Neutrosophic Modal Logic - Smarandache Notions

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90Neutrosophic Sets and Systems, Vol. 15, 2017University of New MexicoNeutrosophic Modal LogicFlorentin SmarandacheUniversity of New Mexico, Mathematics & Science Department, 705 Gurley Ave., Gallup, NM 87301, USA.Email: smarand@unm.eduAbstract: We introduce now for the first time theneutrosophic modal logic. The Neutrosophic Modal Logicincludes the neutrosophic operators that express themodalities. It is an extension of neutrosophic predicatelogic and of neutrosophic propositional logic.Applications of neutrosophic modal logic are toneutrosophic modal metaphysics. Similarly to classicalmodal logic, there is a plethora of neutrosophic modallogics. Neutrosophic modal logics is governed by a set ofneutrosophic axioms and neutrosophic rules.Keywords: neutrosophic operators, neutrosophic predicate logic, neutrosophic propositional logic, neutrosophic epistemology,neutrosophic mereology.1Introduction.The paper extends the fuzzy modal logic [1, 2, and4], fuzzy environment [3] and neutrosophic sets,numbers and operators [5 – 12], together with the lastdevelopments of the neutrosophic environment{including (t, i, f)-neutrosophic algebraic structures,neutrosophic triplet structures, and neutrosophicoverset / underset / offset} [13 - 15] passing throughthe symbolic neutrosophic logic [16], ultimately toneutrosophic modal logic.All definitions, sections, and notions introduced inthis paper were never done before, neither in myprevious work nor in other researchers’.Therefore, we introduce now the NeutrosophicModal Logic and the Refined Neutrosophic ModalLogic.Then we can extend them to SymbolicNeutrosophic Modal Logic and Refined SymbolicNeutrosophic Modal Logic, using labels instead ofnumerical values.There is a large variety of neutrosophic modallogics, as actually happens in classical modal logic too.Similarly, the neutrosophic accessibility relation ns, depending on each particularapplication. Several neutrosophic modal applicationsare also listed.Due to numerous applications of neutrosophicmodal logic (see the examples throughout the paper),the introduction of the neutrosophic modal logic wasneeded.Neutrosophic Modal Logic is a logic where someneutrosophic modalities have been included.Let 𝒫 be a neutrosophic proposition. We have thefollowing types of neutrosophic modalities:Florentin Smarandache, Neutrosophic Modal LogicA) Neutrosophic Alethic Modalities (related totruth) has three neutrosophic operators:i.Neutrosophic Possibility: It is neutrosophically possible that 𝒫.ii.Neutrosophic Necessity: It is neutrosophically necessary that 𝒫.iii. Neutrosophic Impossibility: It is neutrosophically impossible that 𝒫.B) Neutrosophic Temporal Modalities (relatedto 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.C) Neutrosophic Epistemic Modalities (relatedto knowledge):It is neutrosophically known that 𝒫.D) Neutrosophic Doxastic Modalities (relatedto belief):It is neutrosophically believed that 𝒫.E) Neutrosophic Deontic Modalities:It is neutrosophically obligatory that 𝒫.It is neutrosophically permissible that 𝒫.2Neutrosophic Alethic Modal OperatorsThe modalities used in classical (alethic) modallogic can be neutrosophicated by inserting the indeterminacy. We insert the degrees of possibility anddegrees of necessity, as refinement of classical modaloperators.3Neutrosophic Possibility OperatorThe classical Possibility Modal Operator Β« 𝑃 Β»meaning Β«It is possible that PΒ» is extended toNeutrosophic Possibility Operator: 𝑁 𝒫 meaning

91Neutrosophic Sets and Systems, Vol. 15, 2017Β«It is (t, i, f)-possible that 𝒫 Β», using NeutrosophicProbability, where Β«(t, i, f)-possibleΒ» 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(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) is a neutrosophic proposition, with𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 subsets of [0, 1], then the neutrosophic truthvalue of the neutrosophic possibility operator is: 𝑁 𝒫 (sup(𝑑𝑝 ), inf(𝑖𝑝 ), inf(𝑓𝑝 )),which means that if a proposition P is 𝑑𝑝 true, 𝑖𝑝indeterminate, and 𝑓𝑝 false, then the value of theneutrosophic possibility operator 𝑁 𝒫 is: sup(𝑑𝑝 )possibility, inf(𝑖𝑝 ) indeterminate-possibility, andinf(𝑓𝑝 ) impossibility.For example.Let P Β«It will be snowing tomorrowΒ».According to the meteorological center, theneutrosophic 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: 𝑁 𝒫 (sup[0.5, 0.6], inf(0.2, 0.4), inf{0.3, 0.5}) (0.6, 0.2, 0.3),𝒫 Β«It will be snowing tomorrowΒ»,𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) ,thenneutrosophic necessity operator is:withthe 𝑁 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, and0.5 unnecessary.5Connection between NeutrosophicPossibility Operator and NeutrosophicNecessity Operator.In classical modal logic, a modal operator isequivalent to the negation of the other: 𝑃 𝑃, 𝑃 𝑃.In neutrosophic logic one has a class ofneutrosophic negation operators. The most used one is: ̅𝑁𝑃(𝑑, 𝑖, 𝑓) 𝑃(𝑓, 1 𝑖, 𝑑),where t, i, f are real subsets of the interval [0, 1].Let’s check what’s happening in the neutrosophicmodal logic, using the previous example.One had:𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}),then Μ… ({0.3, 0.5}, 1 (0.2, 0.4), [0.5, 0.6]) 𝑁𝒫 𝒫i.e. 0.6 possible, 0.2 indeterminate-possibility, and 0.3 𝒫̅ ({0.3, 0.5}, 1 (0.2, 0.4), [0.5, 0.6]) impossible.𝒫̅ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]). 4 Neutrosophic Necessity OperatorTherefore, denoting by the neutrosophic equiv𝑁The classical Necessity Modal Operator Β« 𝑃 Β» alence, one has:meaning Β«It is necessary that PΒ» is extended to (0.2, 0.4), {0.3, 0.5})Neutrosophic Necessity Operator: 𝑁 𝒫 meaning Β«It 𝑁𝑁𝑁𝒫([0.5, 0.6],𝑁is (t, i, f)-necessary that 𝒫 Β», using again the It is not neutrosophically necessary that Β«It willNeutrosophic Probability, where similarly Β«(t, i, f)𝑁necessityΒ» means t % necessary (surety that 𝒫 occurs), not be snowing tomorrowΒ»i % indeterminate (indeterminate-surety that 𝒫 occurs), It is not neutrosophically necessary thatand f % unnecessary (unsurely that 𝒫 occurs).𝑁̅If 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) is a neutrosophic proposition, with 𝒫 ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 subsets of [0, 1], then the neutrosophic truth It is neutrosophically possible thatvalue of the neutrosophic necessity operator is: 𝑁̅ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])𝑁𝒫 𝑁 𝒫 (inf(𝑑𝑝 ), sup(𝑖𝑝 ), sup(𝑓𝑝 )), It is neutrosophically possible that𝑁which means that if a proposition 𝒫 is 𝑑𝑝 true, 𝑖𝑝indeterminate, and 𝑓𝑝 false, then the value of the 𝒫([0.5, 0.6], 1 (0.6, 0.8), {0.3, 0.5}) neutrosophic necessity operator 𝑁 𝒫 is: inf(𝑑𝑝 )It is neutrosophically possible that𝑁necessary, sup(𝑖𝑝 ) indeterminate-necessity, and 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5})sup(𝑓𝑝 ) unnecessary. 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) 𝑁𝑁Taking the previous example:(0.6, 0.2, 0.3).Florentin Smarandache, Neutrosophic Modal Logic

Neutrosophic Sets and Systems, Vol. 15, 201792Let’s check the second neutrosophic equivalence. (0.2, 0.4), {0.3, 0.5})𝑁𝑁𝑁𝒫([0.5, 0.6],𝑁 It is not neutrosophically possible that Β«It will𝑁not be snowing tomorrowΒ» It is not neutrosophically possible that𝑁𝒫̅ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) It is neutrosophically necessary that 𝑁̅ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])𝑁𝒫 It is neutrosophically necessary that𝑁𝒫([0.5, 0.6], 1 (0.6, 0.8), {0.3, 0.5}) It is neutrosophically necessary that𝑁𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) 𝒫([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) 𝑁𝑁(0.6, 0.2, 0.3).6We can say that the proposition 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) isneutrosophically true if:inf(𝑑𝑝 ) inf(π‘‡π‘‘β„Ž ) and sup(𝑑𝑝 ) sup(π‘‡π‘‘β„Ž );inf(𝑖𝑝 ) inf(πΌπ‘‘β„Ž ) and sup(𝑑𝑝 ) sup(πΌπ‘‘β„Ž );inf(𝑓𝑝 ) inf(πΉπ‘‘β„Ž ) and sup(𝑓𝑝 ) sup(πΉπ‘‘β„Ž ).For the particular case when all π‘‡π‘‘β„Ž , πΌπ‘‘β„Ž , πΉπ‘‘β„Ž and𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 are single-valued numbers from the interval[0, 1], then one has:The proposition 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) is neutrosophicallytrue if:𝑑𝑝 π‘‡π‘‘β„Ž ;𝑖𝑝 πΌπ‘‘β„Ž ;𝑓𝑝 πΉπ‘‘β„Ž .The neutrosophic truth threshold is established byexperts in accordance to each applications.8Neutrosophic SemanticsNeutrosophic Semantics of the NeutrosophicModal Logic is formed by a neutrosophic frame 𝐺𝑁 ,which is a non-empty neutrosophic set, whoseelements are called possible neutrosophic worlds,and a neutrosophic binary relation ℛ𝑁 , calledneutrosophic accesibility relation, between thepossible neutrosophic worlds. By notation, one has:Neutrosophic Modal EquivalencesNeutrosophic Modal Equivalences hold within acertain accuracy, depending on the definitions ofneutrosophic possibility operator and neutrosophicnecessity operator, as well as on the definition of theneutrosophic negation – employed by the expertsβŒ©πΊπ‘ , ℛ𝑁 βŒͺ.depending on each application. Under these conditions,one may have the following neutrosophic modalA neutrosophic world 𝑀′𝑁 that is neutrosophicallyequivalences:accessible from the neutrosophic world 𝑀𝑁 is symbolized as: 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) 𝑁 𝑁𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 )𝑁 𝑁𝑀𝑁 ℛ𝑁 𝑀′𝑁 . 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) 𝑁 𝑁𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 )In a neutrosophic model each neutrosophic𝑁 𝑁proposition𝒫 has a neutrosophic truth-valueFor example, other definitions for the neutrosophic(𝑑𝑀𝑁 , 𝑖𝑀𝑁 , 𝑓𝑀𝑁 ) respectively to each neutrosophicmodal operators may be:world 𝑀𝑁 𝐺𝑁 , where 𝑑𝑀𝑁 , 𝑖𝑀𝑁 , 𝑓𝑀𝑁 are subsets of [0, 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) (sup(𝑑𝑝 ), sup(𝑖𝑝 ), inf(𝑓𝑝 )), or1]. 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) (sup(𝑑𝑝 ),while𝑖𝑝2, inf(𝑓𝑝 ))etc., 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) (inf(𝑑𝑝 ), inf(𝑖𝑝 ), sup(𝑓𝑝 )), or 𝑁 𝒫(𝑑𝑝 , 𝑖𝑝 , 𝑓𝑝 ) (inf(𝑑𝑝 ), 2𝑖𝑝 [0,1], sup(𝑓𝑝 ))etc.7Neutrosophic Truth ThresholdIn neutrosophic logic, first we have to introduce aneutrosophic truth threshold, 𝑇𝐻 βŒ©π‘‡π‘‘β„Ž , πΌπ‘‘β„Ž , πΉπ‘‘β„Ž βŒͺ ,where π‘‡π‘‘β„Ž , πΌπ‘‘β„Ž , πΉπ‘‘β„Ž are subsets of [0, 1]. We use uppercase letters (T, I, F) in order to distinguish theneutrosophic components of the threshold from thoseof a proposition in general.Florentin Smarandache, Neutrosophic Modal LogicA neutrosophic actual world can be similarlynoted as in classical modal logic as 𝑀𝑁 .Formalization.Let 𝑆𝑁 be a set of neutrosophic propositionalvariables.9Neutrosophic Formulas1)Every neutrosophic propositional variable𝒫 𝑆𝑁 is a neutrosophic formula. 2)If A, B are neutrosophic formulas, then 𝑁𝐴, 𝐴 𝐡 , 𝐴 𝐡 , 𝐴 𝐡 , 𝐴 𝐡 , and 𝐴 , 𝐴 , are also𝑁𝑁𝑁𝑁𝑁 𝑁 neutrosophic formulas, where 𝑁, , , , , and ,𝑁 𝑁 𝑁 𝑁𝑁

93Neutrosophic Sets and Systems, Vol. 15, 2017 represent the neutrosophic negation, neutrosophic𝑁intersection, neutrosophic union, dneutrosophic possibility operator, neutrosophicnecessity operator respectively.10 Accesibility Relation in a NeutrosophicTheoryLet 𝐺𝑁 be a set of neutrosophic worlds 𝑀𝑁 such thateach 𝑀𝑁 chracterizes the propositions (formulas) of agiven neutrosophic theory 𝜏.We say that the neutrosophic world 𝑀′𝑁 is accesiblefrom the neutrosophic world 𝑀𝑁 , and we write:𝑀𝑁 ℛ𝑁 𝑀′𝑁 or ℛ𝑁 (𝑀𝑁 , 𝑀′𝑁 ) , if for any proposition(formula) 𝒫 𝑀𝑁 , meaning the neutrosophic truthvalue of 𝒫 with respect to 𝑀𝑁 is𝑀𝑀𝑀𝒫(𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 ),one has the neutrophic truth-value of 𝒫 with respect ��𝑀′𝑁and𝑀sup(𝑑𝑝𝑀′𝑁) inf(𝑖𝑝 𝑁 ) and sup(𝑖𝑝𝑀′𝑀inf(𝑓𝑝 𝑁 ) inf(𝑓𝑝 𝑁 )𝑀sup(𝑓𝑝 𝑁 )𝑀) sup(𝑖𝑝 𝑁 );𝑀′𝑁andsup(𝑓𝑝𝑀) 𝑀) 𝑀(in the general case when 𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 and𝑀′𝑀′𝑀′𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 are subsets of the interval [0, 1]).𝑀𝑀𝑀𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁But in the instant ofand𝑀′𝑀′𝑀′𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 as single-values in [0, 1], the aboveinequalities become:𝑀′𝑁 𝑑𝑝 𝑁 ,𝑀′𝑁 𝑖𝑝 𝑁 ,𝑑𝑝𝑖𝑝𝑀′𝑁𝑓𝑝𝑀𝑀𝑀 𝑓𝑝 𝑁 .11 ApplicationsIf the neutrosophic theory 𝜏 is the NeutrosophicMereology, or Neutrosophic Gnosisology, orNeutrosophic Epistemology etc., the neutrosophicaccesibility relation is defined as above.12 Neutrosophic n-ary Accesibility RelationWe can also extend the classical binary accesibilityrelation β„› to a neutrosophic n-ary accesibilityrelation(𝑛)ℛ𝑁 , for n integer 2.(𝑛)ℛ𝑁 (𝑀1𝑁 , 𝑀2𝑁 , , 𝑀𝑛𝑁 ; 𝑀𝑁′ ),which means that the neutrosophic world 𝑀𝑁′ isaccesiblefromtheneutrosophicworlds𝑀1𝑁 , 𝑀2𝑁 , , 𝑀𝑛𝑁 all together.13 Neutrosophic Kripke Frameπ‘˜π‘ βŒ©πΊπ‘ , 𝑅𝑁 βŒͺ is a neutrosophic Kripke frame,since:𝑖. 𝐺𝑁 is an arbitrary non-empty neutrosophic set ofneutrosophic worlds, or neutrosophic states, orneutrosophic situations.𝑖𝑖. 𝑅𝑁 𝐺𝑁 𝐺𝑁 is a neutrosophic accesibilityrelation of the neutrosophic Kripke frame. Actually,one has a degree of accesibility, degree ofindeterminacy, and a degree of non-accesibility.14 Neutrosophic (t, i, f)-AssignementThe Neutrosophic (t, i, f)-Assignement is aneutrosophic mapping𝑀′𝑀′𝑀′𝒫(𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 ),inf(𝑑𝑝 𝑁 ) inf(𝑑𝑝 𝑁 )𝑀sup(𝑑𝑝 𝑁 );Instead of the classical 𝑅(𝑀, 𝑀′), which means thatthe world 𝑀′ is accesible from the world 𝑀 , wegeneralize it to:𝑣𝑁 : 𝑆𝑁 𝐺𝑁 [0,1] [0,1] [0,1]where, for any neutrosophic proposition 𝒫 𝑆𝑁 andfor any neutrosophic world 𝑀𝑁 , one defines:𝑀𝑀𝑀𝑣𝑁 (𝑃, 𝑀𝑁 ) (𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 ) [0,1] [0,1] [0,1]which is the neutrosophical logical truth value of theneutrosophic proposition 𝒫 in the neutrosophic world𝑀𝑁 .15 Neutrosophic DeducibilityWe say that the neutrosophic formula 𝒫 isneutrosophically deducible from the neutrosophicKripke frame π‘˜π‘ , the neutrosophic (t, i, f) – assignment𝑣𝑁 , and the neutrosophic world 𝑀𝑁 , and we write as:π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 𝒫.𝑁Let’s make the notation:𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )that denotes the neutrosophic logical value that theformula 𝒫 takes with respect to the neutrosophicKripke frame π‘˜π‘ , the neutrosophic (t, i, f)-assignement𝑣𝑁 , and the neutrosphic world 𝑀𝑁 .We define 𝛼𝑁 by neutrosophic induction:1.𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )𝑀𝑁 𝐺𝑁 .2.3.𝑑𝑒𝑓 (𝒫, )𝑣𝑀𝑁 if 𝒫 𝑆𝑁 and 𝑁 𝑑𝑒𝑓 [𝛼 (𝒫;𝛼𝑁 (𝑁𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )]. 𝑁 𝑁 𝑑𝑒𝑓𝛼𝑁 (𝒫 𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )𝑁 [𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )] [𝛼𝑁 (𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )]𝑁Florentin Smarandache Neutrosophic Modal Logic

Neutrosophic Sets and Systems, Vol. 15, 2017944.5.6. 𝑑𝑒𝑓𝛼𝑁 (𝒫 𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )𝑁 [𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )] [𝛼𝑁 (𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )]𝑁 𝑑𝑒𝑓𝛼𝑁 (𝒫 𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )𝑁 [𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )] [𝛼𝑁 (𝑄; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )]𝑁 𝑑𝑒𝑓𝛼𝑁 ( 𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 )𝑁 β€²β€²βŒ©sup, inf, infβŒͺ{𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀 𝑁 ), 𝑀 𝐺𝑁 and 𝑀𝑁 𝑅𝑁 𝑀′𝑁 }.7.𝑑𝑒𝑓𝛼𝑁 ( 𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀𝑁 ) π‘βŒ©inf, sup, supβŒͺ{𝛼𝑁 (𝒫; π‘˜π‘ , 𝑣𝑁 , 𝑀 β€² 𝑁 ), 𝑀𝑁′ 𝐺𝑁 and 𝑀𝑁 𝑅𝑁 𝑀′𝑁 }. 8.𝒫 if and only if 𝑀𝑁 𝒫 (a formula 𝒫 is𝑁neutrosophically deducible if and only if 𝒫 isneutrosophically deducible in the actual neutrosophicworld).We should remark that 𝛼𝑁 has a degree of truth(𝑑𝛼𝑁 ), a degree of indeterminacy (𝑖𝛼𝑁 ), and a degreeof falsehood (𝑓𝛼𝑁 ) , which are in the general casesubsets of the interval [0, 1].Applying 〈sup, inf, infβŒͺ to 𝛼𝑁 is equivalent tocalculating:〈sup(𝑑𝛼𝑁 ), inf(𝑖𝛼𝑁 ), inf(𝑓𝛼𝑁 )βŒͺ,and similarly〈inf, sup, supβŒͺ𝛼𝑁 〈inf(𝑑𝛼𝑁 ), sup(𝑖𝛼𝑁 ), sup(𝑓𝛼𝑁 )βŒͺ.16 Refined Neutrosophic Modal SingleValued LogicUsing neutrosophic (t, i, f) - thresholds, we refinefor the first time the neutrosophic modal logic as:a) 2𝒫 Β«It is a little bigger necessity (degree of𝑁 (𝑑,𝑖,𝑓)necessity π‘‘π‘š 2 ) that 𝒫 Β», i.e. π‘‘π‘š 1 𝑑 π‘‘π‘š 2 , 𝑖 π‘–π‘š 2 π‘–π‘š 1 , 𝑓 π‘“π‘š 2 π‘“π‘š 1 ; and so on; π‘˜π’« Β«It is a very high necessity (degree of𝑁 (𝑑,𝑖,𝑓)necessity π‘‘π‘š π‘˜ ) that 𝒫», i.e. π‘‘π‘š π‘˜ 1 𝑑 π‘‘π‘š π‘˜ 1,𝑖 π‘–π‘š π‘˜ π‘–π‘š π‘˜ 1 , 𝑓 π‘“π‘š π‘˜ π‘“π‘š π‘˜ 1 .17 Application of the NeutrosophicThresholdWe have introduced the term of (t, i, f)-physical law,meaning that a physical law has a degree of truth (t), adegree 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, maybein laboratory.But our actual world (𝑀𝑁 ) is not perfect and notsteady, but continously changing, varying, fluctuating.For example, there are physicists that have proved auniversal constant (c) is not quite universal (i.e. thereare special conditions where it does not apply, or itsvalue varies between (𝑐 πœ€, 𝑐 πœ€), for πœ€ 0 that canbe a tiny or even a bigger number).Thus, we can say that a proposition 𝒫 isneutrosophically nomological necessary, if 𝒫 isneutrosophically true at all possible neutrosophicworlds that obey the (t, i, f)-physical laws of the actualneutrosophic world 𝑀𝑁 .In other words, at each possible neutrosophic world𝑀𝑁 , neutrosophically accesible from 𝑀𝑁 , one has:𝑀Refined Neutrosophic Possibility Operator. 1𝒫 Β«It is very little possible (degree of𝑁 (𝑑,𝑖,𝑓)possibility 𝑑1 ) that 𝒫», corresponding to the threshold(𝑑1 , 𝑖1 , 𝑓1 ), i.e. 0 𝑑 𝑑1 , 𝑖 𝑖1 , 𝑓 𝑓1 , for 𝑑1 a verylittle number in [0, 1]; 2𝒫 Β«It is little possible (degree of𝑁 (𝑑,𝑖,𝑓)possibility 𝑑2 ) that 𝒫», corresponding to the threshold(𝑑2 , 𝑖2 , 𝑓2 ), i.e. 𝑑1 𝑑 𝑑2 , 𝑖 𝑖2 𝑖1 , 𝑓 𝑓2 𝑓1 ; and so on; π‘šπ’« Β«It is possible (with a degree of𝑁 (𝑑,𝑖,𝑓)possibility π‘‘π‘š ) that 𝒫», corresponding to the threshold(π‘‘π‘š , π‘–π‘š , π‘“π‘š ), i.e. π‘‘π‘š 1 𝑑 π‘‘π‘š , 𝑖 π‘–π‘š π‘–π‘š 1 , 𝑓 π‘“π‘š π‘“π‘š 1 .b)Refined Neutrosophic Necessity Operator. 1𝒫 Β«It is a small necessity (degree of𝑁 (𝑑,𝑖,𝑓)necessity π‘‘π‘š 1 ) that 𝒫 Β», i.e. π‘‘π‘š 𝑑 π‘‘π‘š 1 , 𝑖 π‘–π‘š 1 π‘–π‘š , 𝑓 π‘“π‘š 1 π‘“π‘š ;Florentin Smarandache, Neutrosophic Modal Logic𝑀𝑀𝒫(𝑑𝑝 𝑁 , 𝑖𝑝 𝑁 , 𝑓𝑝 𝑁 ) 𝑇𝐻(π‘‡π‘‘β„Ž , πΌπ‘‘β„Ž , πΉπ‘‘β„Ž ),𝑀𝑀𝑀𝑁i.e. 𝑑𝑝 𝑁 π‘‡π‘‘β„Ž , 𝑖𝑝 𝑁 πΌπ‘‘β„Ž , and 𝑓𝑝 πΉπ‘‘β„Ž .18 Neutrosophic MereologyNeutrosophic Mereology means the theory of theneutrosophic relations among the parts of a whole, andthe neutrosophic relations between the parts and thewhole.A neutrosophic relation between two parts, andsimilarly a neutrosophic relation between a part andthe whole, has a degree of connectibility (t), a degreeof indeterminacy (i), and a degree of disconnectibility(f).19 Neutrosophic Mereological Thresholdas:Neutrosophic Mereological Threshold is definedTH M (min(tM ), max(iM ), max( f M ))where 𝑑𝑀 is the set of all degrees of connectibilitybetween the parts, and between the parts and thewhole;

95Neutrosophic Sets and Systems, Vol. 15, 2017𝑖𝑀 is the set of all degrees of indeterminacy betweenthe parts, and between the parts and the whole;24 ConclusionsWe have introduced for the first time the𝑓𝑀 is the set of all degrees of disconnectibility Neutrosophic Modal Logic and the Refinedbetween the parts, and between the parts and the whole. Neutrosophic Modal Logic.Symbolic Neutrosophic Logic can be connected toWe have considered all degrees as single-valuedthe neutrosophic modal logic too, where instead ofnumbers.numbers we may use labels, or instead of quantitativeneutrosophic logic we may have a quantitative20 Neutrosophic Gnosisologyneutrosophic logic. As an extension, we may introduceNeutrosophic Gnosisology is the theory of (t, i, f)- Symbolic Neutrosophic Modal Logic and Refinedknowledge, because in many cases we are not able to Symbolic Neutrosophic Modal Logic, where thecompletely (100%) find whole knowledge, but only a symbolic neutrosophic modal operators (and thepart of it (t %), another part remaining unknown (f %), symbolic neutrosophic accessibility relation) haveand a third part indeterminate (unclear, vague, qualitative values (labels) instead on numerical valuescontradictory) (i %), where t, i, f are subsets of the (subsets of the interval [0, 1]).Applications of neutrosophic modal logic are tointerval [0, 1].neutrosophic modal metaphysics. Similarly to classicalmodal logic, there is a plethora of neutrosophic modal21 Neutrosophic Gnosisological Thresholdlogics. Neutrosophic modal logics is governed by a setNeutrosophic Gnosisological Threshold is of neutrosophic axioms and neutrosophic rules. Thedefined, similarly, as:neutrosophic accessibility relation has variousinterpretations, depending on the applications.TH G (min(tG ), max(iG ), max( f G )) ,Similarly, the notion of possible neutrosophic worldswhere 𝑑𝐺 is the set of all degrees of knowledge of all has many interpretations, as part of possibleneutrosophic semantics.theories, ideas, propositions etc.,𝑖𝐺 is the set of all degrees of indeterminate-knowledgeReferencesof all theories, ideas, propositions etc.,𝑓𝐺 is the set of all degrees of non-knowledge of all [1] Rod Girle, Modal Logics and Philosophy, 2nd ed.,theories, ideas, propositions etc.McGill-Queen's University Press, 2010.We have considered all degrees as single-valued [2] P. HΓ‘jek, D. HarmancovΓ‘, A comparative fuzzy modalnumbers.logic, in Fuzzy Logic in Artificial Intelligence, Lecture22 Neutrosophic EpistemologyAnd Neutrosophic Epistemology, as part of theNeutrosophic 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 %), anda third part indeterminate (unclear, vague,contradictort) (i %).Of course, t, i, f are subsets of [0, 1].23 Neutrosophic Epistemological ThresholdIt is defined as:TH E (min(tE ), max(iE ), max( f E ))where 𝑑𝐸 is the set of all degrees of scientificknowledge of all scientific theories, ideas, propositionsetc.,𝑖𝐸 is the set of all degrees of indeterminate scientificknowledge of all scientific theories, ideas, propositionsetc.,𝑓𝐸 is the set of all degrees of non-scientific knowledgeof all scientific theories, ideas, propositions etc.We have considered all degrees as single-valuednumbers.Notes in AI 695, 1993, 27-34.[3] K. Hur, H. W. Kang and K. C. Lee, Fuzzy equivalencerelations and fuzzy partitions, Honam Math. J. 28 (2006)291-315.[4] C. J. Liau, B. I-Pen Lin, Quantitative Modal Logic andPossibilistic Reasoning, 10th European Conference onArtificial Intelligence, 1992, 43-47.[5] P.D. Liu, Y.C. Chu, Y.W. Li, Y.B. Chen, on operators and their application to GroupDecision Making, Int. J. Fuzzy Syst., 2014,16(2): 242255.[6] P.D. Liu, L.L. Shi, The Generalized Hybrid WeightedAverage Operator Based on Interval NeutrosophicHesitant Set and Its Application to Multiple AttributeDecision Making, Neural Computing and Applications,2015,26(2): 457-471.[7] P.D. Liu, G.L. Tang, Some power generalizedaggregation operators based on the intervalneutrosophic numbers and their application to decisionmaking, Journal of Intelligent & Fuzzy Systems 30(2016): 2517–2528.[8] P.D. Liu, Y.M. Wang, Interval neutrosophic prioritizedOWA operator and its application to multiple attributedecision making, JOURNAL OF SYSTEMSSCIENCE & COMPLEXITY, 2016, 29(3): 681-697.[9] P.D. Liu, H.G. Li, Multiple attribute decision makingmethod based on some normal neutrosophic Bonferronimean operators, Neural Computing and Applications,2017, 28(1), 179-194.Florentin Smarandache, Neutrosophic Modal Logic

96[10] Peide Liu, Guolin Tang, Multi-criteria group decisionmaking based on interval neutrosophic uncertainlinguistic variables and Choquet integral, CognitiveComputation, 8(6) (2016) 1036-1056.[11] Peide Liu, Lili Zhang, Xi Liu, Peng Wang, Multi-valuedNeutrosophic Number Bonferroni mean Operators andTheir Application in Multiple Attribute GroupDecision Making, INTERNATIONAL JOURNAL OFINFORMATION TECHNOLOGY & DECISIONMAKING 15(5) (2016) 1181–1210.[12] Peide Liu, The aggregation operators based onArchimedean t-conorm and t-norm for the singlevalued neutrosophic numbers and their application toDecision Making, Int. J. Fuzzy Syst., 2016,18(5):849–863.[13] F. Smarandache, (t, i, f)-Physical Laws and (t, i, f)Physical Constants, 47th Annual Meeting of the APSDivision of Atomic, Molecular and Optical Physics,Providence, Rhode Island, Volume 61, Number org/Meeting/DAMOP16/Session/Q1.197Florentin Smarandache, Neutrosophic Modal LogicNeutrosophic Sets and Systems, Vol. 15, 2017[14] F. Smarandache, M. Ali, Neutrosophic Triplet asextension of Matter Plasma, Unmatter Plasma, andAntimatter Plasma, 69th Annual Gaseous ElectronicsConference, Bochum, Germany, Volume 61, tings.aps.org/Meeting/GEC16/Session/HT6.111[15] Florentin Smarandache, Neutrosophic Overset,Neutrosophic Underset, and Neutrosophic Offset.Similarly for Neutrosophic Over-/Under-/Off- Logic,Probability, and Statistics, 168 p., Pons Editions,Brussels, 2016; 16] Florentin Smarandache, Symbolic Neutrosophic Theory,EuropaNova, Brussels, 00047.pdfReceived: February 10, 2017. Accepted: February 24, 2017.

Neutrosophic Modal Logic Florentin Smarandache University of New Mexico, Mathematics & Science Department, 705 Gurley Ave., Gallup, NM 87301, USA. . modalities. It is an extension of neutrosophic predicate logic and of neutrosophic propositional logic. Applications of neutrosophic modal logic are to neutrosophic modal metaphysics. Similarly .

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