , 𝑖 , 𝑓  be a neutrosophic proposition, with 𝑡 , 𝑖 , 𝑓  single-valued of [0, 1], then 

the neutrosophic truth-value of the neutrosophic necessity operator is: 
□ 𝒫 = ( min{𝑡 , 1-𝑓 }, max{𝑖 , 1-𝑖 }, max{𝑓 , 1- 𝑡 } ) 
or 
 □ 𝒫 = ( min{𝑡 , 1-𝑡 }, max{𝑖 , 1-𝑖 }, max{𝑓 , 1- 𝑓 } ) 
or 
 □ 𝒫 = (𝑡 , 𝑖 , 1 − 𝑡 )  
{defined by Anas Al-Masarwah}. 
The above six defined operators may be extended from single-valued numbers of [0, 1] to 

interval and subsets of [0, 1], by simply replacing the subtractions of numbers by subtractions of 
intervals or subsets, and “min” by “inf”, while “max” by “sup”. 

 
3. Connection between Neutrosophic Possibility Operator and Neutrosophic Necessity 

Operator 
In classical modal logic, a modal operator is equivalent to the negation of the other: 
◊ 𝑃 ↔ ¬□¬𝑃,                 (3) 
□𝑃 ↔ ¬ ◊ ¬𝑃.     (4) 
 
In neutrosophic logic one has a class of neutrosophic negation operators. The most used one is: 
¬
𝑁

𝑃(𝑡, 𝑖, 𝑓) = 𝑃(𝑓, 1 − 𝑖, 𝑡),    (5) 
where t, i, f are real subsets of the interval [0, 1]. 
 
Let’s check what’s happening in the neutros-ophic modal logic, using the previous example. 
One had:  



  F. Smarandache - Neutrosophic Modal Logic 

29 

𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}),  
then  

¬
𝑁

𝒫 = 𝒫({0.3, 0.5}, 1 − (0.2, 0.4), [0.5, 0.6]) = 

𝒫({0.3, 0.5}, 1 − (0.2, 0.4), [0.5, 0.6]) = 
𝒫({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]). 
 

Therefore, denoting by 
↔
𝑁

 the neutrosophic equivalence, one has: 
¬
𝑁

□
𝑁

¬
𝑁

𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5})
↔
𝑁

 
↔
𝑁

 It is not neutrosophically necessary that «It will not be snowing tomorrow» 
↔
𝑁

 It is not neutrosophically necessary that 𝒫({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) 
↔
𝑁

 It is neutrosophically possible that 
¬
𝑁

𝒫({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) 
↔
𝑁

 It is neutrosophically possible that 𝒫([0.5, 0.6], 1 − (0.6, 0.8), {0.3, 0.5}) 
↔
𝑁

 It is neutrosophically possible that 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) 
↔
𝑁

◊
𝑁

𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) = (0.6, 0.2, 0.3). 

 
Let’s check the second neutrosophic equivalence. 

¬
𝑁

◊
𝑁

¬
𝑁

𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5})
↔
𝑁

 
↔
𝑁

 It is not neutrosophically possible that «It will not be snowing tomorrow» 
↔
𝑁

 It is not neutrosophically possible that 𝒫({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) 
↔
𝑁

 It is neutrosophically necessary that 
¬
𝑁

𝒫({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) 
↔
𝑁

 It is neutrosophically necessary that 𝒫([0.5, 0.6], 1 − (0.6, 0.8), {0.3, 0.5}) 
↔
𝑁

 It is neutrosophically necessary that 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) 
↔
𝑁

□
𝑁

𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) = (0.6, 0.2, 0.3). 

 
4. Neutrosophic Modal Equivalences  
Neutrosophic Modal Equivalences hold within a certain accuracy, depending on the 

definitions of neutrosophic possibility operator and neutros-ophic 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: 

◊ 𝒫 𝑡 , 𝑖 , 𝑓
↔
𝑁

¬
𝑁

□
𝑁

¬
𝑁

𝒫 𝑡 , 𝑖 , 𝑓   (6) 

□ 𝒫 𝑡 , 𝑖 , 𝑓
↔
𝑁

¬
𝑁

◊
𝑁

¬
𝑁

𝒫 𝑡 , 𝑖 , 𝑓   (7) 

 
For example, other definitions for the neutros-ophic modal operators may be: 

◊ 𝒫 𝑡 , 𝑖 , 𝑓 = sup 𝑡 , sup 𝑖 , inf 𝑓 ,  (8) 

or 

◊ 𝒫 𝑡 , 𝑖 , 𝑓 = sup 𝑡 , , inf 𝑓  etc.,  (9) 

while 

□ 𝒫 𝑡 , 𝑖 , 𝑓 = inf 𝑡 , inf 𝑖 , sup 𝑓 ,  (10) 

or 



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□ 𝒫 𝑡 , 𝑖 , 𝑓 = inf 𝑡 , 2𝑖 ∩ [0,1], sup 𝑓   (11) 

etc. 
 
5. Neutrosophic Truth Threshold 
In neutrosophic logic, first we have to introduce a neutrosophic truth threshold, 𝑇𝐻 =

〈𝑇 , 𝐼 , 𝐹 〉, where 𝑇 , 𝐼 , 𝐹  are subsets of [0, 1]. We use uppercase letters (T, I, F) in order to 
distinguish the neutrosophic components of the threshold from those of a proposition in general. 

 
We can say that the proposition 𝒫 𝑡 , 𝑖 , 𝑓  is neutrosophically true if: 
inf 𝑡 ≥ inf(𝑇 ) and sup 𝑡 ≥ sup(𝑇 );             (12) 

inf 𝑖 ≤ inf(𝐼 ) and sup 𝑡 ≤ sup(𝐼 );    (13) 
inf 𝑓 ≤ inf(𝐹 ) and sup 𝑓 ≤ sup(𝐹 ).  (14) 
 
For the particular case when all 𝑇 , 𝐼 , 𝐹  and 𝑡 , 𝑖 , 𝑓  are single-valued numbers from the 

interval [0, 1], then one has: 
The proposition 𝒫 𝑡 , 𝑖 , 𝑓  is neutrosophically true if: 
𝑡 ≥ 𝑇 ; 
𝑖 ≤ 𝐼 ; 
𝑓 ≤ 𝐹 . 
The neutrosophic truth threshold is established by experts in accordance to each application. 
 
6. Neutrosophic Semantics 
Neutrosophic Semantics of the Neutrosophic Modal Logic is formed by a neutrosophic 

frame 𝐺 , 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: 

〈𝐺 , ℛ 〉. 
 
A neutrosophic world 𝑤′  that is neutrosophically accessible from the neutrosophic world 

𝑤  is symbolized as: 
𝑤 ℛ 𝑤′ . 

In a neutrosophic model each neutrosophic proposition 𝒫 has a neutrosophic truth-value 
𝑡 , 𝑖 , 𝑓  respectively to each neutrosophic world 𝑤 ∈ 𝐺 , where 𝑡 , 𝑖 , 𝑓  are subsets 

of [0, 1]. 
A neutrosophic actual world can be similarly noted as in classical modal logic as 𝑤 ∗ . 
Formalization 
Let 𝑆  be a set of neutrosophic propositional variables. 
 
7. Neutrosophic Formulas 
1. Every neutrosophic propositional variable 𝒫 ∈ 𝑆  is a neutrosophic formula. 

2. If A, B are neutrosophic formulas, then 
¬
𝑁

𝐴, 𝐴
∧
𝑁

𝐵, 𝐴
∨
𝑁

𝐵, 𝐴
→
𝑁

𝐵, 𝐴
↔
𝑁

𝐵, and 
◊
𝑁

𝐴, 
□
𝑁

𝐴, are 

also neutrosophic formulas, where 
¬
𝑁

, 
∧
𝑁

, 
∨
𝑁

, 
→
𝑁

, 
↔
𝑁

, and 
◊
𝑁

, 
□
𝑁

 represent the neutrosophic negation, 

neutrosophic intersection, neutrosophic union, neutros-ophic implication, neutrosophic equivalence, 
and neutrosophic possibility operator, neutrosophic necessity operator respectively. 

 



  F. Smarandache - Neutrosophic Modal Logic 

31 

8. Accesibility Relation in a Neutrosophic Theory 
Let 𝐺  be a set of neutrosophic worlds 𝑤  such that each 𝑤  chracterizes the propositions 

(formulas) of a given neutrosophic theory 𝜏. 
We say that the neutrosophic world 𝑤′  is accesible from the neutrosophic world 𝑤 , and 

we write: 𝑤 ℛ 𝑤′  or ℛ (𝑤 , 𝑤′ ), if for any proposition (formula) 𝒫 ∈ 𝑤 , meaning the 
neutrosophic truth-value of 𝒫 with respect to 𝑤  is 

𝒫 𝑡 , 𝑖 , 𝑓 , 
one has the neutrophic truth-value of 𝒫 with respect to 𝑤′  
𝒫 𝑡 , 𝑖 , 𝑓 , 
where 
inf 𝑡 ≥ inf 𝑡  and sup 𝑡 ≥ sup 𝑡 ;         (15) 

inf 𝑖 ≤ inf 𝑖  and sup 𝑖 ≤ sup 𝑖 ;                (16) 

inf 𝑓 ≤ inf 𝑓  and sup 𝑓 ≤ sup 𝑓                (17) 

(in the general case when 𝑡 , 𝑖 , 𝑓  and 𝑡 , 𝑖 , 𝑓  are subsets of the interval [0, 1]). 
 
But in the instant of 𝑡 , 𝑖 , 𝑓  and 𝑡 , 𝑖 , 𝑓  as single-values in [0, 1], the above 

inequalities become: 
𝑡 ≥ 𝑡 ,      (18) 

𝑖 ≤ 𝑖 ,      (19) 

𝑓 ≤ 𝑓 .                 (20) 
 
9. Applications 
If the neutrosophic theory 𝜏 is the Neutros-ophic Mereology, or Neutrosophic Gnosisology, 

or Neutrosophic Epistemology etc., the neutrosophic accesibility relation is defined as above. 
 
9.1. Neutrosophic n-ary Accesibility Relation 
We can also extend the classical binary accesibility relation ℛ to a neutrosophic n-ary 

accesibility relation 

ℛ
( )

, for n integer ≥ 2. 
Instead of the classical 𝑅(𝑤, 𝑤′), which means that the world 𝑤′ is accesible from the world 

𝑤, we generalize it to: 

ℛ
( )

𝑤 , 𝑤 , … , 𝑤 ; 𝑤 , 
which means that the neutrosophic world 𝑤  is accesible from the neutrosophic worlds 

𝑤 , 𝑤 , … , 𝑤  all together. 
 
9.2. Neutrosophic Kripke Frame 
𝑘 = 〈𝐺 , 𝑅 〉 is a neutrosophic Kripke frame, since: 
𝑖. 𝐺  is an arbitrary non-empty neutrosophic set of neutrosophic worlds, or neutrosophic 

states, or neutrosophic situations. 
𝑖𝑖. 𝑅 ⊆ 𝐺 × 𝐺  is a neutrosophic accesibility relation of the neutrosophic Kripke frame. 

Actually, one has a degree of accessibility, degree of indeterminacy, and a degree of non-acces-
sibility. 

 
9.3. Neutrosophic (t, i, f)-Assignement 
The Neutrosophic (t, i, f)-Assignement is a neutrosophic mapping 
𝑣 : 𝑆 × 𝐺 → [0,1] ⨯ [0,1] ⨯ [0,1]   (21) 
where, for any neutrosophic proposition 𝒫 ∈ 𝑆  and for any neutrosophic world 𝑤 , one 

defines:  
𝑣 𝑃,  𝑤 = 𝑡 , 𝑖 , 𝑓 ∈ [0,1] ⨯ [0,1] ⨯ [0,1]           (22) 



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which is the neutrosophical logical truth value of the neutrosophic proposition 𝒫 in the 
neutros-ophic world 𝑤 . 

 
9.4. Neutrosophic Deducibility 
We say that the neutrosophic formula 𝒫 is neutrosophically deducible from the neutrosophic 

Kripke frame 𝑘 , the neutrosophic (t, i, f) – assignment 𝑣 , and the neutrosophic world 𝑤 , and we 
write as: 

𝑘 , 𝑣 , 𝑤  
⊨
𝑁

 𝒫.                       (23) 

 
Let’s make the notation: 

𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 ) 
that denotes the neutrosophic logical value that the formula 𝒫 takes with respect to the 

neutrosophic Kripke frame 𝑘 , the neutrosophic (t, i, f)-assignment 𝑣 , and the neutrosphic world 
𝑤 . 

We define 𝛼  by neutrosophic induction: 

1. 𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 ) 
𝑑𝑒𝑓

=
 𝑣 (𝒫, 𝑤 ) if 𝒫 ∈ 𝑆  and 𝑤 ∈ 𝐺 . 

2. 𝛼
¬
𝑁

𝒫;  𝑘 , 𝑣 , 𝑤
𝑑𝑒𝑓

=
 
¬
𝑁

[𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 )]. 

3. 𝛼 𝒫
∧
𝑁

𝑄;  𝑘 , 𝑣 , 𝑤  
𝑑𝑒𝑓

=
 [𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 )]

∧
𝑁

 

[𝛼 (𝑄;  𝑘 , 𝑣 , 𝑤 )] 

4. 𝛼 𝒫
∨
𝑁

𝑄;  𝑘 , 𝑣 , 𝑤  
𝑑𝑒𝑓

=
 [𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 )]

∨
𝑁

 

[𝛼 (𝑄;  𝑘 , 𝑣 , 𝑤 )] 

5. 𝛼 𝒫
→
𝑁

𝑄;  𝑘 , 𝑣 , 𝑤  
𝑑𝑒𝑓

=
 [𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 )]

→
𝑁

 

[𝛼 (𝑄;  𝑘 , 𝑣 , 𝑤 )] 

6. 𝛼
◊
𝑁

𝒫;  𝑘 , 𝑣 , 𝑤  
𝑑𝑒𝑓

=
 〈sup, inf, inf〉{𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 ), 𝑤 ∈

𝐺  and 𝑤 𝑅 𝑤′ }. 

7. 𝛼
□
𝑁

𝒫;  𝑘 , 𝑣 , 𝑤
𝑑𝑒𝑓

=
〈inf, sup, sup〉{𝛼 (𝒫;  𝑘 , 𝑣 , 𝑤 ),

𝑤 ∈ 𝐺  and 𝑤 𝑅 𝑤′ }. 

8. 
⊨
𝑁

𝒫 if and only if 𝑤 ∗⊨ 𝒫 (a formula 𝒫 is neutrosophically deducible if and only if 𝒫 

is neutrosophically deducible in the actual neutrosophic world). 
 
We should remark that 𝛼  has a degree of truth 𝑡 , a degree of indeterminacy 𝑖 , and 

a degree of falsehood 𝑓 , which are in the general case subsets of the interval [0, 1]. 
 
Applying 〈sup, inf, inf〉 to 𝛼  is equivalent to calculating: 
〈sup 𝑡 , inf 𝑖 , inf 𝑓 〉, 
and similarly  
〈inf, sup, sup〉𝛼 = 〈inf 𝑡 , sup 𝑖 , sup 𝑓 〉. 
 
10. Refined Neutrosophic Modal Single-Valued Logic 
Using neutrosophic (t, i, f) - thresholds, we refine for the first time the neutrosophic modal 

logic as: 
 
 



  F. Smarandache - Neutrosophic Modal Logic 

33 

10.1. Refined Neutrosophic Possibility Operator 
◊
𝑁

𝒫( , , ) = «It is very little possible (degree of possibility 𝑡 ) that 𝒫», corresponding to the 

threshold (𝑡 , 𝑖 , 𝑓 ), i.e. 0 ≤ 𝑡 ≤ 𝑡 , 𝑖 ≥ 𝑖 , 𝑓 ≥ 𝑓 , for 𝑡  a very little number in [0, 1]; 
 
◊
𝑁

𝒫( , , ) = «It is little possible (degree of pos-sibility 𝑡 ) that 𝒫», corresponding to the 

threshold (𝑡 , 𝑖 , 𝑓 ), i.e. 𝑡 < 𝑡 ≤ 𝑡 , 𝑖 ≥ 𝑖 > 𝑖 , 𝑓 ≥ 𝑓 > 𝑓 ; 
… … …  
and so on; 
◊
𝑁

𝒫( , , ) = «It is possible (with a degree of possibility 𝑡 ) that 𝒫», corresponding to the 

threshold (𝑡 , 𝑖 , 𝑓 ), i.e. 𝑡 < 𝑡 ≤ 𝑡 , 𝑖 ≥ 𝑖 > 𝑖 , 𝑓 ≥ 𝑓 > 𝑓 . 
 
10.2. Refined Neutrosophic Necessity Operator 
□
𝑁

𝒫( , , ) = «It is a small necessity (degree of necessity 𝑡 ) that 𝒫», i.e. 𝑡 < 𝑡 ≤ 𝑡 , 

𝑖 ≥ 𝑖 ≥ 𝑖 , 𝑓 ≥ 𝑓 > 𝑓 ; 
 
□
𝑁

𝒫( , , ) = «It is a little bigger necessity (degree of necessity 𝑡 ) that 𝒫», i.e. 𝑡 <

𝑡 ≤ 𝑡 , 𝑖 ≥ 𝑖 > 𝑖 , 𝑓 ≥ 𝑓 > 𝑓 ; 
… … …  
and so on; 
 
□
𝑁

𝒫( , , ) = «It is a very high necessity (degree of necessity 𝑡 ) that 𝒫», i.e. 𝑡 <

𝑡 ≤ 𝑡 = 1, 𝑖 ≥ 𝑖 > 𝑖 , 𝑓 ≥ 𝑓 > 𝑓 . 
 
11. Application of the Neutrosophic Threshold 
We 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 (𝑤 ∗) 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 
(𝑐 − 𝜀, 𝑐 + 𝜀), for 𝜀 > 0 that can be a tiny or even a bigger number). 

 
Thus, we can say that a proposition 𝒫 is neutrosophically nomological necessary, if 𝒫 is 

neutrosophically true at all possible neutrosophic worlds that obey the (t, i, f)-physical laws of the 
actual neutrosophic world 𝑤 ∗. 

 
In other words, at each possible neutrosophic world 𝑤 , neutrosophically accesible from 

𝑤 ∗, one has: 
𝒫 𝑡 , 𝑖 , 𝑓 ≥ 𝑇𝐻(𝑇 , 𝐼 , 𝐹 ),  (24) 
 
i.e. 𝑡 ≥ 𝑇 , 𝑖 ≤ 𝐼 , and 𝑓 ≥ 𝐹 .  (25) 
 
12. 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. 



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A neutrosophic relation between two parts, and similarly a neutrosophic relation between a 
part and the whole, has a degree of connectibility (t), a degree of indeterminacy (i), and a degree of 
disconnectibility (f). 

 
12.1. Neutrosophic Mereological Threshold 
Neutrosophic Mereological Threshold is def-ined as: 

(min( ),max( ),max( ))M M M MTH t i f   (26) 
where 𝑡  is the set of all degrees of con-nectibility between the parts, and between the parts 

and the whole; 
𝑖  is the set of all degrees of indeterminacy between the parts, and between the parts and the 

whole; 
𝑓  is the set of all degrees of disconnectibility between the parts, and between the parts and 

the whole. 
We have considered all degrees as single-valued numbers. 
 
13. 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]. 

 
13.1. Neutrosophic Gnosisological Threshold 
Neutrosophic Gnosisological Threshold is defined, similarly, as: 

(min( ), max( ), max( ))G G G GTH t i f   (27) 
where 𝑡  is the set of all degrees of knowledge of all theories, ideas, propositions etc., 
𝑖  is the set of all degrees of indeterminate-knowledge of all theories, ideas, propositions 

etc., 
𝑓  is the set of all degrees of non-knowledge of all theories, ideas, propositions etc. 
We have considered all degrees as single-valued numbers. 
 
14. Neutrosophic Epistemology 
And 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%). 
Of course, t, i, f are subsets of [0, 1]. 

 
14.1. Neutrosophic Epistemological Threshold 
Neutrosophic Epistemological Threshold is defined as: 

(min( ), max( ), max( ))E E E ETH t i f   (28) 
where 𝑡  is the set of all degrees of scientific knowledge of all scientific theories, ideas, 

propositions etc., 
𝑖  is the set of all degrees of indeterminate scientific knowledge of all scientific theories, 

ideas, propositions etc., 
𝑓  is the set of all degrees of non-scientific knowledge of all scientific theories, ideas, 

propositions etc. 
We have considered all degrees as single-valued numbers. 



  F. Smarandache - Neutrosophic Modal Logic 

35 

 
15. Conclusions 
We 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, we 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]). 

Applications of neutrosophic modal logic are to neutrosophic modal metaphysics. Similarly 
to classical modal logic, there is a plethora of neutrosophic modal logics. Neutrosophic modal 
logics is governed by a set of neutrosophic axioms and neutrosophic rules. The neutrosophic 
accessibility relation has various interpretations, depending on the applications. Similarly, the 
notion of possible neutrosophic worlds has many interpretations, as part of possible neutrosophic 
semantics. 

 
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