Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
Florentin Smarandache1
1 UniversityofNewMexico 705GurleyAve.,Gallup,NM87301,USA smarand@unm.edu
Abstract
In this chapter we define for the first time three neutrosophic actions and their properties.Wethenintroducetheprevalenceorderon{��,��,��}withrespecttoagiven neutrosophicoperator“o”,whichmaybesubjective asdefinedbytheneutrosophic experts.Andtherefinementofneutrosophicentities<A>,<neutA>,and<antiA>.
Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, and we define the refinement neutrosophicliteralspace.
Keywords neutrosophy, neutrosophics, neutrosophic actions, prevalence order, neutrosophic operator,refinementofneutrosophicentities,neutrosophicliterallogicaloperators, refinedliterallogicaloperators,refinementneutrosophicliteralspace.
1 Introduction
In Boolean Logic, a proposition �� is either true (T), or false (F). In Neutrosophic Logic, a proposition �� is either true (T), false (F), or indeterminate(I).
Forexample,inBooleanLogictheproposition��1: "1+1=2(inbase10)" istrue,whiletheproposition��2: "1+1=3(inbase10)" isfalse.
Florentin Smarandache
Inneutrosophiclogic,besidespropositions��1 (whichistrue)and��2 (whichis false),wemayalsohaveproposition��3: "1+1=?(inbase10)", whichisanincomplete/indeterminateproposition(neithertrue,norfalse).
1.1 Remark
Allconjecturesinscienceareindeterminateatthebeginning(researchersnot knowingiftheyaretrueorfalse),andlatertheyareprovedasbeingeithertrue, orfalse,orindeterminateinthecasetheywereunclearlyformulated.
2 Notations
Inordertoavoidconfusionsregardingtheoperators,wenotethemas: Boolean(classical)logic: ¬, ∧, ∨, ∨, →, ↔
Fuzzylogic: ¬ �� , ∧ ��, ∨ ��, ∨ ��, → �� , ↔ ��
Neutrosophiclogic: ¬ ��, ∧ ��, ∨ ��, ∨ ��, → ��, ↔ ��
3 Three Neutrosophic Actions
Intheframeofneutrosophy,wehaveconsidered[1995]foreachentity〈A〉,its opposite 〈antiA〉, and their neutrality 〈neutA〉 {i.e. neither 〈A〉, nor 〈antiA〉}.
Also,by〈nonA〉wemeanwhatisnot〈A〉,i.e.itsopposite〈antiA〉,togetherwith itsneutral(ity)〈neutA〉;therefore: 〈non��〉=〈neut��〉∨〈anti��〉.
Based on these, we may straightforwardly introduce for the first time the followingneutrosophicactionswithrespecttoanentity<A>:
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators Critical Review Volume XI, 2015
1. Toneutralize(ortoneuter,orsimplytoneut ize)theentity<A>. [As a noun: neutralization, or neuter ization, or simply neut ization.] Wedenoteitby<neutA>orneut(A).
2. To antithetic ize (or to anti ize) the entity <A>. [As a noun: antithetic ization, or anti ization.] We denote it by <antiA> ot anti(A).
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
Thisactionis100%oppositiontoentity<A>(strongopposition,or strongnegation).
3. Tonon izetheentity<A>. [Asanoun:non ization]. Wedenote itby<nonA>ornon(A). Itisanoppositioninapercentagebetween(0,100]%toentity<A> (weakopposition).
Of course, not all entities <A> can be neutralized, or antithetic ized, or non ized.
3.1 Example Let
〈A〉="PhoenixCardinalsbeatsTexasCowboys". Then, 〈neutA〉="\"Phoenix Cardinals has a tie game with Texas Cowboys\""; 〈antiA〉="\"PhoenixCardinalsisbeatenbyTexasCowboys\""; 〈nonA〉="\"PhoenixCardinalshasatiegamewithTexasCowboys," "orPhoenixCardinalsisbeatenbyTexasCowboys\"."
3.2 Properties of the Three Neutrosophic Actions
neut(〈anti��〉)=neut(〈neutA〉)=neut(��); anti(〈anti��〉)=��; anti(〈neut��〉)=〈��〉or〈anti��〉; non(〈anti��〉)=〈��〉or〈neut��〉; non(〈neut��〉)=〈��〉or〈anti��〉.
4 Neutrosophic Actions’ Truth Value Tables
Let’shavealogicalpropositionP,whichmaybetrue(T),Indeterminate(I),or false(F)asinpreviousexample.Oneappliestheneutrosophicactionsbelow.
4.1 Neutralization (or Indetermination) of P
neut(P) T I F I I I
Critical Review. Volume XI, 2015
Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
4.2 Antitheticization (Neutrosophic Strong Opposition to P)
anti(P) T I F F ��∨�� T
4.3 Non ization (Neutrosophic Weak Opposition to P):
non(P) T I F ��∨�� ��∨�� ��∨��
5 Refinement of Entities in Neutrosophy
In neutrosophy, an entity 〈A〉 has an opposite 〈antiA〉 and a neutral 〈neutA〉. But these three categories can be refined in sub entities〈��〉1,〈��〉2,…,〈��〉��, and respectively 〈neut��〉1,〈neut��〉2,…,〈neut��〉�� , and also 〈anti��〉1 , 〈anti��〉2,…,〈anti��〉�� , where m, n, p are integers ≥1, but ��+��+�� ≥4 (meaningthatatleastoneof〈A〉,〈antiA〉or〈neutA〉isrefinedintwoormore sub entities).
Forexample,if 〈A〉=whitecolor,then 〈antiA〉=blackcolor, while 〈neutA〉=colorsdifferentfromwhiteandblack.
If we refine them, we get various nuances of white color:〈��〉1,〈��〉2, , and variousnuancesofblackcolor:〈anti��〉1,〈anti��〉2,…,andthecolorsinbetween them(red,green,yellow,blue,etc.):〈neut��〉1,〈neut��〉2,….
Similarlyasabove,wewanttopointoutthatnotallentities<A>and/ortheir corresponding(ifany)<neutA>and<antiA>canberefined.
6 The Prevalence Order
Let’sconsidertheclassicalliteral(symbolic)truth(T)andfalsehood(F).
In a similar way, for neutrosophic operators we may consider the literal (symbolic) truth (T), the literal (symbolic) indeterminacy (I), and the literal (symbolic)falsehood(F).
We also introduce the prevalence order on {��,��,��}with respect to a given binaryandcommutativeneutrosophicoperator“o”.
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
The neutrosophic operators are: neutrosophic negation, neutrosophic conjunction, neutrosophic disjunction, neutrosophic exclusive disjunction, neutrosophic Sheffer’s stroke, neutrosophic implication, neutrosophic equivalence,etc.
Theprevalenceorderispartiallyobjective(followingtheclassicallogicforthe relationship between T and F), and partially subjective (when the indeterminacyIinterfereswithitselforwith T or F).
Foritssubjectivepart,theprevalenceorderisdeterminedbytheneutrosophic logicexpertintermsoftheapplication/problemtosolve,andalsodepending onthespecificconditionsoftheapplication/problem.
For�� ≠��, wewrite��℗��, or�� ≻�� ��, andweread X prevailsto Y withrespect to the neutrosophic binary commutative operator “o”, which means that ������ =��.
Let’s see the below examples. We mean by “o”: conjunction, disjunction, exclusivedisjunction,Sheffer’sstroke,andequivalence.
7 Neutrosophic Literal Operators & Neutrosophic Numerical Operators
7.1 Ifwemeanbyneutrosophicliteralproposition,apropositionwhosetruth valueisaletter:either T or I or F Theoperatorsthat dealwithsuchlogical propositionsarecalledneutrosophicliteraloperators.
7.2 And by neutrosophic numerical proposition, a proposition whose truth valueisatripleofnumbers(oringeneralofnumericalsubsetsoftheinterval [0,1]),forexamplesA(0.6,0.1,0.4)orB([0,0.2],{0.3,0.4,0.6},(0.7,0.8)).The operators that deal with such logical propositions are called neutrosophic numericaloperators.
8 Truth Value Tables of Neutrosophic Literal Operators
InBooleanLogic,onehasthefollowingtruth valuetablefornegation:
8.1 Classical Negation ¬ T F F T
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Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
In Neutrosophic Logic, one has the following neutrosophic truth value table fortheneutrosophicnegation:
8.2 Neutrosophic Negation
So,wehavetoconsiderthatthenegationofIisI,whilethenegationsofTand Faresimilarasinclassicallogic.
Inclassicallogic,onehas:
8.3 Classical Conjunction
Inneutrosophiclogic,onehas:
8 4 Neutrosophic Conjunction (��������), version 1
Theobjectivepart(circledliteralcomponentsintheabovetable)remainsas inclassicallogic,butwhenindeterminacy I interferes,theneutrosophicexpert maychoosethemostfitprevalenceorder.
There are also cases when the expert may choose, for various reasons, to entangle the classical logic in the objective part. In this case, the prevalence orderwillbetotallysubjective.
Critical Review Volume XI, 2015
T I F I
T F T T F F F F
N T I F T I I I I I F I
F T T F F FFlorentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
Theprevalenceorderworksforclassicallogictoo.Asanexample,forclassical conjunction, one has �� ≻�� �� , which means that ��∧�� =�� While the prevalenceorderfortheneutrosophicconjunctionintheabovetableswas: �� ≻�� �� ≻�� ��, whichmeansthat��∧�� �� =��,and��∧�� �� =��. Otherprevalenceorderscanbeusedherein,suchas: �� ≻�� �� ≻�� ��, anditscorrespondingtablewouldbe:
∧N
I
8.5 Neutrosophic Conjunction (��������), version 2 whichmeansthat��∧���� =��and��∧���� =��;oranotherprevalenceorder: �� ≻�� �� ≻�� ��, anditscorrespongingtablewouldbe:
8.6 Neutrosophic Conjunction (��������), version 3 whichmeansthat��∧���� =��and��∧���� =��.
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T
F T I I I I F F F
N T I F
T I T I F F F
T F F F
T F F F
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Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
Ifonecomparesthethreeversionsoftheneutrosophicliteralconjunction,one observes that the objective part remains the same, but the subjective part changes.
Thesubjectiveoftheprevalenceordercanbeestablishedinanoptimisticway, orpessimisticway,oraccordingtotheweightsassignedtotheneutrosophic literalcomponents T, I, F bytheexperts.
Inasimilarway,wedofordisjunction.Inclassicallogic,onehas:
8.7 Classical Disjunction Inneutrosophiclogic,onehas: 8.8 Classical Disjunction (������) whereweusedthefollowingprevalenceorder: �� ≻�� �� ≻�� ��, butthereaderisinvited(asanexercise)touseanotherprevalenceorder,such as:
∨
allneutrosophiclogicaloperatorspresentedaboveandbelowinthispaper. Inclassicallogic,onehas:
Critical Review Volume XI, 2015
Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
8.9 Classical Exclusive Disjunction
∨
Inneutrosophiclogic,onehas:
8.10 Neutrosophic Exclusive Disjunction usingtheprevalenceorder �� ≻�� �� ≻�� ��.
Inclassicallogic,onehas:
8.11 Classical Sheffer’s Stroke
Inneutrosophiclogic,onehas:
8.12 Neutrosophic Sheffer’s Stroke
Critical Review. Volume XI, 2015
T F
F T F T F
T I F T T I T I F F F
T F T F T F T T
N T I F T T I T I I F I
F T T T
F T T F
Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
usingtheprevalenceorder
�� ≻�� �� ≻�� ��.
Inclassicallogic,onehas: 8.13 Classical Implication
→
Inneutrosophiclogic,onehas: 8.14 Neutrosophic Implication
using the subjective preference that�� →N ��is true (because in the classical implication��isimpliedbyanything),and�� →N ��isfalse,while�� →N ��istrue because is similar to the classical implications�� →�� and�� →��, which are true.
Thereaderisfreetocheckdifferentsubjectivepreferences.
Inclassicallogic,onehas:
8.15 Classical Equivalence
Inneutrosophiclogic,onehas:
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T F T T F F T T
N T I F T I I T T F F T
T F T T F F F T
T F T T
Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
8.15 Neutrosophic Equivalence
usingthesubjectivepreferencethat�� ↔N ��istrue,becauseitissimilartothe classical equivalences that �� →�� and �� →�� are true, and also using the prevalence: �� ≻�� �� ≻�� ��.
9 Refined Neutrosophic Literal Logic
Eachparticularcasehastobetreatedindividually.
In this paper, we present a simple example. Let’s consider the following neutrosophiclogicalpropositions:
T =Tomorrowitwillrainorsnow.
T issplitinto Tomorrowitwillrain. Tomorrowitwillsnow.
F =Tomorrowitwillneitherrainnorsnow.
F issplitinto Tomorrowitwillnotrain. Tomorrowitwillnotsnow.
I =Donotknowiftomorrowitwillberaining,norifitwillbesnowing. I issplitinto Donotknowiftomorrowitwillberainingornot. Donotknowiftomorrowitwillbesnowingornot.
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Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
Itisclearthatthenegationof��1 (Tomorrowitwillraining)is��1 (Tomorrowit willnotberaining).Similarlyforthenegationof ��2,whichis��2 But,thenegationof ��1 (Donotknowiftomorrowitwillberainingornot)is “Do know if tomorrow it will be raining or not”, which is equivalent to “We knowthattomorrowitwillberaining”(��1),or“Weknowthattomorrowitwill notberaining”(��1).
Whence,thenegationof��1 is��1 ∨��1,andsimilarly,thenegationof��2 is��2 ∨��2.
9.1 Refined Neutrosophic Literal Conjunction Operator
∧N T1 T2 I1 I2 F1 F2 T1 ��1 ��12 ��1 ��2 ��1 ��2
T2 ��12 ��2 ��1 ��2 ��1 ��2
I1 ��1 ��1 ��1 I ��1 ��2 I2 ��2 ��2 I ��2 ��1 ��2 F1 ��1 ��1 ��1 ��1 ��1 F F2 ��2 ��2 ��2 ��2 F ��2
where��12 =��1 ∧��2 =“Tomorrowitwillrainanditwillsnow”.
Ofcourse,otherprevalenceorderscanbestudiedforthisparticularexample. With respect to the neutrosophic conjunction,���� prevail in front of����, which prevailinfrontof����,or���� ≻���� ≻����,forall��,��,�� ∈{1,2}
9.2 Refined Neutrosophic Literal Disjunction Operator
∨N T1 T2 I1 I2 F1 F2 T1 ��1 T ��1 ��1 ��1 ��1
T2 T ��2 ��2 ��2 ��2 ��2
I1 ��1 ��2 ��1 I ��1 ��2
I2 ��1 ��2 I ��2 ��1 ��2
F1 ��1 ��2 ��1 ��1 ��1 ��1 ∨��2
F2 ��1 ��2 ��2 ��2 ��1 ∨��2 ��2
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
With respect to the neutrosophic disjunction,���� prevail in front of����, which prevailinfrontof ����,or���� ≻���� ≻����,forall��,��,�� ∈{1,2}.
Forexample,��1 ∨��2 =��,but��1 ∨��2 ∉{��,����}∪{��1,T2,��1,I2,��1,F2}
9.3 Refined Neutrosophic Literal Space
The Refinement Neutrosophic Literal Space{��1,��2,��1,��2,��1,��2}is not closed under neutrosophic negation, neutrosophic conjunction, and neutrosophic disjunction. The reader can check the closeness under other neutrosophic literaloperations.
Aneutrosophicrefinedliteralspace ���� ={��1,��2,…,����; ��1,��2,…,����; ��1,��2,…,����},
where��,��,��areintegers≥1,issaidtobeclosedunderagivenneutrosophic operator"���� ",ifforanyelements��,�� ∈���� onehas��������∈����.
Let’sdenotetheextensionof���� withrespecttoasingle���� by: ����1 �� =(����,����) If���� is not closed with respect to the given neutrosophic operator����, then ����1 �� ≠���� ,andweextend���� byaddinginthenewelementsresultedfromthe operation��������,let’sdenotethemby��1,��2,…����. Therefore, ����1 �� ≠���� ∪{��1,��2,…����}. ����1 �� encloses����.
Similarly,wecandefinetheclosenessoftheneutrosophicrefinedliteralspace ���� with respect to the two or more neutrosophic operators��1��,��2��,…,������, for�� ≥2 ���� is closed under ��1��,��2��,…,������ if for any ��,�� ∈���� and for any �� ∈ {1,2,…,��}onehas���������� ∈����.
If���� is not closed under these neutrosophic operators, one can extend it as previously.
Let’sconsider:������ �� =(����,��1��,��2��, ,������),whichis���� closedwithrespectto allneutrosophicoperators��1��,��2��,…,������,then������ �� encloses����.
Critical Review. Volume XI, 2015
Florentin Smarandache
Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators
10 Conclusion
We have defined for the first time three neutrosophic actions and their properties.Wehaveintroducedtheprevalenceorderon{��,��,��}withrespect toagivenneutrosophicoperator“o”,therefinementofneutrosophicentities <A>,<neutA>,and<antiA>,andtheneutrosophicliterallogicaloperatorsand refinedliterallogicaloperators,andtherefinementneutrosophicliteralspace.
References
[1] F. Smarandache, Neutrosophy. / Neutrosophic Probability, Set, and Logic,Am.Res.Press,Rehoboth,USA,105p.,1998; Republished in 2000, 2003, 2005, as: A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics (second, third, and respectively fourth edition),AmericanResearchPress,156p.; ChinesetranslationbyF.Liu, A Unifying Field in Logics: Neutrosophic Logic. / Neutrosophy, Neutrosophic Set, Neutrosophic Probability and statistics,XiquanChineseBranch,121p.,2003; Russian partial translation by D. Rabounski: Hexis, Сущность нейтрософии,32p.,2006.
[2] Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub Indeterminacies, in “Neutrosophic Sets and Systems”,Vol.9,2015,pp.58 63.
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