Neutrosophic Modal Logic - Smarandache Notions
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 β 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 researchersβ.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 π« 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 π«.ii.Neutrosophic Necessity: It is neutrosophically necessary that π«.iii. Neutrosophic Impossibility: It is neutrosophically impossible that π«.B) Neutrosophic Temporal Modalities (relatedto time)It was the neutrosophic case that π«.It will neutrosophically be that π«.And similarly:It has always neutrosophically been that π«.It will always neutrosophically be that π«.C) Neutrosophic Epistemic Modalities (relatedto knowledge):It is neutrosophically known that π«.D) Neutrosophic Doxastic Modalities (relatedto belief):It is neutrosophically believed that π«.E) Neutrosophic Deontic Modalities:It is neutrosophically obligatory that π«.It is neutrosophically permissible that π«.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« π Β»meaning Β«It is possible that PΒ» is extended toNeutrosophic Possibility Operator: π π« meaning
91Neutrosophic Sets and Systems, Vol. 15, 2017Β«It is (t, i, f)-possible that π« Β», using NeutrosophicProbability, where Β«(t, i, f)-possibleΒ» means t %possible (chance that π« occurs), i % indeterminate(indeterminate-chance that π« occurs), and f %impossible (chance that π« does not occur).If π«(π‘π , ππ , ππ ) is a neutrosophic proposition, withπ‘π , ππ , ππ subsets of [0, 1], then the neutrosophic truthvalue of the neutrosophic possibility operator is: π π« (sup(π‘π ), inf(ππ ), inf(ππ )),which means that if a proposition P is π‘π true, ππindeterminate, and ππ false, then the value of theneutrosophic possibility operator π π« is: sup(π‘π )possibility, inf(ππ ) indeterminate-possibility, andinf(ππ ) impossibility.For example.Let P Β«It will be snowing tomorrowΒ».According to the meteorological center, theneutrosophic truth-value of π« is:π«([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: π π« (sup[0.5, 0.6], inf(0.2, 0.4), inf{0.3, 0.5}) (0.6, 0.2, 0.3),π« Β«It will be snowing tomorrowΒ»,π«([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) ,thenneutrosophic necessity operator is:withthe π π« (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: π π, π π.In neutrosophic logic one has a class ofneutrosophic negation operators. The most used one is: Μ ππ(π‘, π, π) π(π, 1 π, π‘),where t, i, f are real subsets of the interval [0, 1].Letβs check whatβs happening in the neutrosophicmodal logic, using the previous example.One had:π«([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]) ππ« π«i.e. 0.6 possible, 0.2 indeterminate-possibility, and 0.3 π«Μ ({0.3, 0.5}, 1 (0.2, 0.4), [0.5, 0.6]) impossible.π«Μ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]). 4 Neutrosophic Necessity OperatorTherefore, denoting by the neutrosophic equivπThe classical Necessity Modal Operator Β« π 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: π π« meaning Β«It ππππ«([0.5, 0.6],πis (t, i, f)-necessary that π« Β», using again the It is not neutrosophically necessary that Β«It willNeutrosophic Probability, where similarly Β«(t, i, f)πnecessityΒ» means t % necessary (surety that π« occurs), not be snowing tomorrowΒ»i % indeterminate (indeterminate-surety that π« occurs), It is not neutrosophically necessary thatand f % unnecessary (unsurely that π« occurs).πΜ If π«(π‘π , ππ , ππ ) is a neutrosophic proposition, with π« ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])π‘π , ππ , ππ subsets of [0, 1], then the neutrosophic truth It is neutrosophically possible thatvalue of the neutrosophic necessity operator is: πΜ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])ππ« π π« (inf(π‘π ), sup(ππ ), sup(ππ )), It is neutrosophically possible thatπwhich means that if a proposition π« is π‘π true, ππindeterminate, and ππ false, then the value of the π«([0.5, 0.6], 1 (0.6, 0.8), {0.3, 0.5}) neutrosophic necessity operator π π« is: inf(π‘π )It is neutrosophically possible thatπnecessary, sup(ππ ) indeterminate-necessity, and π«([0.5, 0.6], (0.2, 0.4), {0.3, 0.5})sup(ππ ) unnecessary. π«([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) ππTaking the previous example:(0.6, 0.2, 0.3).Florentin Smarandache, Neutrosophic Modal Logic
Neutrosophic Sets and Systems, Vol. 15, 201792Letβs check the second neutrosophic equivalence. (0.2, 0.4), {0.3, 0.5})ππππ«([0.5, 0.6],π It is not neutrosophically possible that Β«It willπnot be snowing tomorrowΒ» It is not neutrosophically possible thatππ«Μ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6]) It is neutrosophically necessary that πΜ ({0.3, 0.5}, (0.6, 0.8), [0.5, 0.6])ππ« It is neutrosophically necessary thatππ«([0.5, 0.6], 1 (0.6, 0.8), {0.3, 0.5}) It is neutrosophically necessary thatππ«([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) π«([0.5, 0.6], (0.2, 0.4), {0.3, 0.5}) ππ(0.6, 0.2, 0.3).6We can say that the proposition π«(π‘π , ππ , ππ ) isneutrosophically true if:inf(π‘π ) inf(ππ‘β ) and sup(π‘π ) sup(ππ‘β );inf(ππ ) inf(πΌπ‘β ) and sup(π‘π ) sup(πΌπ‘β );inf(ππ ) inf(πΉπ‘β ) and sup(ππ ) sup(πΉπ‘β ).For the particular case when all ππ‘β , πΌπ‘β , πΉπ‘β andπ‘π , ππ , ππ are single-valued numbers from the interval[0, 1], then one has:The proposition π«(π‘π , ππ , ππ ) is neutrosophicallytrue if:π‘π ππ‘β ;ππ πΌπ‘β ;ππ πΉπ‘β .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 πΊπ ,which is a non-empty neutrosophic set, whoseelements are called possible neutrosophic worlds,and a neutrosophic binary relation βπ , 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 β employed by the expertsβ©πΊπ , βπ βͺ.depending on each application. Under these conditions,one may have the following neutrosophic modalA neutrosophic world π€β²π that is neutrosophicallyequivalences:accessible from the neutrosophic world π€π is symbolized as: π π«(π‘π , ππ , ππ ) π ππ«(π‘π , ππ , ππ )π ππ€π βπ π€β²π . π π«(π‘π , ππ , ππ ) π ππ«(π‘π , ππ , ππ )In a neutrosophic model each neutrosophicπ πpropositionπ« has a neutrosophic truth-valueFor example, other definitions for the neutrosophic(π‘π€π , ππ€π , ππ€π ) respectively to each neutrosophicmodal operators may be:world π€π πΊπ , where π‘π€π , ππ€π , ππ€π are subsets of [0, π π«(π‘π , ππ , ππ ) (sup(π‘π ), sup(ππ ), inf(ππ )), or1]. π π«(π‘π , ππ , ππ ) (sup(π‘π ),whileππ2, inf(ππ ))etc., π π«(π‘π , ππ , ππ ) (inf(π‘π ), inf(ππ ), sup(ππ )), or π π«(π‘π , ππ , ππ ) (inf(π‘π ), 2ππ [0,1], sup(ππ ))etc.7Neutrosophic Truth ThresholdIn neutrosophic logic, first we have to introduce aneutrosophic truth threshold, ππ» β©ππ‘β , πΌπ‘β , πΉπ‘β βͺ ,where ππ‘β , πΌπ‘β , πΉπ‘β 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 π€π .Formalization.Let ππ be a set of neutrosophic propositionalvariables.9Neutrosophic Formulas1)Every neutrosophic propositional variableπ« ππ is a neutrosophic formula. 2)If A, B are neutrosophic formulas, then ππ΄, π΄ π΅ , π΄ π΅ , π΄ π΅ , π΄ π΅ , and π΄ , π΄ , are alsoπππππ π neutrosophic formulas, where π, , , , , and ,π π π ππ
93Neutrosophic Sets and Systems, Vol. 15, 2017 represent the neutrosophic negation, neutrosophicπintersection, neutrosophic union, dneutrosophic possibility operator, neutrosophicnecessity operator respectively.10 Accesibility Relation in a NeutrosophicTheoryLet πΊπ be a set of neutrosophic worlds π€π such thateach π€π chracterizes the propositions (formulas) of agiven neutrosophic theory π.We say that the neutrosophic world π€β²π is accesiblefrom the neutrosophic world π€π , and we write:π€π βπ π€β²π or βπ (π€π , π€β²π ) , if for any proposition(formula) π« π€π , meaning the neutrosophic truthvalue of π« with respect to π€π isπ€π€π€π«(π‘π π , ππ π , ππ π ),one has the neutrophic truth-value of π« with respect οΏ½οΏ½π€β²πandπ€sup(π‘ππ€β²π) inf(ππ π ) and sup(πππ€β²π€inf(ππ π ) inf(ππ π )π€sup(ππ π )π€) sup(ππ π );π€β²πandsup(πππ€) π€) π€(in the general case when π‘π π , ππ π , ππ π andπ€β²π€β²π€β²π‘π π , ππ π , ππ π are subsets of the interval [0, 1]).π€π€π€π‘π π , ππ π , ππ πBut in the instant ofandπ€β²π€β²π€β²π‘π π , ππ π , ππ π as single-values in [0, 1], the aboveinequalities become:π€β²π π‘π π ,π€β²π ππ π ,π‘ππππ€β²ππππ€π€π€ ππ π .11 ApplicationsIf the neutrosophic theory π 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(π)βπ , for n integer 2.(π)βπ (π€1π , π€2π , , π€ππ ; π€πβ² ),which means that the neutrosophic world π€πβ² isaccesiblefromtheneutrosophicworldsπ€1π , π€2π , , π€ππ all together.13 Neutrosophic Kripke Frameππ β©πΊπ , π π βͺ is a neutrosophic Kripke frame,since:π. πΊπ is an arbitrary non-empty neutrosophic set ofneutrosophic worlds, or neutrosophic states, orneutrosophic situations.ππ. π π πΊπ πΊπ 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 mappingπ€β²π€β²π€β²π«(π‘π π , ππ π , ππ π ),inf(π‘π π ) inf(π‘π π )π€sup(π‘π π );Instead of the classical π (π€, π€β²), which means thatthe world π€β² is accesible from the world π€ , wegeneralize it to:π£π : ππ πΊπ [0,1] [0,1] [0,1]where, for any neutrosophic proposition π« ππ andfor any neutrosophic world π€π , one defines:π€π€π€π£π (π, π€π ) (π‘π π , ππ π , ππ π ) [0,1] [0,1] [0,1]which is the neutrosophical logical truth value of theneutrosophic proposition π« in the neutrosophic worldπ€π .15 Neutrosophic DeducibilityWe say that the neutrosophic formula π« isneutrosophically deducible from the neutrosophicKripke frame ππ , the neutrosophic (t, i, f) β assignmentπ£π , and the neutrosophic world π€π , and we write as:ππ , π£π , π€π π«.πLetβs make the notation:πΌπ (π«; ππ , π£π , π€π )that denotes the neutrosophic logical value that theformula π« takes with respect to the neutrosophicKripke frame ππ , the neutrosophic (t, i, f)-assignementπ£π , and the neutrosphic world π€π .We define πΌπ by neutrosophic induction:1.πΌπ (π«; ππ , π£π , π€π )π€π πΊπ .2.3.πππ (π«, )π£π€π if π« ππ and π πππ [πΌ (π«;πΌπ (ππ«; ππ , π£π , π€π )ππ , π£π , π€π )]. π π ππππΌπ (π« π; ππ , π£π , π€π )π [πΌπ (π«; ππ , π£π , π€π )] [πΌπ (π; ππ , π£π , π€π )]πFlorentin Smarandache Neutrosophic Modal Logic
Neutrosophic Sets and Systems, Vol. 15, 2017944.5.6. ππππΌπ (π« π; ππ , π£π , π€π )π [πΌπ (π«; ππ , π£π , π€π )] [πΌπ (π; ππ , π£π , π€π )]π ππππΌπ (π« π; ππ , π£π , π€π )π [πΌπ (π«; ππ , π£π , π€π )] [πΌπ (π; ππ , π£π , π€π )]π ππππΌπ ( π«; ππ , π£π , π€π )π β²β²β©sup, inf, infβͺ{πΌπ (π«; ππ , π£π , π€ π ), π€ πΊπ and π€π π π π€β²π }.7.ππππΌπ ( π«; ππ , π£π , π€π ) πβ©inf, sup, supβͺ{πΌπ (π«; ππ , π£π , π€ β² π ), π€πβ² πΊπ and π€π π π π€β²π }. 8.π« if and only if π€π π« (a formula π« isπneutrosophically deducible if and only if π« isneutrosophically deducible in the actual neutrosophicworld).We should remark that πΌπ has a degree of truth(π‘πΌπ ), a degree of indeterminacy (ππΌπ ), and a degreeof falsehood (ππΌπ ) , which are in the general casesubsets of the interval [0, 1].Applying β©sup, inf, infβͺ to πΌπ is equivalent tocalculating:β©sup(π‘πΌπ ), inf(ππΌπ ), inf(ππΌπ )βͺ,and similarlyβ©inf, sup, supβͺπΌπ β©inf(π‘πΌπ ), sup(ππΌπ ), sup(ππΌπ )βͺ.16 Refined Neutrosophic Modal SingleValued LogicUsing neutrosophic (t, i, f) - thresholds, we refinefor the first time the neutrosophic modal logic as:a) 2π« Β«It is a little bigger necessity (degree ofπ (π‘,π,π)necessity π‘π 2 ) that π« Β», i.e. π‘π 1 π‘ π‘π 2 , π ππ 2 ππ 1 , π ππ 2 ππ 1 ; and so on; ππ« Β«It is a very high necessity (degree ofπ (π‘,π,π)necessity π‘π π ) that π«Β», i.e. π‘π π 1 π‘ π‘π π 1,π ππ π ππ π 1 , π ππ π ππ 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 (π€π ) 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 (π π, π π), for π 0 that canbe a tiny or even a bigger number).Thus, we can say that a proposition π« isneutrosophically nomological necessary, if π« isneutrosophically true at all possible neutrosophicworlds that obey the (t, i, f)-physical laws of the actualneutrosophic world π€π .In other words, at each possible neutrosophic worldπ€π , neutrosophically accesible from π€π , one has:π€Refined Neutrosophic Possibility Operator. 1π« Β«It is very little possible (degree ofπ (π‘,π,π)possibility π‘1 ) that π«Β», corresponding to the threshold(π‘1 , π1 , π1 ), i.e. 0 π‘ π‘1 , π π1 , π π1 , for π‘1 a verylittle number in [0, 1]; 2π« Β«It is little possible (degree ofπ (π‘,π,π)possibility π‘2 ) that π«Β», corresponding to the threshold(π‘2 , π2 , π2 ), i.e. π‘1 π‘ π‘2 , π π2 π1 , π π2 π1 ; and so on; ππ« Β«It is possible (with a degree ofπ (π‘,π,π)possibility π‘π ) that π«Β», corresponding to the threshold(π‘π , ππ , ππ ), i.e. π‘π 1 π‘ π‘π , π ππ ππ 1 , π ππ ππ 1 .b)Refined Neutrosophic Necessity Operator. 1π« Β«It is a small necessity (degree ofπ (π‘,π,π)necessity π‘π 1 ) that π« Β», i.e. π‘π π‘ π‘π 1 , π ππ 1 ππ , π ππ 1 ππ ;Florentin Smarandache, Neutrosophic Modal Logicπ€π€π«(π‘π π , ππ π , ππ π ) ππ»(ππ‘β , πΌπ‘β , πΉπ‘β ),π€π€π€πi.e. π‘π π ππ‘β , ππ π πΌπ‘β , and ππ πΉπ‘β .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 π‘π is the set of all degrees of connectibilitybetween the parts, and between the parts and thewhole;
95Neutrosophic Sets and Systems, Vol. 15, 2017ππ 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 theππ 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 π‘πΊ is the set of all degrees of knowledge of all has many interpretations, as part of possibleneutrosophic semantics.theories, ideas, propositions etc.,ππΊ is the set of all degrees of indeterminate-knowledgeReferencesof all theories, ideas, propositions etc.,ππΊ 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 π‘πΈ is the set of all degrees of scientificknowledge of all scientific theories, ideas, propositionsetc.,ππΈ is the set of all degrees of indeterminate scientificknowledge of all scientific theories, ideas, propositionsetc.,ππΈ 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): 2517β2528.[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) 1181β1210.[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):849β863.[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 .
Laws of Classical Logic That Do Not Hold in The Interval Neutrosophic Logic 184 Modal Contexts 186 Neutrosophic Score Function 186 Applications. 187 Neutrosophic Lattices 188 Conclusion 191 CHAPTER XII Neutrosophic Predicate Logic 196 Neutrosophic Quantifiers 199 Neutrosophic Existential Quantifier. 199 Neutrosophic Universal Quantifier. 199
Content THEORY Definition of Neutrosophy A Short Historyyg of the Logics Introduction to Non-Standard Analysis Operations with Classical Sets Neutrosophic Logic (NL) Refined Neutrosophic Logic and Set Classical Mass and Neutrosophic Mass Differences between Neutrosophic Logic and Intuitionistic Fuzzy Logic Neutrosophic Logic generalizes many Logics
2.4 Homomorphism and quotient rings 28 2.5 Special rings 30 2.6 Modules 34 2.7 Rings with chain conditions 35 3. Smarandache rings and its properties 3.1 Definition of Smarandache ring with examples 38 3.2 Smarandache units in rings 41 3.3 Smarandache zero divisors in rings 46
An Introduction to Modal Logic 2009 Formosan Summer School on Logic, Language, and Computation 29 June-10 July, 2009 ; 9 9 B . : The Agenda Introduction Basic Modal Logic Normal Systems of Modal Logic Meta-theorems of Normal Systems Variants of Modal Logic Conclusion ; 9 9 B . ; Introduction Let me tell you the story ; 9 9 B . Introduction Historical overview .
integer neutrosophic complex numbers, rational neutrosophic complex numbers and real neutrosophic complex numbers and derive interesting properties related with them. Throughout this chapter Z denotes the set of integers, Q the rationals and R the reals. I denotes the indeterminacy and I 2 I. Further i is the complex number and i 2 -1 or .
rst, a brief run-through of the basic concepts in modal logic. This rst chapter follows the outline of Martin Otto in [7], while expanding on the arguments presented. 2.1 The Fundamentals of Modal Logic The usual way to de ne modal logic in mathematical logic is as an extension of proposi-tional logic, with the de nition of structures in the .
seminear-rings using loops and groupoids which we call as groupoid near-rings, near loop rings, groupoid seminear-rings and loop seminear-ring. For all these concepts a Smarandache analogue is defined and several Smarandache properties are introduced and studied. The ninth chapter deals with fuzzy concepts in near-rings and gives 5
An Introduction to Literary Criticism and Theory Before we begin our examination and study of literary theory, it is important that we define exactly what literary theory is and is not, identify some of the main characteristics of such, as well as identify some of the key differences between traditional βliterary criticismβ and βliterary theory.β While literary criticism since the late .
